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atom.xml
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<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
<id>https://hkr04.github.io</id>
<title>hkr04‘s blogs</title>
<updated>2020-04-01T00:33:15.683Z</updated>
<generator>https://github.com/jpmonette/feed</generator>
<link rel="alternate" href="https://hkr04.github.io"/>
<link rel="self" href="https://hkr04.github.io/atom.xml"/>
<subtitle>撒野奔跑，踩碎迷茫。</subtitle>
<logo>https://hkr04.github.io/images/avatar.png</logo>
<icon>https://hkr04.github.io/favicon.ico</icon>
<rights>All rights reserved 2020, hkr04‘s blogs</rights>
<entry>
<title type="html"><![CDATA[最长不下降子序列]]></title>
<id>https://hkr04.github.io/LIS/</id>
<link href="https://hkr04.github.io/LIS/">
</link>
<updated>2020-04-01T00:32:22.000Z</updated>
<content type="html"><![CDATA[<p>第一问，求出最长不下降子序列长度<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>。我知道你会用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><mi>n</mi><mi>l</mi><mi>o</mi><mi>g</mi><mi>n</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">O(nlogn)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span>的复杂度求它，但是仔细想想，这么做以后，后面怎么建图呢？我怎么知道两个点是否都在最长的不下降子序列上呢？回想一下朴素的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>n</mi><mn>2</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">O(n^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>的做法，状态<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>表示以<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>结尾的最长子序列长度。假如<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi><mo>≤</mo><mi>j</mi><mo separator="true">,</mo><msub><mi>a</mi><mi>i</mi></msub><mo>≤</mo><msub><mi>a</mi><mi>j</mi></msub><mo separator="true">,</mo><msub><mi>f</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn><mo>=</mo><msub><mi>f</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">i\le j,a_i\le a_j,f_i+1=f_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.79549em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>，说明<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">f_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>是由<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>转移而来的，进而说明在以<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>a</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">a_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>结尾的最长不下降子序列的前一位可以是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">a_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。</p>
<p>第二问，限制每个点只能用一次，求最长不下降子序列数目。这里需要拆点，为了防止出现以下情况：<br>
<img src="https://cdn.luogu.com.cn/upload/image_hosting/euq4420k.png" alt="example" loading="lazy"><br>
只需要加上一个虚点限制该点的出流只能为1，即可解决：<br>
<img src="https://cdn.luogu.com.cn/upload/image_hosting/j4t0pge8.png" alt="example2" loading="lazy"><br>
考虑源点与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f_i=1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>的点相连，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub><mo>=</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">f_i=s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>的点与汇点相连；点与点之间根据上述转移条件连边。将所有边的边权都设为1。这样，从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>的单位流量就表示存在1个经过的每个点只用1次的合法方案，每条路径之间一定不会有交点。</p>
<p>第三问，由于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>和<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>可以重复使用，且它们只能作为第一个或最后一个存在于序列中，所以若它们与源点或汇点之间有连边，将边权设为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi><mi>n</mi><mi>f</mi></mrow><annotation encoding="application/x-tex">inf</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">i</span><span class="mord mathdefault">n</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span></span></span></span>即可。</p>
<p>代码：</p>
<pre><code class="language-cpp">#include <cstdio>
#include <cstring>
const int maxn=500+10;
const int maxm=1e6+10;
const int INF=0x3f3f3f3f;
int a[maxn],f[maxn],dep[maxn<<1];
int cur[maxn<<1],head[maxn<<1],to[maxm],nxt[maxm],val[maxm];
int tot=1,cnt=0;
int n,s,t;
struct Queue
{
int a[maxn<<1];
int l,r;
Queue() {l=1,r=0;}
void push(int x) {a[++r]=x;}
void pop() {l++;}
int front() {return a[l];}
bool empty() {return l>r;}
}q;
int min(int x,int y) {return x<y?x:y;}
int max(int x,int y) {return x>y?x:y;}
void add(int u,int v,int w)
{
nxt[++tot]=head[u];
head[u]=tot;
to[tot]=v;
val[tot]=w;
}
bool bfs()
{
memset(dep, 0x3f, sizeof(dep));
dep[s]=0;
q=Queue();
q.push(s);
while(!q.empty())
{
int u=q.front();
q.pop();
for (int i=head[u];i;i=nxt[i])
{
int v=to[i];
if (val[i]&&dep[v]>dep[u]+1)
{
dep[v]=dep[u]+1;
q.push(v);
}
}
}
return dep[t]!=INF;
}
int dfs(int u,int minf)
{
if (u==t||!minf)
return minf;
int used=0;
for (int &i=cur[u];i;i=nxt[i])
{
int v=to[i];
if (val[i]&&dep[v]==dep[u]+1)
{
int flow=dfs(v, min(minf-used, val[i]));
if (!flow)
continue;
used+=flow;
val[i]-=flow;
val[i^1]+=flow;
if (used==minf)
break;
}
}
return used;
}
int main()
{
scanf("%d",&n);
s=0,t=2*n+1;
for (int i=1;i<=n;i++)
scanf("%d",&a[i]);
for (int i=1;i<=n;i++)
f[i]=1;
int sum=1;
for (int i=1;i<=n;i++)
for (int j=i-1;j>=1;j--)
if (a[j]<=a[i])
f[i]=max(f[i], f[j]+1);
for (int i=1;i<=n;i++)
sum=max(sum, f[i]);
if (sum==1)
{
printf("%d\n%d\n%d\n",sum,n,n);
return 0;
}
printf("%d\n",sum);
for (int i=1;i<=n;i++)
{
add(i, i+n, 1),add(i+n, i, 0);
if (f[i]==1)
add(s, i, 1),add(i, s, 0);
if (f[i]==sum)
add(i+n, t, 1),add(t, i+n, 0);
for (int j=i+1;j<=n;j++)
if (a[i]<=a[j]&&f[i]+1==f[j])
add(i+n, j, 1),add(j, i+n, 0);
}
int ans=0;
while(bfs())
{
for (int i=s;i<=t;i++)
cur[i]=head[i];
ans+=dfs(s, INF);
}
printf("%d\n",ans);
ans=0;
for (int i=2;i<=tot;i++)
{
if (i&1)
val[i]=0;
else
val[i]=to[i]==1||to[i]==n*2||to[i^1]==1||to[i^1]==n*2?INF:1;
}
while(bfs())
{
for (int i=s;i<=t;i++)
cur[i]=head[i];
ans+=dfs(s, INF);
}
printf("%d\n",ans);
return 0;
}
</code></pre>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[AC自动机]]></title>
<id>https://hkr04.github.io/Aho-Corasick-automaton/</id>
<link href="https://hkr04.github.io/Aho-Corasick-automaton/">
</link>
<updated>2020-04-01T00:30:38.000Z</updated>
<content type="html"><![CDATA[<p><strong>AC自动机</strong>利用<strong>trie树</strong>可以高效解决有关多个字符串的问题。</p>
<h3 id="trie树">Trie树:</h3>
<p>也称<strong>字典树</strong>，它的本质是使得字符串集合<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>构成一棵树，其中边权记录字符信息。<br>
它的根到任意节点的路径对应集合<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>中某一字符串的前缀。<br>
任意节点向深度增大的方向经过的路径对应<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>中某一字符串的子串。<br>
比如下面这一棵<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>trie</mtext></mrow><annotation encoding="application/x-tex">\text{trie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66786em;vertical-align:0em;"></span><span class="mord text"><span class="mord">trie</span></span></span></span></span>，就记录了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>why,who,when</mtext></mrow><annotation encoding="application/x-tex">\text{why,who,when}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">why,who,when</span></span></span></span></span>这几个字符串所构成的集合的信息:<br>
<img src="https://cdn.luogu.com.cn/upload/image_hosting/85d9l1e8.png" alt="trie示例" loading="lazy"><br>
简单来说，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>trie</mtext></mrow><annotation encoding="application/x-tex">\text{trie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66786em;vertical-align:0em;"></span><span class="mord text"><span class="mord">trie</span></span></span></span></span>的插入是这样的:</p>
<ol>
<li>从根节点进入，从插入的字符串的首位开始考虑，若属于这位字符的边走向的点已存在，就继续往下走;否则，就为这条边创建一个新的节点;</li>
<li>向下走后重复以上过程;</li>
<li>直到走完所有字符，最后所处的位置进行标记，表明从根节点到这的路径属于集合中。</li>
</ol>
<pre><code class="language-cpp">int tot=0;
void insert(char *s,int l_s)
{
int p=0;
for (int i=1;i<=l_s;i++)
{
if (!trie[p][s[i]-'a'])
trie[p][s[i]-'a']=++tot;
p=trie[p][s[i]-'a'];
}
end[p]++;
}
</code></pre>
<h3 id="关于trie树的定义">关于trie树的定义:</h3>
<ul>
<li>状态:从根节点到任何一个字典树上的节点的路径表示的字符串都代表着一个状态</li>
<li>转移:任意一个状态添加一个新的字符得到新状态</li>
<li>可识别:若一个字符串能对应字典树上的某一状态，则称其为可识别</li>
<li>标记状态:特指属于集合S的状态</li>
</ul>
<p>查询字符串<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>是否在集合中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>trie</mtext></mrow><annotation encoding="application/x-tex">\text{trie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66786em;vertical-align:0em;"></span><span class="mord text"><span class="mord">trie</span></span></span></span></span>上遍历，在结尾处查询是否有相应状态(遍历与插入的过程类似)<br>
查询是否有字符串<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>的某个<strong>前缀</strong>在集合中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span>遍历的时候每经过一个节点就检查一遍是否为标记状态<br>
查询是否有字符串<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>的某个<strong>子串</strong>在集合中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span>因为子串都是<strong>某个前缀的后缀</strong>,所以只需查询<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>的<strong>每个前缀</strong>有多少<strong>可识别并被标记的后缀</strong><br>
一般敏感词屏蔽就可以用这样的方法，不过有可能会误伤一些连起来的正常词语。</p>
<h2 id="下面我们来解决查询是否有字符串s的某个子串在集合中的这个问题">下面我们来解决查询是否有字符串<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>的某个子串在集合中的这个问题</h2>
<p>我们定义<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>fail[s[1..i]]</mtext></mrow><annotation encoding="application/x-tex">\text{fail[s[1..i]]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">fail[s[1..i]]</span></span></span></span></span>表示<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>s[1..i]</mtext></mrow><annotation encoding="application/x-tex">\text{s[1..i]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">s[1..i]</span></span></span></span></span>的<strong>最长可识别后缀</strong>(你可以将它理解为一个字符串指针，虽然实际上我们会用数组实现，使它指向一个状态的编号)。容易发现的是,<strong>可识别后缀</strong>具有传递性，即:若<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>的可识别后缀,<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span></span>的可识别后缀，则<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>也为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span></span>的可识别后缀.<br>
依据这个性质，设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>s[1..i]</mtext></mrow><annotation encoding="application/x-tex">\text{s[1..i]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">s[1..i]</span></span></span></span></span> <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>=</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">=t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>,其所有可识别后缀为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>fail[t],fail[fail[t]],fail[fail[fail[t]]]</mtext></mrow><annotation encoding="application/x-tex">\text{fail[t],fail[fail[t]],fail[fail[fail[t]]]}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">fail[t],fail[fail[t]],fail[fail[fail[t]]]</span></span></span></span></span> <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.31em;vertical-align:0em;"></span><span class="minner">⋯</span></span></span></span>，且长度递减。</p>
<p>那么，怎么处理这个<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>fail</mtext></mrow><annotation encoding="application/x-tex">\text{fail}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">fail</span></span></span></span></span>指针呢?<br>
假设有字符串<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span>,<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>fail[x]=y</mtext></mrow><annotation encoding="application/x-tex">\text{fail[x]=y}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">fail[x]=y</span></span></span></span></span>.考虑<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span>加上一个字符<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span>后的新字符串<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">xc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mord mathdefault">c</span></span></span></span>的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>fail</mtext></mrow><annotation encoding="application/x-tex">\text{fail}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">fail</span></span></span></span></span>指针。显然,<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">xc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mord mathdefault">c</span></span></span></span>的可识别后缀必然为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span>的某个可识别后缀加上<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">c</span></span></span></span>的形式。所以，我们为了求出<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">xc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mord mathdefault">c</span></span></span></span>的最长可识别后缀，只需查询<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>yc,fail[y]c,fail[fail[y]]c</mtext></mrow><annotation encoding="application/x-tex">\text{yc,fail[y]c,fail[fail[y]]c}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">yc,fail[y]c,fail[fail[y]]c</span></span></span></span></span> <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⋯</mo></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.31em;vertical-align:0em;"></span><span class="minner">⋯</span></span></span></span>中最长的可识别串即可.</p>
<p>接下来考虑依赖关系。由上，我们知道想要一个字符串的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>fail</mtext></mrow><annotation encoding="application/x-tex">\text{fail}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">fail</span></span></span></span></span>指针与比其长度少一的可识别后缀有关，长度在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>trie</mtext></mrow><annotation encoding="application/x-tex">\text{trie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66786em;vertical-align:0em;"></span><span class="mord text"><span class="mord">trie</span></span></span></span></span>树上的表现即为深度。所以，我们应按照深度从小到大计算每个节点所代表的状态的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>fail</mtext></mrow><annotation encoding="application/x-tex">\text{fail}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">fail</span></span></span></span></span>指针。可用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>BFS</mtext></mrow><annotation encoding="application/x-tex">\text{BFS}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">BFS</span></span></span></span></span>实现。</p>
<p>建立AC自动机伪代码:</p>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>b</mi><mi>u</mi><mi>i</mi><mi>l</mi><mi>d</mi><mtext> </mtext><mi>t</mi><mi>r</mi><mi>i</mi><mi>e</mi><mtext> </mtext><mi>T</mi><mtext> </mtext><mi>f</mi><mi>r</mi><mi>o</mi><mi>m</mi><mtext> </mtext><mi>S</mi><mo>=</mo><mrow><mo fence="true">{</mo><mi>s</mi><mi>t</mi><msub><mi>r</mi><mn>1</mn></msub><mo separator="true">,</mo><mi>s</mi><mi>t</mi><msub><mi>r</mi><mn>2</mn></msub><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo fence="true">}</mo></mrow></mrow><annotation encoding="application/x-tex">build\ trie\ T\ from\ S=\left\{str_1,str_2...\right\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">b</span><span class="mord mathdefault">u</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">d</span><span class="mspace"> </span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">i</span><span class="mord mathdefault">e</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">o</span><span class="mord mathdefault">m</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;">{</span><span class="mord mathdefault">s</span><span class="mord mathdefault">t</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">s</span><span class="mord mathdefault">t</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mclose delimcenter" style="top:0em;">}</span></span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi><mi>o</mi><mi>r</mi><mi>t</mi><mtext> </mtext><mi>e</mi><mi>v</mi><mi>e</mi><mi>r</mi><mi>y</mi><mtext> </mtext><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>i</mi><mi>n</mi><mtext> </mtext><mi>T</mi><mtext> </mtext><mi>b</mi><mi>y</mi><mtext> </mtext><mi>d</mi><mi>e</mi><mi>p</mi><mi>t</mi><mi>h</mi></mrow><annotation encoding="application/x-tex">sort\ every\ state\ in\ T\ by\ depth</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">s</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">t</span><span class="mspace"> </span><span class="mord mathdefault">e</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">e</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace"> </span><span class="mord mathdefault">s</span><span class="mord mathdefault">t</span><span class="mord mathdefault">a</span><span class="mord mathdefault">t</span><span class="mord mathdefault">e</span><span class="mspace"> </span><span class="mord mathdefault">i</span><span class="mord mathdefault">n</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mspace"> </span><span class="mord mathdefault">b</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace"> </span><span class="mord mathdefault">d</span><span class="mord mathdefault">e</span><span class="mord mathdefault">p</span><span class="mord mathdefault">t</span><span class="mord mathdefault">h</span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi><mo>[</mo><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mo>]</mo><mo>=</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">fail[root]=\phi</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">o</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">ϕ</span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>e</mi><mi>a</mi><mi>c</mi><mi>h</mi><mtext> </mtext><mi>c</mi><mo separator="true">,</mo><mi>t</mi><mi>r</mi><mi>a</mi><mi>n</mi><mi>s</mi><mo>(</mo><mi>ϕ</mi><mo separator="true">,</mo><mi>c</mi><mo>)</mo><mo>=</mo><mi>r</mi><mi>o</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mo>(</mo><mi>t</mi><mi>o</mi><mtext> </mtext><mi>a</mi><mi>v</mi><mi>o</mi><mi>i</mi><mi>d</mi><mtext> </mtext><mi>s</mi><mi>o</mi><mi>m</mi><mi>e</mi><mtext> </mtext><mi>e</mi><mi>r</mi><mi>r</mi><mi>o</mi><mi>r</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">for\ each\ c,trans(\phi, c)=root\ (to\ avoid\ some\ error)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mspace"> </span><span class="mord mathdefault">e</span><span class="mord mathdefault">a</span><span class="mord mathdefault">c</span><span class="mord mathdefault">h</span><span class="mspace"> </span><span class="mord mathdefault">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">a</span><span class="mord mathdefault">n</span><span class="mord mathdefault">s</span><span class="mopen">(</span><span class="mord mathdefault">ϕ</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">o</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mspace"> </span><span class="mopen">(</span><span class="mord mathdefault">t</span><span class="mord mathdefault">o</span><span class="mspace"> </span><span class="mord mathdefault">a</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">o</span><span class="mord mathdefault">i</span><span class="mord mathdefault">d</span><span class="mspace"> </span><span class="mord mathdefault">s</span><span class="mord mathdefault">o</span><span class="mord mathdefault">m</span><span class="mord mathdefault">e</span><span class="mspace"> </span><span class="mord mathdefault">e</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mclose">)</span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>s</mi><mi>t</mi><mi>a</mi><mi>t</mi><mi>e</mi><mtext> </mtext><mi>x</mi><mtext> </mtext><mi>i</mi><mi>n</mi><mtext> </mtext><mi>T</mi></mrow><annotation encoding="application/x-tex">for\ state\ x\ in\ T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mspace"> </span><span class="mord mathdefault">s</span><span class="mord mathdefault">t</span><span class="mord mathdefault">a</span><span class="mord mathdefault">t</span><span class="mord mathdefault">e</span><span class="mspace"> </span><span class="mord mathdefault">x</span><span class="mspace"> </span><span class="mord mathdefault">i</span><span class="mord mathdefault">n</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext> </mtext><mi>f</mi><mi>o</mi><mi>r</mi><mtext> </mtext><mi>x</mi><mi>c</mi><mtext> </mtext><mi>i</mi><mi>n</mi><mtext> </mtext><mi>T</mi></mrow><annotation encoding="application/x-tex">\ \ \ \ \ \ for\ xc\ in\ T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">o</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mspace"> </span><span class="mord mathdefault">x</span><span class="mord mathdefault">c</span><span class="mspace"> </span><span class="mord mathdefault">i</span><span class="mord mathdefault">n</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi><mo>[</mo><mi>x</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">\ \ \ \ \ \ \ \ \ \ \ \ \ y=fail[x]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mopen">[</span><span class="mord mathdefault">x</span><span class="mclose">]</span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext> </mtext><mi>w</mi><mi>h</mi><mi>i</mi><mi>l</mi><mi>e</mi><mtext> </mtext><mi>y</mi><mi mathvariant="normal">≠</mi><mi>ϕ</mi><mtext> </mtext><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>y</mi><mi>c</mi><mtext> </mtext><mi>n</mi><mi>o</mi><mi>t</mi><mtext> </mtext><mi>i</mi><mi>n</mi><mtext> </mtext><mi>T</mi></mrow><annotation encoding="application/x-tex">\ \ \ \ \ \ \ \ \ \ \ \ \ while \ y\neq\phi\ and \ yc\ not\ in\ T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord mathdefault">h</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">e</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mrel"><span class="mord"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mrel"></span></span><span class="fix"></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.19444em;"><span></span></span></span></span></span></span><span class="mrel">=</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">ϕ</span><span class="mspace"> </span><span class="mord mathdefault">a</span><span class="mord mathdefault">n</span><span class="mord mathdefault">d</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord mathdefault">c</span><span class="mspace"> </span><span class="mord mathdefault">n</span><span class="mord mathdefault">o</span><span class="mord mathdefault">t</span><span class="mspace"> </span><span class="mord mathdefault">i</span><span class="mord mathdefault">n</span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi><mo>[</mo><mi>y</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ y=fail[y]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">]</span></span></span></span><br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext> </mtext><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi><mo>[</mo><mi>x</mi><mi>c</mi><mo>]</mo><mo>=</mo><mi>y</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">\ \ \ \ \ \ \ \ \ \ \ \ \ fail[xc]=yc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace"> </span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mopen">[</span><span class="mord mathdefault">x</span><span class="mord mathdefault">c</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord mathdefault">c</span></span></span></span></p>
<p>注意到，当新加进来字符的状态不能直接被当前的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">fail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span></span></span></span>匹配时，需要一直在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">fail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span></span></span></span>链上跳，直到能扩展匹配或跳到空集。有一个小优化可以在我们不需要访问或更改<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>trie</mtext></mrow><annotation encoding="application/x-tex">\text{trie}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66786em;vertical-align:0em;"></span><span class="mord text"><span class="mord">trie</span></span></span></span></span>树的信息使用：若<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">yc</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord mathdefault">c</span></span></span></span>不存在，则将其直接指向<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi><mo>[</mo><mi>y</mi><mo>]</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">fail[y]c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">]</span><span class="mord mathdefault">c</span></span></span></span>。这样以来，就可以避免多余的跳动，直接指到最长可匹配的状态。<br>
结合代码理解：</p>
<pre><code class="language-cpp">void build()
{
for (int i=0;i<26;i++)
if (trie[0][i])
{
fail[trie[0][i]]=0;
q.push(trie[0][i]);
}
while(!q.empty())//借助队列依照深度从小到大的关系对状态进行更新
{
int u=q.front();
q.pop();
for (int i=0;i<26;i++)
{
if (trie[u][i])
{
fail[trie[u][i]]=trie[fail[u]][i];//不需要考虑是否真的有对应状态，若有，直接匹配；若没有，得到之前深度最大的匹配
q.push(trie[u][i]);//继续更新深度更大的状态
}
else
trie[u][i]=trie[fail[u]][i];//相当于一个虚点，指向它相当于指向前一个点，这是一个传递的关系，减少了尝试匹配的操作
}
}
}
</code></pre>
<p>接下来，求<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>被<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>中的字符串匹配的次数只需在于AC自动机上遍历<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>的过程中求出每个状态<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">fail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span></span></span></span>链上的标记数即可。需要注意的是，如果我们要求的是<strong>有多少个字符串出现在s中</strong>，那么每个标记状态应只被记录一次。为避免重复记录，应将访问过的标记状态的贡献记为-1，就像这样：</p>
<pre><code class="language-cpp">int solve(char *s,int l)
{
int u=0,ans=0;
for (int i=1;i<=l;i++)
{
u=trie[u][s[i]-'a'];
for (int t=u;t&&end[t]!=-1;t=fail[t])//由于fail的传递性，fail链上的状态应是连续的被访问过再到未被访问过
ans+=end[t],end[t]=-1;
}
return ans;
}
</code></pre>
<p>如果我想要知道所有字符串分别被匹配的总次数呢？改变一下状态的记录方式，把每个字符串的终点编号记录下来，每次访问到该状态就将这个次数加1。这是非常直观的想法，但是想想我们求解<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">fail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span></span></span></span>的过程，往上跳真的很需要时间！接下来好好想想，怎么样才能避免那么多的跳跃？<br>
由于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mi>a</mi><mi>i</mi><mi>l</mi></mrow><annotation encoding="application/x-tex">fail</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord mathdefault">a</span><span class="mord mathdefault">i</span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span></span></span></span>是具有唯一前驱状态的，这符合一棵树的形态，其根节点为空集。当我们访问某一个节点，其到根的链上的节点都被访问一次。所以，为了避免每次都通过跳跃来更新信息，我们应该在每次第一个访问的节点上打上标记，当所有标记都打完之后自底向上地更新信息。可以用拓扑排序，也可以<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>dfs</mtext></mrow><annotation encoding="application/x-tex">\text{dfs}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">dfs</span></span></span></span></span>回溯时更新信息，方法多样。</p>
<h4 id="相关练习">相关练习</h4>
<p><a href="https://www.luogu.com.cn/problem/P3808">AC自动机 模板1</a><br>
<a href="https://www.luogu.com.cn/problem/P5357">AC自动机 模板2</a></p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[网络流]]></title>
<id>https://hkr04.github.io/network-flows/</id>
<link href="https://hkr04.github.io/network-flows/">
</link>
<updated>2020-03-31T15:06:06.000Z</updated>
<content type="html"><![CDATA[<h1 id="网络流简述">网络流简述</h1>
<p><strong>网络流</strong>主要可以拿来解决一些跟有向关系相关的问题，例如液体在管道中的流动、货物的运载、网络中的信息波动等。<br>
简单介绍一下它：在一个<strong>有向图</strong>上选择一个<strong>源点s</strong>、一个<strong>汇点t</strong>。源点只流出，汇点只流进。同时，一条边<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>经过的<strong>流量</strong>记为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>,也有<strong>允许通过的最大流量</strong>称为<strong>容量</strong>,记为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">c(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>。（若该边不存在，则<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c(u,v)=0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>）。除了源点和汇点以外的每个店入流和出流都相等。一条边上的剩余流量（没有用完的）称为<strong>残量</strong>，即 <strong>容量-流量</strong>。<br>
因此网络流模型可以形象地描述为：在每一条边都不超过容量限制的前提下，“流”从源点源源不断地产生，最终全部归于汇点。</p>
<h3 id="基本性质">基本性质</h3>
<ol>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>≤</mo><mi>c</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(u,v)\le c(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span><strong>（容量限制）</strong></li>
<li>对于任何一个不是源点或汇点的点<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>u</mtext></mrow><annotation encoding="application/x-tex">\text{u}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord">u</span></span></span></span></span>,总有 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mo>∑</mo><mrow><mi>p</mi><mo>∈</mo><mi>E</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>p</mi><mo separator="true">,</mo><mi>u</mi><mo>)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>q</mi><mo>∈</mo><mi>E</mi></mrow></msub><mi>f</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>q</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\sum_{p\in E}f(p,u)=\sum_{q\in E}f(u,q)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.185818em;vertical-align:-0.43581800000000004em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.17862099999999992em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.43581800000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">u</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.185818em;vertical-align:-0.43581800000000004em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.17862099999999992em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">q</span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.43581800000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mclose">)</span></span></span></span><br>
<strong>因为入流和出流相等（流量平衡）</strong></li>
<li>对于任何一条有向边<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>,总有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>=</mo><mo>−</mo><mi>f</mi><mo>(</mo><mi>v</mi><mo separator="true">,</mo><mi>u</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(u,v)=-f(v,u)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">u</span><span class="mclose">)</span></span></span></span> <strong>（斜对称性）</strong></li>
</ol>
<h2 id="最大流">最大流</h2>
<p>先讲一下比较简单的<strong>最大流问题</strong>。<br>
直接从字面意思即可理解，从源点流到汇点的流量最大就是最大流。</p>
<p><strong>算法思想</strong>：从零流（所有边的流量均为0）开始不断增加流量，保持每次增加流量后都满足以上性质（增加流量，也就是减去容量时要相应的给反向边加上等量的容量，便于反悔）。计算每条边上的残量，得到<strong>残量网络</strong>，再继续在残量网络中尝试增加流量。</p>
<p><strong>增广路算法</strong>基于：残量网络中任何一条从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>的有向道路都对应一条原图中的增广路——只要求出该道路中所有容量的最小值<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>minf</mtext></mrow><annotation encoding="application/x-tex">\text{minf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">minf</span></span></span></span></span>，把对应的所有边上的流量减去<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>minf</mtext></mrow><annotation encoding="application/x-tex">\text{minf}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">minf</span></span></span></span></span>，在答案里加上它即可，这个过程被称为<strong>增广</strong>。<br>
<strong>显然只要残量网络中存在增广路，流量就可以增大；反之，如果不存在增广路，流量就已经最大。（增广路定理）</strong></p>
<p>顺便也讲一下<strong>最大流最小割</strong>定理。</p>
<h2 id="最小割">最小割</h2>
<ul>
<li>割：对于一个网络流图<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo separator="true">,</mo><mi>E</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">G=(V,E)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">G</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mclose">)</span></span></span></span>，其割的定义为一种<strong>点的划分方式</strong>：将所有的点划分为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>和<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi><mo>=</mo><mi>V</mi><mo>−</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">T=V-S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>两个集合，其中源点<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s\in S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>，汇点<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi><mo>∈</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t\in T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65418em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span>。</li>
<li>割的容量：定义割的容量<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">c(S,T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span></span>表示所有从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span>的边的容量之和，即<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi>u</mi><mo>∈</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>∈</mo><mi>T</mi></mrow></msub><mi>c</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">c(S,T)=\sum_{u\in S,v\in T}c(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.185818em;vertical-align:-0.43581800000000004em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.17862099999999992em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">u</span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight" style="margin-right:0.05764em;">S</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span><span class="mrel mtight">∈</span><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.43581800000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>。当然也可以用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">c(s,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span>表示割的容量。</li>
<li>最小割：使得容量最小的割<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(S,T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span></span>。也可以理解为使得<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>和<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span>不联通所需要删去边权最小的割。</li>
</ul>
<h3 id="最大流最小割">最大流最小割</h3>
<ul>
<li><strong>定理</strong>：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>c</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f(s,t)_{max}=c(s,t)_{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></li>
<li><strong>证明</strong>：对于任意一个可行流<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(s,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span>和任意割<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(S,T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span></span>，有：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mi>S</mi><mtext>出边的总流量</mtext></msub><mo>−</mo><msub><mi>S</mi><mtext>入边的总流量</mtext></msub><mo>≤</mo><msub><mi>S</mi><mtext>出边的总流量</mtext></msub><mo>=</mo><mi>c</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(s,t)=S_\text{出边的总流量}-S_\text{入边的总流量}\le S_\text{出边的总流量}=c(s,t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">出边的总流量</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">入边的总流量</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord cjk_fallback mtight">出边的总流量</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose">)</span></span></span></span>。而当达到最大流时，残量网络中不存在从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>的增广路，所以<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>的出边都是满流，上式等号成立。同时，上式的另一表达形式为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>≤</mo><mi>c</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f(s,t)_{max}\le c(s,t)_{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。又因等号可以取到，则<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>c</mi><mo>(</mo><mi>s</mi><mo separator="true">,</mo><mi>t</mi><msub><mo>)</mo><mrow><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f(s,t)_{max}=c(s,t)_{min}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">a</span><span class="mord mathdefault mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">t</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>，证毕。</li>
</ul>
<h3 id="问题模型">问题模型</h3>
<p>一般在最小割的问题中，割掉一条边表示<strong>选择某个条件</strong>，而边权表示的是<strong>选择这个条件的价值</strong>。至于该价值的贡献是好是坏，则根据你如何使用这个价值而定。</p>
<p>举个例子，对于二分图，有点集<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>U</mi><mo separator="true">,</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">U,V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span>。<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∀</mi><mi>e</mi><mo>=</mo><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>(</mo><mi>u</mi><mo>∈</mo><mi>U</mi><mo separator="true">,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\forall e=(u,v) (u\in U,v\in V)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">∀</span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mclose">)</span></span></span></span>表示<strong>u、v中至少选择一个</strong>，保证无孤立点，求<strong>最小点权覆盖集</strong>。<br>
<img src="https://cdn.luogu.com.cn/upload/image_hosting/2fdq4eo2.png" alt="example1" loading="lazy"><br>
设左边为点集<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span></span></span></span>，右边为点集<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span>，如图所示连边。割掉红色边表示选择<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span></span></span></span>中某个点，其点权设为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi><mi>a</mi><msub><mi>l</mi><mi>u</mi></msub></mrow><annotation encoding="application/x-tex">val_u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">a</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>；割掉蓝色边表示选择该边两端的点，其点权设为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi><mi>a</mi><msub><mi>l</mi><mi>u</mi></msub><mo>+</mo><mi>v</mi><mi>a</mi><msub><mi>l</mi><mi>v</mi></msub></mrow><annotation encoding="application/x-tex">val_u+val_v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">a</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">u</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">a</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>；割掉橙色边表示选择<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span>中某个点，边权设为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi><mi>a</mi><msub><mi>l</mi><mi>v</mi></msub></mrow><annotation encoding="application/x-tex">val_v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">a</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.01968em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。这样一来，若一条边不从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>连通到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>，则表示至少有一个点被选中，求得的最小割即为最小点权之和，覆盖集可通过边的使用情况求得。<br>
事实上，我们都清楚，为了使点权之和最小化，蓝色的边是不会被割掉的。而仍然要设置它的原因，一是为了结合问题背景来建模，二是让它承担<strong>通道</strong>的角色，保证网络流图的性质。问题建模是非常值得琢磨的。</p>
<p>如果转换一下，边的限制变成<strong>u、v中至多选一个</strong>，求<strong>最大点权独立集</strong>。其本质相同，因为至多选一个等价于至少删去一个。由此可以看出，最大点权独立集为最小点权覆盖集的补集，其解法自然也就不言而喻了。</p>
<p>这里只是提供了一个参考的思路，具体的问题还要具体分析。</p>
<h4 id="最小割的边数">最小割的边数</h4>
<p>将边权设为1重新求一遍最小割即可。</p>
<h4 id="最小割集的求解">最小割集的求解</h4>
<p>由于使用网络流解决此类问题效率较为低下，普遍使用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Stoer-Wagner</mtext></mrow><annotation encoding="application/x-tex">\texttt{Stoer-Wagner}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Stoer-Wagner</span></span></span></span></span>算法进行求解，有兴趣可自行查阅。</p>
<h3 id="一些小细节">一些小细节</h3>
<ul>
<li>
<p>存图的时候要从偶数开始存，同时存正向边和反向边（这样就可以保证正向边编号全为偶数，反向边编号为 i^1（奇偶性相反））</p>
</li>
<li>
<p>反向边怎么用？因为找到的增广路不一定是最优的，反边给你“反悔”的机会。如果一条边边权为0，那么往回走的时候并不会对流量有所影响（走不回去）。所以一开始反边的边权应该存0。当正向边减去<strong>流过这条边的增广路上容量最小值d</strong>（此时这个值已经被加入到了答案）的时候，反向边应该加上d（因为对于源点和汇点来说中间流量的这些变化都是无差别的，为了保证反向正向相加得<strong>原边权</strong>，也就是不改变原本的条件就得这么做）</p>
</li>
</ul>
<p><a href="https://hkr04.github.io/Edmunds-Karp/">Edmonds-Karp(EK)</a> 便是不断用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>BFS</mtext></mrow><annotation encoding="application/x-tex">\text{BFS}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">BFS</span></span></span></span></span>来寻找增广路，直到图中不存在增广路的算法。</p>
<p>但是如果一条一条地找出增广路，万一有一些极（毒）端（瘤）数据（比如几条相邻的边容量相差特别大），这个时间复杂度就是无法承受的。<a href="https://hkr04.github.io/Dinic/">Dinic</a>的高效之处在于它能够同时找出几条增广路.</p>
<p>关于最大流，我还没有讲完！当然，实际上在大部分情况下以上两个算法已经够用（我认为）。可以先跳过剩下有关最大流的算法。</p>
<p><s>（这里应该有ISAP和HLPP）</s></p>
<h3 id="最大流最小割-练习题">最大流/最小割 练习题</h3>
<p><a href="https://www.luogu.com.cn/problem/P2740">USACO4.2 草地排水</a></p>
<p><a href="https://www.luogu.com.cn/problem/P2756">飞行员配对方案问题</a></p>
<p><a href="https://www.luogu.com.cn/problem/P1344">USACO4.4 追查坏牛奶</a></p>
<p><a href="https://www.luogu.com.cn/problem/P3254">圆桌问题</a></p>
<p><a href="https://www.luogu.com.cn/problem/P2764">最小路径覆盖问题</a></p>
<p><a href="https://www.luogu.com.cn/problem/P2765">魔术球问题</a></p>
<p><a href="https://www.luogu.com.cn/problem/P2766">最长不下降子序列问题</a></p>
<p><a href="https://www.luogu.com.cn/problem/P2774">方格取数问题</a></p>
<h2 id="费用流">费用流</h2>
<p>假如流经一条边有对应流量的花费，那么问题就可以有更多的变式了。</p>
<ul>
<li>定义一条边的费用<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>w</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">w(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>表示单位流量流经所需花费的费用。即，当边<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>的流量为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>时，需要花费<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>×</mo><mi>w</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(u,v)\times w(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>的费用。</li>
</ul>
<p>类似的，我们先从<strong>最小费用最大流</strong>引入。即在最大化流量的基础上使得总花费最小。<br>
这当然和上面的最大流相关。之前寻找增广路的方式是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">BFS</mtext></mrow><annotation encoding="application/x-tex">\texttt{BFS}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">BFS</span></span></span></span></span>，对于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">EK</mtext></mrow><annotation encoding="application/x-tex">\texttt{EK}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">EK</span></span></span></span></span>来说，就是在边权为1的图上找到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi><mo>→</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">s\rightarrow t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>的一条最短路。那么，将 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">BFS</mtext></mrow><annotation encoding="application/x-tex">\texttt{BFS}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">BFS</span></span></span></span></span> 更换为寻找最短路的算法，边权设为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>w</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">w(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>，会产生什么样的效果呢？</p>
<p>这样的方式，既保证了最短路的性质，也保证了增广路的性质。由于每单位流到汇点的流量都要花费途经的边权之和，则最短路的性质使得了这些每次从最短路流过的流量是<strong>最划算</strong>的。在这样的情境下，反向边相当于<strong>退钱</strong>，所以要将边权设为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>−</mo><mi>w</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">-w(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>。</p>
<p>而 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Dinic</mtext></mrow><annotation encoding="application/x-tex">\texttt{Dinic}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Dinic</span></span></span></span></span> 是同时找到多条增广路的算法，只需将 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">BFS</mtext></mrow><annotation encoding="application/x-tex">\texttt{BFS}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">BFS</span></span></span></span></span> 更改为最短路算法的同时，限制流量只能由当前点流向到汇点的最短路上的点即可。</p>
<p>因为有负权边的存在，所以不能直接采用 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Dijkstra</mtext></mrow><annotation encoding="application/x-tex">\texttt{Dijkstra}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Dijkstra</span></span></span></span></span> ，应该采用 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">SPFA</mtext></mrow><annotation encoding="application/x-tex">\texttt{SPFA}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">SPFA</span></span></span></span></span> 或经由 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Primal-Dual（原始对偶算法）</mtext></mrow><annotation encoding="application/x-tex">\texttt{Primal-Dual（原始对偶算法）}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Primal-Dual</span><span class="mord texttt cjk_fallback">（原始对偶算法）</span></span></span></span></span> 处理的 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Dijkstra</mtext></mrow><annotation encoding="application/x-tex">\texttt{Dijkstra}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.22222em;"></span><span class="mord text"><span class="mord texttt">Dijkstra</span></span></span></span></span> 。</p>
<h3 id="reference">Reference</h3>
<ul>
<li><a href="https://www.luogu.org/blog/ONE-PIECE/wang-lao-liu-jiang-xie-zhi-dinic">EK不够快？ 再学个Dinic吧 by 钱逸凡</a></li>
<li><a href="https://www.cnblogs.com/SYCstudio/p/7260613.html">Dinic算法 by SYCstudio</a></li>
<li><a href="https://artofproblemsolving.com/community/c1368h1020435">从入门到精通：最小费用流的“zkw”算法 by zkw</a></li>
<li><a href="https://oi-wiki.org/">OI Wiki</a></li>
<li>《算法竞赛入门经典第二版》</li>
<li>《算法竞赛进阶指南》</li>
<li>《算法导论》</li>
</ul>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[线段树入门]]></title>
<id>https://hkr04.github.io/segment-tree/</id>
<link href="https://hkr04.github.io/segment-tree/">
</link>
<updated>2020-02-28T14:04:15.000Z</updated>
<content type="html"><![CDATA[<blockquote>
<p><strong>这是百度百科上的图，线段树定义可以看这里-><a href="https://baike.baidu.com/item/%E7%BA%BF%E6%AE%B5%E6%A0%91/10983506?fr=aladdin">定义戳我</a></strong><img src="https://gss1.bdstatic.com/-vo3dSag_xI4khGkpoWK1HF6hhy/baike/crop%3D120%2C37%2C1504%2C992%3Bc0%3Dbaike180%2C5%2C5%2C180%2C60/sign=ec67ee96cc5c1038303194828f20a123/0e2442a7d933c895d47476f7db1373f082020037.jpg" alt="线段树图示" loading="lazy"></p>
</blockquote>
<p>任何一个区间都能被分成两个小区间，从而能够把对于大区间的查询转换为对几个小区间和小小区间和小小小区间……的查询。<br>
这是我对线段树的朴素理解，实际上就是一种分治的思想。</p>
<p>从图上也可以明显地看出任意一个长度大于1的区间都由两个小的子区间组成，子区间再往下分，直到区间内只有一个元素无法再分。<strong>因此，对于每一个长度大于1的区间[l,r]，有mid=(l+r)/2,<br>
分为左子区间[l,mid]，右子区间[mid+1,r]；且满足二叉树的性质。</strong></p>
<p>当我们在考虑用线段树解题时，要思考：</p>
<blockquote>
<p><strong>1. 是否能分成多个区间；</strong><br>
<strong>2. 区间是否具有可加性；</strong><br>
<strong>3. 叶子节点存储什么信息。</strong></p>
</blockquote>
<p>有一个小地方需要注意：<br>
通常线段树底层都不会满，但是它就是要多用那一层<br>
我们假设线段树底层是满的且叶子数为n（空间利用最优情况）（易证最后一层叶子数等于原数组元素个数），那么显而易见的，叶子总数为<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>+</mo><mi>n</mi><mi mathvariant="normal">/</mi><mn>2</mn><mo>+</mo><mi>n</mi><mi mathvariant="normal">/</mi><mn>2</mn><mi mathvariant="normal">/</mi><mn>2</mn><mo>…</mo><mo>+</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>=</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+n/2+n/2/2…+2+1=2n-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">n</span><span class="mord">/</span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">n</span><span class="mord">/</span><span class="mord">2</span><span class="mord">/</span><span class="mord">2</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">2</span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span><br>
凡事要往坏的地方多去想想，我们的空间不会运用得那么彻底，那我们还要再加一层防止溢出，这一层的叶子数就为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord mathdefault">n</span></span></span></span>。<br>
加上预留的一层空间，我们给线段树的空间就为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>4</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">4n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">4</span><span class="mord mathdefault">n</span></span></span></span>（舍去了影响较小的-1），这样就能防止因为数组不够大而re了( ´▽｀)</p>
<p>把基础的线段树掰开来理解，主要有三个部分：</p>
<ol>
<li>构建</li>
<li>修改</li>
<li>查询</li>
</ol>
<h2 id="注以下代码以基础的区间修改和查询区间和的线段树为例">注：以下代码以基础的区间修改和查询区间和的线段树为例</h2>
<p><a href="https://www.luogu.org/problemnew/show/P3372">模板：p3372线段树1</a></p>
<h3 id="1-构建">1. 构建：</h3>
<p>我采用结构体来存储每个区间的信息（这里我用数组，也有指针的写法，看个人喜好）</p>
<pre><code class="language-cpp">typedef long long ll;
int a[maxn];
struct SegmentTree
{
int l,r;//l、r分别代表这一区间的左右边界
ll dat,add;//dat为存储的信息，add是使用的lazytag（可以理解为修改标记，等下修改的时候要用到，现在可以先不管它）
SegmentTree(){add=0;}//先默认不修改
}t[maxn*4];//maxn通常在函数开头定义，方便修改大小。maxn*4是防止数组越界
</code></pre>
<pre><code class="language-cpp">void build(int p, int l, int r)//初始化函数
{
//p是当前构造的线段树数组下标，l、r分别为当前构造的点代表的区间的左右边界
t[p].l=l;t[p].r=r;//先把边界定好
if (l==r){t[p].dat=a[l];return;}
//如果l==r说明不能再分，把原数组上的信息存一下，返回
int mid=(l+r)>>1;//mid=(l+r)/2
build(p*2, l, mid);//构建左子区间
build(p*2+1, mid+1, r);//构建右子区间
t[p].dat=t[p*2].dat+t[p*2+1].dat;
//这是一个求区间和的线段树，所以该区间所存的信息等于两个子区间所存信息之和
}
</code></pre>
<p>当a数组的信息完全输入以后，build（1，1，n）即可<br>
（n为数组中元素个数）</p>
<h3 id="2-修改">2. 修改：</h3>
<p>前面出现了个神奇的<strong>lazy tag</strong>，它的作用是什么呢？</p>
<p><strong>跟你想的一样，就是用来偷懒的</strong></p>
<p>首先我们知道，修改带来的影响是对于好几个区间的。如果我们耿直地每一次修改都敬职敬责地改到最小的区间，但是很多修改根本不会被询问到，无疑是吃力不讨好的。当修改特别多的时候就可能超时。这个时候我们就得想办法<s>偷工减料</s>优化了。<br>
要是你修改了却不用，和不修改是没有区别的。所以我们先把工作攒起来，当查询的时候再去执行~~（本质就是偷工减料嘛（逃~~<br>
lazy tag的不严谨定义~~（因为是我自己写的）~~：</p>
<blockquote>
<p>当lazy tag不为0时，说明当前区间已经被这个影响所作用，这个区间的所有子区间全都攒着工作没有修改</p>
</blockquote>
<p>代码</p>
<pre><code class="language-cpp">void spread(int p)//先看下面那个函数
{
if (t[p].add)//如果当前位置为p的点的下一层要改
{
t[p*2].dat+=t[p].add*(t[p*2].r-t[p*2].l+1);
//修改左子区间（r-l+1得到区间内元素个数，我每一个都要加上从上一层传过来的add的值，那么对于这个区间的影响就是加上【元素个数*add】
//注意是+=
t[p*2+1].dat+=t[p].add*(t[p*2+1].r-t[p*2+1].l+1);//右子区间同理
t[p*2].add+=t[p].add;
//把影响给加上，同样要注意+=
t[p*2+1].add+=t[p].add;
t[p].add=0;//已经把工作传给子区间了，这一层的信息已经是最新的了，消除lazy tag
}
}
void change(int p, int v, int l, int r)
{
//p是需要修改的位置，v是要加上的值，l、r为修改区间边界
if (l<=t[p].l&&t[p].r<=r)//如果当前的点区间包含于[l,r]，执行修改并打上lazy tag
{
t[p].add+=(ll)v;
t[p].dat+=(ll)v*(t[p].r-t[p].l+1);
return;
}
spread(p);//将修改向下传，见上一个函数
int mid=(t[p].l+t[p].r)>>1;
if (l<=mid)change(p*2, v, l, r);//如果l<=mid，说明在左子区间中有地方需要更改
if (r>mid)change(p*2+1, v, l, r);
//右子区间同理
t[p].dat=t[p*2].dat+t[p*2+1].dat;
//更新当前信息
}
</code></pre>
<h3 id="3-查询">3. 查询</h3>
<p>直接上代码吧</p>
<pre><code class="language-cpp">ll ask(int p, int l, int r)
{
if (l<=t[p].l&&t[p].r<=r)//如果当前区间包含于需要查询的区间，直接征用该区间的信息，返回
return t[p].dat;
spread(p);//先看看有没有积压的工作要在查询前做了
int mid=(t[p].l+t[p].r)>>1;
ll val=0;
if (l<=mid)val+=ask(p*2, l, r);
//如果l<=mid，说明左子区间有地方的信息需要用到
if (r>mid)val+=ask(p*2+1, l, r);//同理
//因为两边的子区间中都有可能有某个子区间有信息需要征用，所以是两个if
return val;
}
</code></pre>
<p>线段树大概就是这样……</p>
<p>因为个人水平有限，难免有疏漏，欢迎批评指正(=ﾟωﾟ)ﾉ</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[线段树？树状数组！]]></title>
<id>https://hkr04.github.io/fenwick-tree-plus/</id>
<link href="https://hkr04.github.io/fenwick-tree-plus/">
</link>
<updated>2020-02-28T14:02:12.000Z</updated>
<content type="html"><![CDATA[<p>众所周知，线段树可以实现<strong>区间修改+区间查询</strong>。但实际上，树状数组也可以，并且在较为一般的情况下<strong>常数更小、占用空间更少、码量更小</strong><br>
<img src="https://cdn.luogu.com.cn/upload/image_hosting/abmoosr6.png" alt="比较图片" loading="lazy"><br>
上面为树状数组实现，下面为线段树实现（已经用了lazy tag），可以看到明显的碾压</p>
<p>设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>[</mo><mi>i</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">d[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>为记录修改的差分数组，那么若我们想查询区间1~p的修改时，我们需要把每个位置的差分数组进行求和，即：<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>i</mi></msubsup><mi>d</mi><mo>[</mo><mi>j</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">\sum_{i=1}^{p}\sum_{j=1}^id[j]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.400382em;vertical-align:-0.43581800000000004em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029000000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.964564em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.43581800000000004em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mclose">]</span></span></span></span></p>
<p>由另一个角度来说，1~p中每个<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>[</mo><mi>i</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">d[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>会被计算<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi><mo>−</mo><mi>i</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p-i+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>次（从i一直到p，每次都会被计算进去）<br>
所以上式可转化为：<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><mi>d</mi><mo>[</mo><mi>i</mi><mo>]</mo><mo>∗</mo><mo>(</mo><mi>p</mi><mo>−</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>=</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><mi>d</mi><mo>[</mo><mi>i</mi><mo>]</mo><mo>∗</mo><mi>i</mi><mo>+</mo><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>∗</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>p</mi></msubsup><mi>d</mi><mo>[</mo><mi>i</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">\sum_{i=1}^{p}d[i]*(p-i+1)=\sum_{i=1}^pd[i]*i+(p+1)*\sum_{i=1}^pd[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.104002em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029000000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.104002em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029000000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.74285em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.104002em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.804292em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.2029000000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">p</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span></p>
<p>注意到，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>是一个对数组来说的已知常量，而<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span>为每次询问的一个变量。所以我们将它们分离后可以将<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>[</mo><mi>i</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">d[i]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span></span></span></span>与<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>[</mo><mi>i</mi><mo>]</mo><mo>∗</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">d[i]*i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>作为两种不同的树状数组来维护</p>
<blockquote>
<p><em>这种<strong>分离包含有多个变量的项，使公式中不同变量之间互相独立</strong>的思想非常重要</em><br>
——《算法竞赛进阶指南》</p>
</blockquote>
<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>[</mo><mi>i</mi><mo>]</mo><mo>∗</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">d[i]*i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">d</span><span class="mopen">[</span><span class="mord mathdefault">i</span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>怎么更新？简单！就跟基本的差分的修改差不多，不过要在修改时乘上当前位置的下标而已。具体请看代码</p>
<pre><code class="language-cpp">#include <cstdio>
const int maxn=1e5+10;
typedef long long ll;
ll d[2][maxn],s[maxn];//d[1]即为保存d[i]*i的数组，s为前缀和数组
int n,m;
int lowbit(int x) {return x&(-x);}
void init()
{
for (int i=1;i<=n;i++)
{
scanf("%lld",&s[i]);
s[i]+=s[i-1];
}
}
void change(int t,int p,ll del)
{
while(p<=n)
{
d[t][p]+=del;
p+=lowbit(p);
}
}
ll ask(ll *a,int p)
{
ll res=0;
while(p)
{
res+=a[p];
p-=lowbit(p);
}
return res;
}
ll ask(int p)
{
return s[p]+ask(d[0], p)*(p+1)-ask(d[1], p);
}
int main()
{
scanf("%d%d",&n,&m);
init();
for (int i=1;i<=m;i++)
{
int op;
scanf("%d",&op);
if (op==1)
{
int l,r,d;
scanf("%d%d%d",&l,&r,&d);
change(0, l, d),change(0, r+1, -d);
change(1, l, (ll)l*d),change(1, r+1, -(ll)(r+1)*d);
}
else
{
int l,r;
scanf("%d%d",&l,&r);
printf("%lld\n",ask(r)-ask(l-1));
}
}
return 0;
}
</code></pre>
<p>参考资料：《算法竞赛进阶指南》</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[取整函数的性质]]></title>
<id>https://hkr04.github.io/Integer-valued-function/</id>
<link href="https://hkr04.github.io/Integer-valued-function/">
</link>
<updated>2020-02-28T13:59:05.000Z</updated>
<content type="html"><![CDATA[<p>我们通常将<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><mo>[</mo><mi>x</mi><mo>]</mo></mrow><annotation encoding="application/x-tex">y=[x]</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord mathdefault">x</span><span class="mclose">]</span></span></span></span>或<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>=</mo><mo>⌊</mo><mi>x</mi><mo>⌋</mo></mrow><annotation encoding="application/x-tex">y=\lfloor x \rfloor</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord mathdefault">x</span><span class="mclose">⌋</span></span></span></span>记作关于<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span>的<strong>取整函数</strong>,也称为<strong>高斯函数</strong>,其意义是<strong>不超过x的最大整数</strong></p>
<h3 id="textlemma-0"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>Lemma 0:</mtext></mrow><annotation encoding="application/x-tex">\text{Lemma 0:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">Lemma 0:</span></span></span></span></span></h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⌊</mo><mi>b</mi><mo>⌋</mo><mo>≤</mo><mi>b</mi><mo><</mo><mo>⌊</mo><mi>b</mi><mo>⌋</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\lfloor b \rfloor \le b<\lfloor b \rfloor+1
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord mathdefault">b</span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord mathdefault">b</span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span></span></p>
<h3 id="textlemma-1"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>Lemma 1:</mtext></mrow><annotation encoding="application/x-tex">\text{Lemma 1:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">Lemma 1:</span></span></span></span></span></h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mo>∈</mo><mi>Z</mi><mo separator="true">,</mo><mi>b</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">a\in Z,b\in R
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.00773em;">R</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mo>≤</mo><mo>⌊</mo><mi>b</mi><mo>⌋</mo><mo>⇔</mo><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\le\lfloor b \rfloor \Leftrightarrow a\le b
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord mathdefault">b</span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span></span></span></span></span></p>
<h4 id="textttproof"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Proof:</mtext></mrow><annotation encoding="application/x-tex">\texttt{Proof:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Proof:</span></span></span></span></span></h4>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mo>≤</mo><mo>⌊</mo><mi>b</mi><mo>⌋</mo><mo separator="true">,</mo><mo>⌊</mo><mi>b</mi><mo>⌋</mo><mo>≤</mo><mi>b</mi><mo>⇒</mo><mi>a</mi><mo>≤</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a\le\lfloor b \rfloor,\lfloor b \rfloor\le b \Rightarrow a\le b
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord mathdefault">b</span><span class="mclose">⌋</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">⌊</span><span class="mord mathdefault">b</span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>a</mi><mo>≤</mo><mi>b</mi><mo>⇒</mo><mi>a</mi><mo><</mo><mo>⌊</mo><mi>b</mi><mo>⌋</mo><mo>+</mo><mn>1</mn><mo>⇔</mo><mi>a</mi><mo>≤</mo><mo>⌊</mo><mi>b</mi><mo>⌋</mo></mrow><annotation encoding="application/x-tex">a\le b \Rightarrow a<\lfloor b \rfloor+1 \Leftrightarrow a\le \lfloor b \rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord mathdefault">b</span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">⌊</span><span class="mord mathdefault">b</span><span class="mclose">⌋</span></span></span></span></span></p>
<p>(整数的<strong>离散性</strong>:<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>Z</mi><mo separator="true">,</mo><mi>x</mi><mo><</mo><mi>y</mi><mo>⇔</mo><mi>x</mi><mo>≤</mo><mi>y</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x,y\in Z,x<y\Leftrightarrow x\le y-1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>)</p>
<h3 id="textlemma-2"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>Lemma 2:</mtext></mrow><annotation encoding="application/x-tex">\text{Lemma 2:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">Lemma 2:</span></span></span></span></span></h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">x,y\in Z
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>y</mi></mfrac><mo>⌋</mo><mo>⇔</mo><mi>y</mi><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">x\le \lfloor \frac{n}{y} \rfloor\Leftrightarrow y\le\lfloor \frac{n}{x} \rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.9880000000000002em;vertical-align:-0.8804400000000001em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<h4 id="textttproof-2"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Proof:</mtext></mrow><annotation encoding="application/x-tex">\texttt{Proof:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Proof:</span></span></span></span></span></h4>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>By lemma1:</mtext><mi>x</mi><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>y</mi></mfrac><mo>⌋</mo><mo>⇔</mo><mi>x</mi><mo>≤</mo><mfrac><mi>n</mi><mi>y</mi></mfrac><mo>⇔</mo><mi>y</mi><mo>≤</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⇔</mo><mi>y</mi><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">\text{By lemma1:}x\le\lfloor \frac{n}{y} \rfloor\Leftrightarrow x\le \frac{n}{y} \Leftrightarrow y\le\frac{n}{x}\Leftrightarrow y\le \lfloor \frac{n}{x} \rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">By lemma1:</span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.9880000000000002em;vertical-align:-0.8804400000000001em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.9880000000000002em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8304100000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<h3 id="textproposition-3"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>Proposition 3:</mtext></mrow><annotation encoding="application/x-tex">\text{Proposition 3:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">Proposition 3:</span></span></span></span></span></h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo separator="true">,</mo><mi>n</mi><mo>∈</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">x,n\in Z
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">x\le\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.13856em;vertical-align:-1.0310000000000001em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0310000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<h4 id="textttproof-3"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Proof:</mtext></mrow><annotation encoding="application/x-tex">\texttt{Proof:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Proof:</span></span></span></span></span></h4>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>By lemma2: </mtext><mi>x</mi><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo><mo>⇔</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">\text{By lemma2: }x\le\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor\Leftrightarrow\lfloor\frac{n}{x}\rfloor\le\lfloor\frac{n}{x}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">By lemma2: </span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.13856em;vertical-align:-1.0310000000000001em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0310000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇔</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<h3 id="texttheorem-4"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>Theorem 4:</mtext></mrow><annotation encoding="application/x-tex">\text{Theorem 4:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord text"><span class="mord">Theorem 4:</span></span></span></span></span></h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Z</mi><mo separator="true">,</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo><mo>=</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">x\in Z,\lfloor\frac{n}{\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor}\rfloor=\lfloor\frac{n}{x}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.37936em;vertical-align:-1.2717999999999998em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">⌊</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915428571428572em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mclose mtight">⌋</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5857999999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2717999999999998em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<h4 id="textttproof-4"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Proof:</mtext></mrow><annotation encoding="application/x-tex">\texttt{Proof:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Proof:</span></span></span></span></span></h4>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>By prosition3: </mtext><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo><mo>−</mo><mo>−</mo><mo>(</mo><mn>1</mn><mo>)</mo><mo separator="true">,</mo><mi>x</mi><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">\text{By prosition3: }\lfloor\frac{n}{x}\rfloor\le\lfloor\frac{n}{\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor}\rfloor--(1),x\le\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord text"><span class="mord">By prosition3: </span></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.37936em;vertical-align:-1.2717999999999998em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">⌊</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915428571428572em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mclose mtight">⌋</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5857999999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2717999999999998em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.13856em;vertical-align:-1.0310000000000001em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0310000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⇒</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>≥</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow></mfrac><mo>≥</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">\Rightarrow\frac{n}{x}\ge\frac{n}{\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor}\ge\lfloor\frac{n}{\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.37936em;vertical-align:-1.2717999999999998em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">⌊</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915428571428572em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mclose mtight">⌋</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5857999999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2717999999999998em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.37936em;vertical-align:-1.2717999999999998em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">⌊</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915428571428572em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mclose mtight">⌋</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5857999999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2717999999999998em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>By lamma1: </mtext><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo><mo>−</mo><mo>−</mo><mo>(</mo><mn>2</mn><mo>)</mo></mrow><annotation encoding="application/x-tex">\text{By lamma1: }\lfloor\frac{n}{\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor}\rfloor\le\lfloor\frac{n}{x}\rfloor--(2)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.37936em;vertical-align:-1.2717999999999998em;"></span><span class="mord text"><span class="mord">By lamma1: </span></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">⌊</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915428571428572em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mclose mtight">⌋</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5857999999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2717999999999998em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mn>1</mn><mo>)</mo><mtext> and </mtext><mo>(</mo><mn>2</mn><mo>)</mo><mo>⇒</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo><mo>=</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">(1)\text{ and }(2)\Rightarrow\lfloor\frac{n}{\lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor}\rfloor=\lfloor\frac{n}{x}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mclose">)</span><span class="mord text"><span class="mord"> and </span></span><span class="mopen">(</span><span class="mord">2</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⇒</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.37936em;vertical-align:-1.2717999999999998em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">⌊</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915428571428572em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span><span class="mclose mtight">⌋</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.5857999999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2717999999999998em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<h3 id="textcorollary-5"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>Corollary 5:</mtext></mrow><annotation encoding="application/x-tex">\text{Corollary 5:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">Corollary 5:</span></span></span></span></span></h3>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>y</mi><mo>∈</mo><msub><mi>Z</mi><mo>+</mo></msub><mo separator="true">,</mo><mi>max</mi><mo></mo><mrow><mo fence="true">{</mo><mi>x</mi><mo>∈</mo><msub><mi>Z</mi><mo>+</mo></msub><mi mathvariant="normal">∣</mi><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo><mo>=</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>y</mi></mfrac><mo>⌋</mo><mo fence="true">}</mo></mrow><mo>=</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>y</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">y\in Z_+,\max\left\{x\in Z_+|\lfloor\frac{n}{x}\rfloor=\lfloor\frac{n}{y}\rfloor\right\}=\lfloor\frac{n}{\lfloor\frac{n}{y}\rfloor}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.40003em;vertical-align:-0.95003em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">max</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">{</span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mord">∣</span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">}</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.274668em;vertical-align:-1.167108em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">y</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.481108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.167108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<h4 id="textttproof-5"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext mathvariant="monospace">Proof:</mtext></mrow><annotation encoding="application/x-tex">\texttt{Proof:}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61111em;vertical-align:0em;"></span><span class="mord text"><span class="mord texttt">Proof:</span></span></span></span></span></h4>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∀</mi><mi>x</mi><mo>∈</mo><msub><mi>Z</mi><mo>+</mo></msub><mo separator="true">,</mo><mtext>that </mtext><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo><mo>=</mo><mo>⌊</mo><mfrac><mi>n</mi><mi>y</mi></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">\forall x\in Z_+ ,\text{that } \lfloor\frac{n}{x}\rfloor=\lfloor\frac{n}{y}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord">∀</span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.7935600000000003em;vertical-align:-0.686em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.25833100000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord">that </span></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">x</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.9880000000000002em;vertical-align:-0.8804400000000001em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>By proposition3: </mtext><mi>x</mi><mo>≤</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>x</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo><mo>=</mo><mo>⌊</mo><mfrac><mi>n</mi><mrow><mo>⌊</mo><mfrac><mi>n</mi><mi>i</mi></mfrac><mo>⌋</mo></mrow></mfrac><mo>⌋</mo></mrow><annotation encoding="application/x-tex">\text{By proposition3: }x \le \lfloor\frac{n}{\lfloor\frac{n}{x}\rfloor}\rfloor=\lfloor\frac{n}{\lfloor\frac{n}{i}\rfloor}\rfloor
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">By proposition3: </span></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.13856em;vertical-align:-1.0310000000000001em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0310000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.13856em;vertical-align:-1.0310000000000001em;"></span><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">⌊</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.695392em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.0310000000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">⌋</span></span></span></span></span></p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[Edmunds-Karp算法]]></title>
<id>https://hkr04.github.io/Edmunds-Karp/</id>
<link href="https://hkr04.github.io/Edmunds-Karp/">
</link>
<updated>2020-02-28T13:57:58.000Z</updated>
<content type="html"><![CDATA[<p>简单介绍一下,<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>EK</mtext></mrow><annotation encoding="application/x-tex">\text{EK}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">EK</span></span></span></span></span>是每次找到一条<strong>经过的边数最少</strong>的增广路进行流量增广的算法.在每轮寻找增广路的过程中，<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>EK</mtext></mrow><annotation encoding="application/x-tex">\text{EK}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">EK</span></span></span></span></span>算法只考虑图中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo><</mo><mi>c</mi><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">f(u, v)<c(u, v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>的边，任意一条能从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>通到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>的路径都是一条增广路。根据<strong>斜对称性</strong>，反边都是可以走的。记录下该路径上的最小残量和前驱，到达<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>时可以退出<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>BFS</mtext></mrow><annotation encoding="application/x-tex">\text{BFS}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">BFS</span></span></span></span></span>，然后从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord mathdefault">t</span></span></span></span>回溯到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>更新经过的边的容量。时间复杂度：<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><mi>n</mi><msup><mi>m</mi><mn>2</mn></msup><mo>)</mo></mrow><annotation encoding="application/x-tex">O(nm^2)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">n</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>，一般能处理<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>3</mn></msup></mrow><annotation encoding="application/x-tex">10^3</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span></span></span></span>~<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span>规模的网络。</p>
<p>下面证明一下<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>EK</mtext></mrow><annotation encoding="application/x-tex">\text{EK}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">EK</span></span></span></span></span>的复杂度(可以跳过直接看下方代码).</p>
<h4 id="引理1">引理1:</h4>
<p>设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>为增广<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>次之后的容许流(即已经选择流过的合法网络),<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\lambda^k(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>表示<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">u</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span>的最短路长度,则:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>≤</mo><msup><mi>λ</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo separator="true">,</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>v</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>≤</mo><msup><mi>λ</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>v</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\lambda^k(S,v)\le \lambda^{k+1}(S,v),\lambda^k(v,T)\le\lambda^{k+1}(v,T)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.149108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.149108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.149108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span></span></span></p>
<h4 id="证明">证明:</h4>
<p>假设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span>中一条从<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span>的最短路为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi><mo>→</mo><msub><mi>u</mi><mn>1</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><mo>→</mo><msub><mi>u</mi><mrow><mi>x</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><msub><mi>u</mi><mi>x</mi></msub><mo separator="true">,</mo><msub><mi>u</mi><mi>x</mi></msub><mo>=</mo><mi>v</mi><mo separator="true">,</mo><msup><mi>λ</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">S\rightarrow u_1,\cdots,\rightarrow u_{x-1}\rightarrow u_x,u_x=v,\lambda^{k+1}(S,v)=x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">x</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0991079999999998em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span>.<br>
记<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>e</mi><mi>i</mi></msub><mo>=</mo><mo>(</mo><msub><mi>u</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>u</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">e_i=(u_{i-1},u_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>.<br>
若<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>e</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">e_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>中同样可用,即<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><msub><mi>u</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>u</mi><mi>i</mi></msub><mo>)</mo><mo><</mo><mi>c</mi><mo>(</mo><msub><mi>u</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo separator="true">,</mo><msub><mi>u</mi><mi>i</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">f(u_{i-1}, u_i)<c(u_{i-1}, u_i)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span>,则<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><msub><mi>u</mi><mi>i</mi></msub><mo>)</mo><mo>≤</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><msub><mi>u</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\lambda^k(S,u_i)\le \lambda^k(S,u_{i-1})+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>;<br>
若<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>e</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">e_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>中不可用,则<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>e</mi><mi>i</mi><mo mathvariant="normal">′</mo></msubsup></mrow><annotation encoding="application/x-tex">e_i'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.010556em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span>(<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>e</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">e_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>的反向边)必然可用.而且因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>e</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">e_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>中不可用,在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span>中变成可用,说明<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>e</mi><mi>i</mi><mo mathvariant="normal">′</mo></msubsup></mrow><annotation encoding="application/x-tex">e_i'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.010556em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>中被进行了增广使得<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>e</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">e_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>可用.也就说明了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msubsup><mi>e</mi><mi>i</mi><mo mathvariant="normal">′</mo></msubsup></mrow><annotation encoding="application/x-tex">e_i'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.010556em;vertical-align:-0.258664em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.441336em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span></span></span></span>到<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span>的最短路上,即<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><msub><mi>u</mi><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo><mo>=</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><msub><mi>u</mi><mi>i</mi></msub><mo>)</mo><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\lambda^k(S,u_{i-1})= \lambda^k(S,u_{i})+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span>,也满足上面的不等式.<br>
综上所述,<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>=</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><msub><mi>u</mi><mi>x</mi></msub><mo>)</mo><mo>≤</mo><mi>x</mi><mo>=</mo><msup><mi>λ</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\lambda^k(S,v)=\lambda^k(S,u_x)\le x=\lambda^{k+1}(S,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0991079999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8491079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span></p>
<h4 id="引理2">引理2:</h4>
<p>设边<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">e</span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>k</mi></msub></mrow><annotation encoding="application/x-tex">f_k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>变为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f_{k+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span>的增广路中,<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>e</mi><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">e'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">f_j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>变成<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>f</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">f_{j+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.980548em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mbin mtight">+</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span>的增广路中<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>k</mi><mo><</mo><mi>j</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(k<j)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mclose">)</span></span></span></span>,则有:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mi>j</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>≥</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\lambda^{j}(S,T)\ge \lambda^{k}(S,T)+2
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.124664em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.874664em;"><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.149108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span></span></p>
<h4 id="证明-2">证明:</h4>
<p>假设<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>e</mi><mo>=</mo><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">e=(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">e</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>,则:<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>=</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>u</mi><mo>)</mo><mo>+</mo><mn>1</mn><mo separator="true">,</mo><msup><mi>λ</mi><mi>j</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>=</mo><msup><mi>λ</mi><mi>j</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>+</mo><mn>1</mn><mo>+</mo><msup><mi>λ</mi><mi>j</mi></msup><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\lambda^{k}(S,v)=\lambda^{k}(S,u)+1,\lambda^{j}(S,T)=\lambda^{j}(S,v)+1+\lambda^{j}(u,T)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">u</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span></span></span></span><br>
由<strong>引理1</strong>:<br>
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>λ</mi><mi>j</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>≥</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo><mo>+</mo><mn>1</mn><mo>+</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>=</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>u</mi><mo>)</mo><mo>+</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>=</mo><msup><mi>λ</mi><mi>k</mi></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo>+</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\lambda^{j}(S,T)\ge \lambda^{k}(S,v)+1+\lambda^{k}(u,T)=\lambda^{k}(S,u)+\lambda^{k}(u,T)+2=\lambda^{k}(S,T)+2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0746639999999998em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.824664em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">u</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span></p>
<p>若<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">e</span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>k</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>k</mi><mn>2</mn></msub><mo separator="true">,</mo><mo>⋯</mo><mtext> </mtext><mo separator="true">,</mo><msub><mi>k</mi><mi>x</mi></msub></mrow><annotation encoding="application/x-tex">k_1,k_2,\cdots,k_x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>中在最短增广路上,则必有<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>j</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>j</mi><mn>2</mn></msub><mo>⋯</mo></mrow><annotation encoding="application/x-tex">j_1,j_2\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05724em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05724em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span></span></span></span>使得<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>k</mi><mn>1</mn></msub><mo><</mo><msub><mi>j</mi><mn>1</mn></msub><mo><</mo><msub><mi>k</mi><mn>2</mn></msub><mo><</mo><msub><mi>j</mi><mn>2</mn></msub><mo><</mo><mo>⋯</mo></mrow><annotation encoding="application/x-tex">k_1<j_1<k_2<j_2<\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05724em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03148em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05724em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.31em;vertical-align:0em;"></span><span class="minner">⋯</span></span></span></span>,且<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>e</mi><mo mathvariant="normal">′</mo></msup></mrow><annotation encoding="application/x-tex">e'</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span></span></span></span>在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>j</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>j</mi><mn>2</mn></msub><mo>⋯</mo></mrow><annotation encoding="application/x-tex">j_1,j_2\cdots</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05724em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05724em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">⋯</span></span></span></span>中在最短增广路上.因为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mo>≤</mo><msup><mi>λ</mi><msub><mi>k</mi><mn>1</mn></msub></msup><mo>(</mo><mi>S</mi><mo separator="true">,</mo><mi>T</mi><mo>)</mo><mo separator="true">,</mo><msup><mi>λ</mi><msub><mi>k</mi><mi>x</mi></msub></msup><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1\le \lambda^{k_1}(S,T),\lambda^{k_x}\le n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.78041em;vertical-align:-0.13597em;"></span><span class="mord">1</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:-0.03148em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">λ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03148em;">k</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.03148em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">n</span></span></span></span>,所以<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>≤</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mn>4</mn></mfrac></mrow><annotation encoding="application/x-tex">x\le\frac{n+2}{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7719400000000001em;vertical-align:-0.13597em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>.即每条边最多被增广<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mn>4</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac{n+2}{4}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>次,而每次增广的复杂度是<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><mi>m</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">O(m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord mathdefault">m</span><span class="mclose">)</span></span></span></span>的,总的复杂度即为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow><mn>4</mn></mfrac><mo>∗</mo><mi>m</mi><mo>∗</mo><mi>m</mi><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msup><mi>m</mi><mn>2</mn></msup><mi>n</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">O(\frac{n+2}{4}*m*m)=O(m^2n)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">4</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span><span class="mbin mtight">+</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.46528em;vertical-align:0em;"></span><span class="mord mathdefault">m</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">m</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathdefault">n</span><span class="mclose">)</span></span></span></span>.<br>
证毕.</p>
<p>代码：</p>
<pre><code class="language-cpp">#include <cstdio>
#include <cstring>
const int maxn=10000+10;
const int maxm=100000+10;
const int INF=0x3f3f3f3f;
int head[maxn],to[maxm<<1],nxt[maxm<<1],val[maxm<<1];
int tot=1,maxflow=0;
int pre[maxn],minf[maxn];
int n,m,s,t;
struct Queue
{
int a[maxn];
int l,r;
Queue() {l=1,r=0;}
void push(int x) {a[++r]=x;}
void pop() {l++;}
int front() {return a[l];}
bool empty() {return l>r;}
}q;
int min(int x,int y) {return x<y?x:y;}
void add(int u,int v,int w)
{
nxt[++tot]=head[u];
head[u]=tot;
to[tot]=v;
val[tot]=w;
}
bool bfs()
{
memset(pre, 0, sizeof(pre));
pre[s]=-1;
minf[s]=INF;
q=Queue();
q.push(s);
while(!q.empty())
{
int u=q.front();
q.pop();
for (int i=head[u];i;i=nxt[i])
{
int v=to[i];
if (pre[v]||!val[i])
continue;
pre[v]=i;
minf[v]=min(minf[u], val[i]);
q.push(v);
if (v==t)
return 1;
}
}
return 0;
}
void update()
{
int u=t,d=minf[t];
while(u!=s)
{
int i=pre[u];
val[i]-=d;
val[i^1]+=d;
u=to[i^1];
}
maxflow+=d;
}
int main()
{
scanf("%d%d%d%d",&n,&m,&s,&t);
for (int i=1;i<=m;i++)
{
int u,v,w;
scanf("%d%d%d",&u,&v,&w);
add(u, v, w),add(v, u, 0);
}
while(bfs())
update();
printf("%d\n",maxflow);
return 0;
}
</code></pre>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[Dinic算法]]></title>
<id>https://hkr04.github.io/Dinic/</id>
<link href="https://hkr04.github.io/Dinic/">
</link>
<updated>2020-02-28T13:56:59.000Z</updated>
<content type="html"><![CDATA[<p><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>Dinic</mtext></mrow><annotation encoding="application/x-tex">\text{Dinic}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">Dinic</span></span></span></span></span>相对<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>EK</mtext></mrow><annotation encoding="application/x-tex">\text{EK}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">EK</span></span></span></span></span>的高效之处在于运用了<strong>分层图</strong>(即由满足<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>dep[v]=dep[u]+1</mtext></mrow><annotation encoding="application/x-tex">\text{dep[v]=dep[u]+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">dep[v]=dep[u]+1</span></span></span></span></span>的边<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mi>u</mi><mo separator="true">,</mo><mi>v</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">(u,v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathdefault">u</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span>构成的子图，为有向无环图)，当考虑流向的点在分层图中深度比当前点大1 时才向那个点走，去尝试找增广路。不用担心联通性可能在这个分层图中被破坏，它在之后的分层中还是会考虑到的；不需要在意这条增广路是否为最优，只要走就是了，反正还是有反悔的机会的。<br>
时间复杂度:<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>O</mi><mo>(</mo><msup><mi>n</mi><mn>2</mn></msup><mi>m</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">O(n^2m)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord mathdefault">m</span><span class="mclose">)</span></span></span></span>，实际运用远达不到这个上界，一般能处理<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>4</mn></msup></mrow><annotation encoding="application/x-tex">10^4</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span></span></span></span></span></span></span></span>~<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><msup><mn>0</mn><mn>5</mn></msup></mrow><annotation encoding="application/x-tex">10^5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8141079999999999em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">5</span></span></span></span></span></span></span></span></span></span></span>规模的网络。</p>
<p>怎么实现呢？</p>
<ol>
<li>在残量网络上使用BFS构造分层图</li>
<li>在分层图上使用DFS尝试寻找增广路，并且实时更新每条边的容量</li>
<li>重复执行1.2直到分层图中s不能到达t（没有增广路）为止</li>
</ol>
<p>在优化<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>BFS</mtext></mrow><annotation encoding="application/x-tex">\text{BFS}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">BFS</span></span></span></span></span>次数之后，我们还可以进行优化——<strong>当前弧优化</strong>。</p>
<p>对于在不同的分层图进行<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>DFS</mtext></mrow><annotation encoding="application/x-tex">\text{DFS}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord text"><span class="mord">DFS</span></span></span></span></span>的过程中，不重复走之前走过的<strong>满流</strong>的边，因为再走下去终究会卡住，是白做工。可以用一个数组<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mtext>cur</mtext></mrow><annotation encoding="application/x-tex">\text{cur}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord text"><span class="mord">cur</span></span></span></span></span>记录一下当前点更新到哪条边了，具体看代码。</p>
<pre><code class="language-cpp">#include <cstdio>
#include <cstring>
const int maxn=10000+10;
const int maxm=100000+10;
const int INF=0x3f3f3f3f;
int cur[maxn],head[maxn],nxt[maxm<<1],to[maxm<<1],val[maxm<<1];//因为还要存反向边，所以要开两倍
int dep[maxn],inq[maxn];
int n,m,s,t;
int tot=1;
struct Queue
{
int a[maxn];
int l,r;
Queue() {l=1,r=0;}
void push(int x) {a[++r]=x;}
void pop() {l++;}
int front() {return a[l];}
bool empty() {return l>r;}
}q;
inline int min(int x,int y) {return x<y?x:y;}
void add(int u,int v,int w)
{
nxt[++tot]=head[u];
head[u]=tot;
to[tot]=v;
val[tot]=w;
}
bool bfs()
{
memset(dep, 0x3f, sizeof(dep));
dep[s]=0;
q=Queue();
q.push(s);
while(!q.empty())
{
int u=q.front();
q.pop();
for (int i=head[u];i;i=nxt[i])
{
int v=to[i];//通向的点
if (val[i]&&dep[v]>dep[u]+1)//如果容量不为0且在u点之前还没有被搜到
{
dep[v]=dep[u]+1;
q.push(v);
}
}
}
return dep[t]<INF;//只要汇点被搜到了，就还有增广路
}
int dfs(int u,int minf)//当前位置和目前搜到的最小剩余容量
{
if (u==t)//到达汇点
return minf;//返回值不为0即说明可以增广
int used=0;//该点已经使用了的流量
for (int &i=cur[u];i;i=nxt[i])//这里取址是顺便更新cur
{
int v=to[i];
if (val[i]&&dep[v]==dep[u]+1)
{
int flow=dfs(v, min(minf-used, val[i]));//能流到t的流量
if (flow)
{
used+=flow;
val[i]-=flow;
val[i^1]+=flow;
if (used==minf)//该点已达最大流量，不用继续找了
break;
}
}
}
return used;//返回该点已使用流量
}
int main()
{
scanf("%d%d%d%d",&n,&m,&s,&t);
for (int i=1;i<=m;i++)
{
int u,v,w;
scanf("%d%d%d",&u,&v,&w);
add(u, v, w),add(v, u, 0);
}
int flag=0,maxflow=0;
while(bfs())
{
for (int i=1;i<=n;i++)//新的分层图要重新开始
cur[i]=head[i];
maxflow+=dfs(s, INF);
}
printf("%d\n",maxflow);
return 0;
}
</code></pre>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[飞行员配对方案]]></title>
<id>https://hkr04.github.io/pilots-matching/</id>
<link href="https://hkr04.github.io/pilots-matching/">
</link>
<updated>2020-02-28T09:18:51.000Z</updated>
<content type="html"><![CDATA[<h4 id="题解">题解</h4>
<p>二分图匹配。<br>
设一个超级源点和一个超级汇点，建立源点->外籍飞行员->英国飞行员->汇点的网络。所有边的容量设为1。此时，每单位流到汇点的流量即代表一对合法配对，配对间互不干扰。</p>
<p>所以跑最大流即可得到最大匹配数。若想得到谁和谁配对的方案，则看看哪条边流量为0即可。</p>
<h4 id="代码">代码</h4>
<pre><code class="language-cpp">#include <cstdio>
#include <cstring>
const int maxn=200+10;
const int maxm=10000+10;
const int INF=0x3f3f3f3f;
int cur[maxn],head[maxn],nxt[maxm<<1],to[maxm<<1],val[maxm<<1];
int dep[maxn];
int n,m,s,t;
int tot=1,cnt=0;
struct edge
{
int u,v;
edge() {}
edge(int x,int y) {u=x,v=y;}
}e[maxm];
struct Queue
{
int a[maxn];
int l,r;
Queue() {l=1,r=0;}
void push(int x) {a[++r]=x;}
void pop() {l++;}
int front() {return a[l];}
bool empty() {return l>r;}
}q;
inline int min(int x,int y) {return x<y?x:y;}
void add(int u,int v,int w)
{
nxt[++tot]=head[u];
head[u]=tot;
to[tot]=v;
val[tot]=w;
}
bool bfs()
{
memset(dep, 0x3f, sizeof(dep));
dep[s]=0;
q=Queue();
q.push(s);
while(!q.empty())
{
int u=q.front();
q.pop();
for (int i=head[u];i;i=nxt[i])
{
int v=to[i];
if (val[i]&&dep[v]>dep[u]+1)
{
dep[v]=dep[u]+1;
q.push(v);
}
}
}
return dep[t]<INF;
}
int dfs(int u,int minf)
{
if (u==t)
return minf;
int used=0;
for (int &i=cur[u];i;i=nxt[i])
{
int v=to[i];
if (val[i]&&dep[v]==dep[u]+1)
{
int flow=dfs(v, min(minf-used, val[i]));
if (flow)
{
used+=flow;
val[i]-=flow;
val[i^1]+=flow;
if (used==minf)
break;
}
}
}
return used;
}
int main()
{
scanf("%d%d",&m,&n);
s=0,t=m+n+1;
int u=-1,v=-1;
while(scanf("%d%d",&u,&v)&&u!=-1)
{
add(u, v, 1),add(v, u, 0);
e[++cnt]=edge(u, v);
}
for (int i=1;i<=m;i++)
add(s, i, 1),add(i, s, 0);
for (int i=m+1;i<=m+n;i++)
add(i, t, 1),add(t, i, 0);
int maxflow=0;
while(bfs())
{
for (int i=0;i<=m+n;i++)
cur[i]=head[i];
maxflow+=dfs(s, INF);
}
printf("%d\n",maxflow);
for (int i=1;i<=cnt;i++)
if (!val[i<<1])
printf("%d %d\n",e[i].u,e[i].v);
return 0;
}
</code></pre>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[圆桌问题]]></title>
<id>https://hkr04.github.io/round-table-problem/</id>
<link href="https://hkr04.github.io/round-table-problem/">
</link>
<updated>2020-02-28T09:14:48.000Z</updated>
<content type="html"><![CDATA[<h4 id="题解">题解</h4>
<p>建立超级源超级汇，建立<strong>源点->国际单位->圆桌->汇点</strong>的网络。其中，源点连边容量为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>r</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">r_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>（INF也可以，反正限制主要是后面），国际单位连边容量为1，圆桌连边容量为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>c</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">c_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>。跑出最大流即可得到最大匹配，与人数总数相比较即可知道是否有合法方案。方案统计只需对每个单位查询有哪些边的流量不为0即可。</p>
<p>有一个小技巧是，先连<strong>国际单位->圆桌</strong>，后面统计时，查验边是否流量不为0的顺序就和存边时相等。由于同时存正反边（编号从2开始），第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord mathdefault">i</span></span></span></span>个单位和第<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span></span></span></span>个圆桌之间的边的编号即为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>(</mo><mo>(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>×</mo><mi>n</mi><mo>+</mo><mi>j</mi><mo>)</mo><mo>×</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">((i-1)\times n+j)\times 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mopen">(</span><span class="mord mathdefault">i</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">n</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.05724em;">j</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">2</span></span></span></span>。</p>
<h4 id="代码">代码</h4>
<pre><code class="language-cpp">#include <cstdio>
const int maxn=500;
const int maxm=1e5;
const int INF=0x3f3f3f3f;
int cur[maxn],head[maxn],to[maxm<<1],nxt[maxm<<1],val[maxm<<1];
int tot=1;
int c[maxn],r[maxn];
int s,t,n,m;
int dep[maxn];
struct Queue
{
int a[maxn];
int l,r;
Queue() {l=1,r=0;}
void push(int x) {a[++r]=x;}
void pop() {l++;}
int front() {return a[l];}
bool empty() {return l>r;}
}q;
int min(int x,int y) {return x<y?x:y;}
void add(int u,int v,int w)