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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>2020-02-28T14:08:06.801Z</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[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)
{
nxt[++tot]=head[u];
head[u]=tot;
to[tot]=v;
val[tot]=w;
}
bool bfs()
{
for (int i=1;i<=n+m+1;i++)
dep[i]=INF;
q=Queue();
dep[s]=0;
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);
int sum=0;
for (int i=1;i<=m;i++)
scanf("%d",&r[i]),sum+=r[i];
for (int i=1;i<=n;i++)
scanf("%d",&c[i]);
for (int i=1;i<=m;i++)
for (int j=1;j<=n;j++)
add(i, m+j, 1),add(m+j, i, 0);
s=0,t=n+m+1;
for (int i=1;i<=m;i++)
add(s, i, r[i]),add(i, s, 0);
for (int i=1;i<=n;i++)
add(m+i, t, c[i]),add(t, m+i, 0);
int cnt=0;
while(bfs())
{
for (int i=0;i<=n+m;i++)
cur[i]=head[i];
cnt+=dfs(s, INF);
}
if (cnt<sum)
printf("0\n");
else
{
printf("1\n");
for (int i=1;i<=m;i++)
{
for (int j=1;j<=n;j++)
if (!val[((i-1)*n+j)<<1])
printf("%d ",j);
putchar('\n');
}
}
return 0;
}
</code></pre>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[最小路径覆盖问题]]></title>
<id>https://hkr04.github.io/the-smallest-path-cover/</id>
<link href="https://hkr04.github.io/the-smallest-path-cover/">
</link>
<updated>2020-02-28T09:09:13.000Z</updated>
<content type="html"><![CDATA[<h4 id="题意简述">题意简述</h4>
<p>给出一个有向图<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>，求出简单路径（你可以理解为一条链）数量最少的覆盖集，使得覆盖集中所有简单路径之间无相交节点，且集合中所有节点集合等于<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>。简单路径可以只包含一个节点。</p>
<h4 id="题解">题解</h4>
<p>考虑每一个节点处在简单路径中存在两个状态：前驱和后继。由这两个状态可唯一地确定一条经过该节点的简单路径。且只要满足前驱和后继都是唯一的，那么所有的简单路径之间都不会相交。</p>
<p>所以考虑将一个节点<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>拆分成<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>和<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>u</mi><mo mathvariant="normal">′</mo></msup></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:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">u</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>，对于边<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>，由<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>向<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>v</mi><mo mathvariant="normal">′</mo></msup></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:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</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>连边。这样一来，若<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>选择了<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>v</mi><mo mathvariant="normal">′</mo></msup></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:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">v</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>，则说明<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>的后继是<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>；若<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>u</mi><mo mathvariant="normal">′</mo></msup></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:0.751892em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault">u</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>被<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>选择，说明<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>的前驱是<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>。如此一来，该题就转化成了二分图匹配问题。</p>
<p>假设一开始我所拥有的覆盖集为所有的单独节点构成的简单路径，若我能成功地增加一个匹配，那么简单路径的数量就会减一。所以，只要求出了最大二分图匹配数，假设为<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>，点数为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</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">n</span></span></span></span>，那么最小覆盖集的元素个数即为<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi><mo>−</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">n-x</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:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span></span></span></span>。</p>
<p>方案的求解大致就是这样：</p>
<ol>
<li>找到一个没有前驱的节点；</li>
<li>向后继走，走到没有后继为止。</li>
</ol>
<h4 id="代码">代码</h4>
<pre><code class="language-cpp">#include <cstdio>
#include <cstring>
const int maxn=300+10;
const int maxm=10000+10;
const int INF=0x3f3f3f3f;
int cur[maxn],head[maxn],to[maxm<<1],nxt[maxm<<1],val[maxm<<1];
int tot=1;
int n,m;
int s,t;
int dep[maxn];
int l[maxn],r[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)
{
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]>0&&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]>0&&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",&n,&m);
for (int i=1;i<=m;i++)
{
int u,v;
scanf("%d%d",&u,&v);
add(u, n+v, 1),add(n+v, u, 0);
}
s=0,t=2*n+1;
for (int i=1;i<=n;i++)
{
add(s, i, 1),add(i, s, 0);
add(n+i, t, 1),add(t, n+i, 0);
}
int ans=0;
while(bfs())
{
for (int i=s;i<=t;i++)
cur[i]=head[i];
ans+=dfs(s, INF);
}
for (int u=1;u<=n;u++)
for (int i=head[u];i;i=nxt[i])
{
if (val[i])
continue;
int v=to[i];
if (v!=s)
{
r[u]=v-n;
l[v-n]=u;
break;
}
}
for (int i=1;i<=n;i++)
if (!l[i])
{
int u=r[i];
printf("%d",i);
while(u)
{
printf(" %d",u);
u=r[u];
}
putchar('\n');
}
printf("%d\n",n-ans);
return 0;
}
</code></pre>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[魔术球问题]]></title>
<id>https://hkr04.github.io/magic-balls-problem/</id>
<link href="https://hkr04.github.io/magic-balls-problem/">
</link>
<updated>2020-02-28T09:01:03.000Z</updated>
<content type="html"><![CDATA[<h4 id="题意简述">题意简述</h4>
<p>题目希望我们在<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</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">n</span></span></span></span>个柱子放上最多的球，并且要求相邻的球之间满足相加为完全平方数的条件。</p>
<h4 id="题解">题解</h4>
<p>较显然的一点是，对于递增的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</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">n</span></span></span></span>来说，其所能放的最多球的数量一定是单调上升的。因为新增加的柱子至少可以比前一个的最大方案多放一个。同时，如果编号为<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>的球已经找不到放下的方案，根据题目要求，大于<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>的球数也一定是不合法的。</p>
<p>再从另一个角度考虑，球数增加时所需要的最少柱子数量也是单调不增的。于是就可以将这题转化为<strong>求最大的球数，使得其最小覆盖恰不大于n</strong>。如果你还没有做过最小覆盖相关的题目，简单来说，就是你需要把一些元素分成互不相交的集合，且使得集合数最小。参考<a href="https://hkr04.github.io/post/the-smallest-path-cover/">最小路径覆盖问题</a>，这两者的建模方式是类似的，在本题中每尝试增加一个球就让它尝试和之前的球连边即可。<br>
这题就这么解决啦，关键的地方在于找出单调性和由最大转化为最小可行问题。</p>
<p>（关于球的枚举上界问题，可以先设一个较大的值再跑一下极限数据看看）</p>
<h4 id="代码">代码</h4>
<pre><code class="language-cpp">#include <cstdio>
#include <cstring>
#include <cmath>
const int maxn=8000;
const int maxm=5e5+10;
const int INF=0x3f3f3f3f;
const double eps=1e-6;
int cur[maxn],head[maxn],to[maxm],nxt[maxm],val[maxm];
int tot=1;
int n,s,t;
int dep[maxn],l[maxn],r[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;}
bool check(int x,int y)
{
double tmp=sqrt(x+y);
return tmp-(int)tmp<eps;
}
void add(int u,int v,int w)
{
nxt[++tot]=head[u];
head[u]=tot;
to[tot]=v;
val[tot]=w;
}
void cancel()
{
int u=to[tot],v=to[tot^1];
head[u]=nxt[head[u]];
head[v]=nxt[head[v]];
tot-=2;
}
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]>0&&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]>0&&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;
}
}
}