Added Knapsack, CoinChange and Maximum path sum problems

Author: Muhammad-Magdi

Dynamic Programming/CoinChange.cpp

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+#include <bits/stdc++.h>
+
+using namespace std;
+const int N = (1<<20) , M = (1<<16), OO = 0x3f3f3f3f;
+
+int Coins[] = {5, 10, 25, 50, 100};
+long long mem[N];
+int n;
+
+/*
+long long solve(int i = 0, int rem = n){			//O(n*5)
+	if(i==5)	return rem==0;
+	return solve(i+1, rem) + (rem>=Coins[i]? solve(i, rem-Coins[i]) : 0);
+}
+*/
+
+long long coinChange(int rem = n){						//O(n*5)
+	if(!rem)	return 1;
+	long long & ret = mem[rem];
+	if(~ret)	return ret;
+	ret = 0;
+	for(int i = 0 ; i < 5 ; ++i)
+		ret += (rem>=Coins[i]? coinChange(rem-Coins[i]) : 0);
+	return ret;
+}
+
+int main(){
+  // freopen("i.in", "rt", stdin);
+  // freopen("o.out", "wt", stdout);
+  memset(mem, -1, sizeof mem);
+	scanf("%d", &n);
+	printf("%lld\n", coinChange());
+	return 0;
+}
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Dynamic Programming/Knapsack.cpp

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+#include <bits/stdc++.h>
+
+using namespace std;
+const int N = (1<<10) , M = (1<<16), OO = 0x3f3f3f3f;
+
+int n, W[N], C[N], k;
+int mem[N][N];										//O(n*k)
+
+int KS(int i = 0, int rem = k){		//O(n*k)
+	if(i == n)	return 0;
+	int & ret = mem[i][rem];
+	if(~ret)	return ret;
+	return ret = max(rem>=W[i]? KS(i+1, rem-W[i])+C[i] : -OO, KS(i+1, rem));
+}
+
+int main(){
+  // freopen("i.in", "rt", stdin);
+  // freopen("o.out", "wt", stdout);
+	scanf("%d %d", &n, &k);
+	memset(mem, -1, sizeof mem);
+	for(int i = 0 ; i < n ; ++i)
+		scanf("%d %d", W+i, C+i);
+	printf("%d\n", KS());
+	return 0;
+}
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Dynamic Programming/MaximumPathSum.cpp

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+#include <bits/stdc++.h>
+
+using namespace std;
+const int N = (1<<10) , M = (1<<16), OO = 0x3f3f3f3f;
+
+int n, m, A[N][N];
+int mem[N][N];										
+
+int maximumPathSum(int i=0, int j=0){
+	if(i==n-1 && j==m-1)	return 0;
+	int & ret = mem[i][j];
+	if(ret != OO)	return ret;
+	return ret=max(i+1<n? maximumPathSum(i+1, j) : -OO, j+1<m? maximumPathSum(i, j+1) : -OO) + A[i][j];
+}
+
+int main(){
+  // freopen("i.in", "rt", stdin);
+  // freopen("o.out", "wt", stdout);
+	scanf("%d %d", &n, &m);
+	memset(mem, OO, sizeof mem);
+	for(int i = 0 ; i < n ; ++i)
+		for(int j = 0 ; j < m ; ++j)
+			scanf("%d", A[i]+j);
+	printf("%d\n", maximumPathSum());
+	return 0;
+}
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