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Maximum sum path in a matrix from top to bottom in C++
Problem statement
Consider a n*n matrix. Suppose each cell in the matrix has a value assigned. We can go from each cell in row i to a diagonally higher cell in row i+1 only [i.e from cell(i, j) to cell(i+1, j-1) and cell(i+1, j+1) only]. Find the path from the top row to the bottom row following the aforementioned condition such that the maximum sum is obtained
Example
If given input is:
{
{5, 6, 1, 17},
{-2, 10, 8, -1},
{ 3, -7, -9, 4},
{12, -4, 2, 2}
}the maximum sum is (17 + 8 + 4 + 2) = 31
Algorithm
The idea is to find maximum sum, or all paths starting with every cell of first row and finally return maximum of all values in first row.
We use Dynamic Programming as results of many sub problems are needed again and again
Example
#include <bits/stdc++.h>
using namespace std;
#define SIZE 10
int getMaxMatrixSum(int mat[SIZE][SIZE], int n){
if (n == 1) {
return mat[0][0];
}
int dp[n][n];
int maxSum = INT_MIN, max;
for (int j = 0; j < n; j++) {
dp[n - 1][j] = mat[n - 1][j];
}
for (int i = n - 2; i >= 0; i--) {
for (int j = 0; j < n; j++) {
max = INT_MIN;
if (((j - 1) >= 0) && (max < dp[i + 1][j - 1])) {
max = dp[i + 1][j - 1];
}
if (((j + 1) < n) && (max < dp[i + 1][j + 1])) {
max = dp[i + 1][j + 1];
}
dp[i][j] = mat[i][j] + max;
}
}
for (int j = 0; j < n; j++) {
if (maxSum < dp[0][j]) {
maxSum = dp[0][j];
}
}
return maxSum;
}
int main(){
int mat[SIZE][SIZE] = {
{5, 6, 1, 17},
{-2, 10, 8, -1},
{3, -7, -9, 4},
{12, -4, 2, 2}
};
int n = 4;
cout << "Maximum Sum = " << getMaxMatrixSum(mat, n) << endl;
return 0;
}Output
When you compile and execute above program. It generates following output−
Maximum Sum = 31
Updated on: 30-Jan-2020
172 Views
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