Maximum sum path in a matrix from top to bottom

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.

Examples:

Input : mat[][] = { {5, 6, 1, 7},
                {-2, 10, 8, -1},
                {3, -7, -9, 11},
                {12, -4, 2, 6} }
Output : 28

{5, 6, 1, 7},
{-2, 10, 8, -1},
{3, -7, -9, 11},
{12, -4, 2, 6} }

The highlighted numbers from top to bottom
gives the required maximum sum path.
(7 + 8 + 11 + 2) = 28

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 subproblems are needed again and again.

Implementation:

C++

// C++ implementation to find the maximum sum
// path in a matrix
#include <bits/stdc++.h>
using namespace std;

#define SIZE 10

// function to find the maximum sum
// path in a matrix
int maxSum(int mat[SIZE][SIZE], int n)
{
    // if there is a single element only
    if (n == 1)
        return mat[0][0];

    // dp[][] matrix to store the results
    // of each iteration
    int dp[n][n];
    int maxSum = INT_MIN, max;

    // base case, copying elements of
    // last row
    for (int j = 0; j < n; j++)
        dp[n - 1][j] = mat[n - 1][j];

    // building up the dp[][] matrix from
    // bottom to the top row
    for (int i = n - 2; i >= 0; i--) {
        for (int j = 0; j < n; j++) {
            max = INT_MIN;

            // finding the maximum diagonal element in the
            // (i+1)th row if that cell exists
            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];

            // adding that 'max' element to the
            // mat[i][j] element
            dp[i][j] = mat[i][j] + max;
        }
    }

    // finding the maximum value from the
    // first row of dp[][]
    for (int j = 0; j < n; j++)
        if (maxSum < dp[0][j])
            maxSum = dp[0][j];

    // required maximum sum
    return maxSum;
}

// Driver program to test above
int main()
{
    int mat[SIZE][SIZE] = { { 5, 6, 1, 7 },
                            { -2, 10, 8, -1 },
                            { 3, -7, -9, 11 },
                            { 12, -4, 2, 6 } };
    int n = 4;

    cout << "Maximum Sum = "
         << maxSum(mat, n);

    return 0;
}

Java

// Java implementation to find the
// maximum sum path in a matrix
import java.io.*;

class MaxSumPath {
//int mat[][];

// function to find the maximum
// sum path in a matrix
static int maxSum(int[][] mat, int n)
{
    // if there is a single element only
    if (n == 1)
        return mat[0][0];

    // dp[][] matrix to store the results
    // of each iteration
    int dp[][] = new int[n][n];
    int maxSum = Integer.MIN_VALUE, max;

    // base case, copying elements of
    // last row
    for (int j = 0; j < n; j++)
        dp[n - 1][j] = mat[n - 1][j];

    // building up the dp[][] matrix
    // from bottom to the top row
    for (int i = n - 2; i >= 0; i--) {
        for (int j = 0; j < n; j++) {
            max = Integer.MIN_VALUE;

            // finding the maximum diagonal
            // element in the (i+1)th row
            // if that cell exists
            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];

            // adding that 'max' element
            // to the mat[i][j] element
            dp[i][j] = mat[i][j] + max;
        }
    }

    // finding the maximum value from
    // the first row of dp[][]
    for (int j = 0; j < n; j++)
        if (maxSum < dp[0][j])
            maxSum = dp[0][j];

    // required maximum sum
    return maxSum;
}

    // Driver code
    public static void main (String[] args) {

    int mat[][] = { { 5, 6, 1, 7 },
                    { -2, 10, 8, -1 },
                    { 3, -7, -9, 11 },
                    { 12, -4, 2, 6 } };
    int n = 4;

    System.out.println("Maximum Sum = "+
                        maxSum(mat , n));

    }
}

// This code is contributed by Prerna Saini

Python3

# Python3 implementation to find
# the maximum sum path in a matrix

SIZE=10
INT_MIN=-10000000

# function to find the maximum sum
# path in a matrix
def maxSum(mat,n):

    #if there is a single elementif
    #there is a single element only
    if n==1:
        return mat[0][0]

    # dp[][] matrix to store the results
    # of each iteration
    dp=[[0 for i in range(n)]for i in range(n)]
    maxSum=INT_MIN

    # base case, copying elements of
    # last row
    for j in range(n):
        dp[n - 1][j] = mat[n - 1][j]

    # building up the dp[][] matrix from
    # bottom to the top row
    for i in range(n-2,-1,-1):
        for j in range(n):
            maxi=INT_MIN

            # finding the maximum diagonal
            # element in the
            # (i+1)th row if that cell exists
            if ((((j - 1) >= 0) and
               (maxi < dp[i + 1][j - 1]))):
                   maxi = dp[i + 1][j - 1]
            if ((((j + 1) < n) and
               (maxi < dp[i + 1][j + 1]))):
                   maxi = dp[i + 1][j + 1]

            # adding that 'max' element to the
            # mat[i][j] element
            dp[i][j] = mat[i][j] + maxi

    # finding the maximum value from the
    # first row of dp[][]
    for j in range(n):
        if (maxSum < dp[0][j]):
            maxSum = dp[0][j]

    # required maximum sum
    return maxSum

# Driver program to test above
if __name__=='__main__':
    mat=[[5, 6, 1, 7],
        [-2, 10, 8, -1],
        [ 3, -7, -9, 11 ],
        [12, -4, 2, 6 ]]

    n=4
    print("Maximum Sum=",maxSum(mat,n))

#This code is contributed by sahilshelangia

C#

// C# implementation to find the
// maximum sum path in a matrix
using System;

class MaxSumPath
{
    //int mat[][];

    // function to find the maximum
    // sum path in a matrix
    static int maxSum(int[,] mat, int n)
    {
        // if there is a single element only
        if (n == 1)
            return mat[0, 0];

        // dp[][] matrix to store the results
        // of each iteration
        int [,]dp = new int[n, n];
        int maxSum = int.MinValue, max;

        // base case, copying elements of
        // last row
        for (int j = 0; j < n; j++)
            dp[n - 1, j] = mat[n - 1, j];

        // building up the dp[][] matrix
        // from bottom to the top row
        for (int i = n - 2; i >= 0; i--)
        {
            for (int j = 0; j < n; j++)
            {
                max = int.MinValue;

                // finding the maximum diagonal
                // element in the (i+1)th row
                // if that cell exists
                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];

                // adding that 'max' element
                // to the mat[i][j] element
                dp[i, j] = mat[i, j] + max;
            }
        }

        // finding the maximum value from
        // the first row of dp[][]
        for (int j = 0; j < n; j++)
            if (maxSum < dp[0, j])
                maxSum = dp[0, j];

        // required maximum sum
        return maxSum;
    }

    // Driver code
    public static void Main () {

    int [,]mat = { { 5, 6, 1, 7 },
                    { -2, 10, 8, -1 },
                    { 3, -7, -9, 11 },
                    { 12, -4, 2, 6 } };
    int n = 4;

    Console.WriteLine("Maximum Sum = "+
                        maxSum(mat , n));

    }
}

// This code is contributed by vt_m

PHP

<?php
// PHP implementation to find
// the maximum sum path in a matrix

$SIZE = 10;

// function to find the maximum sum
// path in a matrix
function maxSum( $mat, $n)
{

    // if there is a single
    // element only
    if ($n == 1)
        return $mat[0][0];

    // dp[][] matrix to store the results
    // of each iteration
    $dp = array(array());
    $maxSum = PHP_INT_MIN;
    $max;

    // base case, copying elements of
    // last row
    for($j = 0; $j < $n; $j++)
        $dp[$n - 1][$j] = $mat[$n - 1][$j];

    // building up the dp[][] matrix from
    // bottom to the top row
    for ( $i = $n - 2; $i >= 0; $i--)
    {
        for ( $j = 0; $j < $n; $j++)
        {
            $max = PHP_INT_MIN;

            // finding the maximum
            // diagonal element in the
            // (i+1)th row if that cell
            // exists
            if ((($j - 1) >= 0) and
                ($max < $dp[$i + 1][$j - 1]))

                $max = $dp[$i + 1][$j - 1];

            if ((($j + 1) < $n) and ($max <
                      $dp[$i + 1][$j + 1]))

                $max = $dp[$i + 1][$j + 1];

            // adding that 'max' element to the
            // mat[i][j] element
            $dp[$i][$j] = $mat[$i][$j] + $max;
        }
    }

    // finding the maximum value from the
    // first row of dp[][]
    for ( $j = 0; $j < $n; $j++)
        if ($maxSum < $dp[0][$j])
            $maxSum = $dp[0][$j];

    // required maximum sum
    return $maxSum;
}

    // Driver Code
    $mat = array(array(5, 6, 1, 7),
                 array(-2, 10, 8, -1),
                 array(3, -7, -9, 11),
                 array(12, -4, 2, 6));
    $n = 4;
    echo "Maximum Sum = "
        , maxSum($mat, $n);

// This code is contributed by anuj_67.
?>

Javascript

<script>

// JavaScript implementation to find the
// maximum sum path in a matrix

// Function to find the maximum
// sum path in a matrix
function maxSum(mat, n)
{

    // If there is a single element only
    if (n == 1)
        return mat[0][0];

    // dp[][] matrix to store the results
    // of each iteration
    let dp = new Array(n);

    // Loop to create 2D array using 1D array
    for(var i = 0; i < dp.length; i++)
    {
        dp[i] = new Array(2);
    }

    let maxSum = Number.MIN_VALUE, max;

    // Base case, copying elements of
    // last row
    for(let j = 0; j < n; j++)
        dp[n - 1][j] = mat[n - 1][j];

    // Building up the dp[][] matrix
    // from bottom to the top row
    for(let i = n - 2; i >= 0; i--)
    {
        for(let j = 0; j < n; j++)
        {
            max = Number.MIN_VALUE;

            // Finding the maximum diagonal
            // element in the (i+1)th row
            // if that cell exists
            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];

            // Adding that 'max' element
            // to the mat[i][j] element
            dp[i][j] = mat[i][j] + max;
        }
    }

    // Finding the maximum value from
    // the first row of dp[][]
    for(let j = 0; j < n; j++)
        if (maxSum < dp[0][j])
            maxSum = dp[0][j];

    // Required maximum sum
    return maxSum;
}

// Driver Code
let mat = [ [ 5, 6, 1, 7 ],
            [ -2, 10, 8, -1 ],
            [ 3, -7, -9, 11 ],
            [ 12, -4, 2, 6 ] ];
let n = 4;

document.write("Maximum Sum = "+
               maxSum(mat , n));

// This code is contributed by sanjoy_62

</script>

Output

Maximum Sum = 28

Time Complexity: O(n²). 
Auxiliary Space: O(n²).

Exercise: Try to solve it in auxiliary space of O(n).