chore(CPlusPlus): add maximum sum rectangle (#550)

Co-authored-by: Ming Tsai <37890026+ming-tsai@users.noreply.github.com>

algorithms/CPlusPlus/Dynamic-Programming/maximum-sum-rectangle.cpp

@@ -0,0 +1,80 @@
+/*
+Maximum Sum Rectangle
+Given a 2D array, find the maximum sum rectangle in it. In other words find maximum sum over all rectangles in the matrix.
+Input Format:
+First line of input will contain T(number of test case), each test case follows as.
+First line contains 2 numbers n and m denoting number of rows and number of columns. Next n lines contain m space separated integers denoting elements of matrix nxm.
+Output Format:
+Output a single integer, maximum sum rectangle for each test case in a newline.
+Constraints
+1<=n,m<=100
+-10^5 <= mat[i][j] <= 10^5
+
+Sample Input
+4 5
+1 2 -1 -4 -20
+-8 -3 4 2 1
+3 8 10 1 3
+-4 -1 1 7 -6
+Sample Output
+29
+Approach:
+1) Calculate pre sum for every row of the matrix
+   The above matrix will look like:
+   1 3 2 -2 -22
+   -8 -11 -7 -5 -4
+   3 11 21 22 25
+   -4 -5 -4 3 -3
+2) Same approach as kadane algorithms take any two columns of the matrix and calculate maximum subarray sum in the row array.
+for example between column 0 and 1, the row array will be,
+   [4, -19, 14, -9]
+   find max subarray sum.
+3) Do it for every pair of column and find the maximum of all. And that will be the final answer.
+
+Time Complexity: O(n*m*m)
+*/
+
+#include <iostream>
+#include <vector>
+#include <climits>
+using namespace std;
+
+
+int main() {
+
+    int n, m;
+    cin >> n >> m;
+    vector<vector<int>>mat(n, vector<int>(m));
+    for (int i = 0; i < n; i++) {
+        for (int j = 0; j < m; j++) {
+            cin >> mat[i][j];
+            if (j > 0) {
+                mat[i][j] += mat[i][j - 1];
+            }
+        }
+    }
+    int ans = INT_MIN;
+    for (int i = 0; i < m; i++) {
+        for (int j = i; j < m; j++) {
+            //array explained in point (2).
+            vector<int>v(n, 0);
+            v[0] = mat[0][j] - (i > 0 ? mat[0][i - 1] : 0);
+            for (int k = 1; k < n; k++) {
+                v[k] = mat[k][j] - (i > 0 ? mat[k][i - 1] : 0);
+            }
+            int cur = 0, mx = INT_MIN;
+            for (auto p : v) {
+                cur += p;
+                if (cur > mx) {
+                    mx = cur;
+                }
+                if (cur < 0) {
+                    cur = 0;
+                }
+            }
+            ans = max(ans, mx);
+        }
+    }
+    cout << ans << endl;
+
+}

algorithms/CPlusPlus/README.md

@@ -26,9 +26,9 @@
 - [0/1-knapsack](Dynamic-Programming/01-knapsack.cpp)
 - [Matrix chain Multiplication](Dynamic-Programming/matrix-chain-multiplication.cpp)
 - [Edit Distance](Dynamic-Programming/edit-distance.cpp)
+- [Maximum sum rectangle](Dynamic-Programming/maximum-sum-rectangle.cpp)
 - [Coin Change](Dynamic-Programming/coin-change-problem.cpp)
 
-
 ## Graphs
 - [Bellman Ford Algorithm](Graphs/bellman-ford.cpp)
 - [kruskal Algorithm](Graphs/kruskal-algorithm.cpp)