Merge pull request OpenGenus#6610 from Princeton21/dp

Added Unique Paths DP problem solution

Author: AdiChat

code/dynamic_programming/src/unique_paths/unique_paths.cpp

@@ -0,0 +1,65 @@
+// Part of Cosmos by OpenGenus Foundation
+
+#include<bits/stdc++.h>
+using namespace std;
+
+
+    //Tabulation method
+    // Time complexity: O(m*n)
+    int uniquePaths(int m, int n) {
+        int dp[m][n];
+        for(int i=0;i<m;i++){
+            for(int j=0;j<n;j++){
+                if(i==0 || j==0)
+                    dp[i][j]=1;
+                else
+                    dp[i][j]=dp[i-1][j]+dp[i][j-1];
+            }
+        }
+        return dp[m-1][n-1];
+    }
+
+
+    //nCr method
+    // Time complexity: O(m+n)
+    int uniquePaths1(int m, int n) {
+        int N=m+n-2;
+        int r=m-1;
+        double res=1;
+        for(int i=1;i<=r;i++){
+            res=res*(N-r+i)/i;
+        }
+        return (int)res;
+    }
+    
+    
+    // Brute force recursive method
+    // Time complexity: O(2^(m+n))
+    int uniquePaths2(int m, int n) {
+        if(m==1 || n==1)
+            return 1;
+        return uniquePaths2(m-1,n)+uniquePaths2(m,n-1);
+    }
+
+    //Time complexity: O(m*n)
+    int solve(int i, int j, vector<vector<int>> &dp, int &m, int &n){
+        if (i == m - 1 || j == n - 1) return 1;
+        if (dp[i][j] != 0) return dp[i][j];
+        dp[i][j] = solve(i + 1, j, dp, m, n) + solve(i, j + 1, dp, m, n);
+        return dp[i][j];
+    }
+    int uniquePaths3(int m, int n){
+        vector<vector<int>> dp(m, vector<int>(n, 0));
+        return solve(0, 0, dp, m, n);
+    }
+
+    int main(){
+        int m,n;
+        cin>>m>>n;
+        cout<<uniquePaths(m,n)<< endl;
+        cout<<uniquePaths1(m,n) << endl;
+        cout<<uniquePaths2(m,n) << endl;
+        cout<<uniquePaths3(m,n) << endl;
+        return 0;
+    }
+