Maximum path sum that starting with any cell of 0-th row and ending with any cell of (N-1)-th row

Given a N X N matrix Mat[N][N] of positive integers. There are only three possible moves from a cell (i, j) 

1. (i+1, j)
2. (i+1, j-1)
3. (i+1, j+1)

Starting from any column in row 0, return the largest sum of any of the paths up to row N-1.

Examples:

Input : mat[4][4] = { {4, 2, 3, 4},
                                 {2, 9, 1, 10},
                                 {15, 1, 3, 0},
                                 {16, 92, 41, 44} };
Output :120
path : 4 + 9 + 15 + 92 = 120 

Asked in: Amazon interview 

The above problem can be recursively defined.
Let initial position be MaximumPathSum(N-1, j), where j varies from 0 to N-1. We return maximum value between all path that we start traversing (N-1, j) [ where j varies from 0 to N-1] 

i = N-1, j = 0 to N -1
int MaximumPath(Mat[][N], I, j)
 
  // IF we reached to first row of 
  // matrix then return value of that 
  // element  
  IF ( i == 0 && j = 0 )
  return    Mat[i][j]

  // out of matrix bound 
  IF( i = N || j < 0 )
   return 0;

  // call all rest position that we reached
  // from current position and find maximum 
  // between them and add current value in 
  // that path 
  return max(MaximumPath(Mat, i-1, j), 
             MaximumPath(Mat, i-1, j-1), 
             MaximumPath(Mat, i-1, j+1)))
             + Mat[i][j];

If we draw recursion tree of above recursive solution, we can observe overlapping subproblems. Since the problem has overlapping subproblems, we can solve it efficiently using Dynamic Programming. Below is Dynamic Programming based solution. 

Implementation:

C++

// C++ program to find Maximum path sum
// start any column in row '0' and ends
// up to any column in row 'n-1'
#include<bits/stdc++.h>
using namespace std;
#define N 4

// function find maximum sum path
int MaximumPath(int Mat[][N])
{
    int result = 0 ;

    // create 2D matrix to store the sum
    // of the path
    int dp[N][N+2];

    // initialize all dp matrix as '0'
    memset(dp, 0, sizeof(dp));

    // copy all element of first row into
    // 'dp' first row
    for (int i = 0 ; i < N ; i++)
        dp[0][i+1] = Mat[0][i];

    for (int i = 1 ; i < N ; i++)
        for (int j = 1 ; j <= N ; j++)
            dp[i][j] = max(dp[i-1][j-1],
                           max(dp[i-1][j],
                               dp[i-1][j+1])) +
                       Mat[i][j-1] ;

    // Find maximum path sum that end ups
    // at  any column of last row 'N-1'
    for (int i=0; i<=N; i++)
        result = max(result, dp[N-1][i]);

    // return maximum sum path
    return result ;
}

// driver program to test above function
int main()
{
    int Mat[4][4] = { { 4, 2 , 3 , 4 },
        { 2 , 9 , 1 , 10},
        { 15, 1 , 3 , 0 },
        { 16 ,92, 41, 44 }
    };

    cout << MaximumPath ( Mat ) <<endl ;
    return 0;
}

Java

// Java program to find Maximum path sum
// start any column in row '0' and ends
// up to any column in row 'n-1'
import java.util.*;

public class GFG {

    private static int N = 4;

    // function find maximum sum path
    private static int maximumPath(int Mat[][]){
        int result = 0;

        // create 2D matrix to store the sum
        // of the path
        int dp[][] = new int[N][N + 2];

        // copy all element of first row into
        // 'dp' first row
        for (int i = 0; i < N; i++)
            dp[0][i + 1] = Mat[0][i];

        for (int i = 1; i < N; i++)
            for (int j = 1; j <= N; j++)
                dp[i][j] = Math.max(dp[i - 1][j - 1],
                                    Math.max(dp[i - 1][j],
                                    dp[i - 1][j + 1])) +
                                    Mat[i][j - 1];

        // Find maximum path sum that end ups
        // at any column of last row 'N-1'
        for (int i = 0; i <= N; i++)
            result = Math.max(result, dp[N - 1][i]);

        // return maximum sum path
        return result;
    }

    // driver code
    public static void main(String arg[]){
        int Mat[][] = { { 4, 2, 3, 4 },
                        { 2, 9, 1, 10 },
                        { 15, 1, 3, 0 },
                        { 16, 92, 41, 44 } };

        System.out.println(maximumPath(Mat));
    }
}

// This code is contributed by Anant Agarwal.

Python3

# Python3 program to find
# Maximum path sum
# start any column in
# row '0' and ends
# up to any column in row 'n-1'

N = 4

# function find maximum sum path
def MaximumPath(Mat):

    result = 0

    # create 2D matrix to store the sum
    # of the path
    # initialize all dp matrix as '0'
    dp = [[0 for i in range(N+2)] for j in range(N)]

    # copy all element of first row into
    # dp first row
    for i in range(N):
        for j in range(1, N+1):
            dp[i][j] = max(dp[i-1][j-1],
                           max(dp[i-1][j],
                               dp[i-1][j+1])) + \
                       Mat[i][j-1]

    # Find maximum path sum that end ups
    # at any column of last row 'N-1'
    for i in range(N+1):
        result = max(result, dp[N-1][i])

    # return maximum sum path
    return result

# driver program to test above function
Mat = [[4, 2, 3, 4],
       [2, 9, 1, 10],
       [15, 1, 3, 0],
       [16, 92, 41, 44]]

print(MaximumPath(Mat))

# This code is contributed by Soumen Ghosh.

C#

// C# program to find Maximum path sum
// start any column in row '0' and ends
// up to any column in row 'n-1'
using System;

class GFG {

    static int N = 4;

    // function find maximum sum path
    static int MaximumPath(int [,] Mat)
    {
        int result = 0;

        // create 2D matrix to store the sum
        // of the path
        int [,]dp = new int[N,N + 2];

        // initialize all dp matrix as '0'
        //for (int[] rows : dp)
        //    Arrays.fill(rows, 0);

        // copy all element of first row into
        // 'dp' first row
        for (int i = 0; i < N; i++)
            dp[0,i + 1] = Mat[0,i];

        for (int i = 1; i < N; i++)
            for (int j = 1; j <= N; j++)
                dp[i,j] = Math.Max(dp[i - 1,j - 1],
                                    Math.Max(dp[i - 1,j],
                                    dp[i - 1,j + 1])) +
                                    Mat[i,j - 1];

        // Find maximum path sum that end ups
        // at any column of last row 'N-1'
        for (int i = 0; i <= N; i++)
            result = Math.Max(result, dp[N - 1,i]);

        // return maximum sum path
        return result;
    }

    // driver code
    public static void Main()
    {
        int [,]Mat = { { 4, 2, 3, 4 },
                        { 2, 9, 1, 10 },
                        { 15, 1, 3, 0 },
                        { 16, 92, 41, 44 } };

        Console.WriteLine(MaximumPath(Mat));
    }
}

// This code is contributed by Ryuga.

PHP

<?php
// PHP program to find Maximum path sum
// start any column in row '0' and ends
// up to any column in row 'n-1'
$N = 4;

// function find maximum sum path
function MaximumPath(&$Mat)
{
    global $N;
    $result = 0;

    // create 2D matrix to store the sum
    // of the path
    $dp = array_fill(0, $N,
          array_fill(0, $N + 2, NULL));

    // copy all element of first row
    // into 'dp' first row
    for ($i = 0 ; $i < $N ; $i++)
        $dp[0][$i + 1] = $Mat[0][$i];

    for ($i = 1 ; $i < $N ; $i++)
        for ($j = 1 ; $j <= $N ; $j++)
            $dp[$i][$j] = max($dp[$i - 1][$j - 1],
                          max($dp[$i - 1][$j],
                              $dp[$i - 1][$j + 1])) +
                              $Mat[$i][$j - 1] ;

    // Find maximum path sum that end ups
    // at any column of last row 'N-1'
    for ($i = 0; $i <= $N; $i++)
        $result = max($result, $dp[$N - 1][$i]);

    // return maximum sum path
    return $result ;
}

// Driver Code
$Mat = array(array(4, 2 , 3 , 4),
             array(2 , 9 , 1 , 10),
             array(15, 1 , 3 , 0),
             array(16 ,92, 41, 44));

echo MaximumPath ($Mat) . "\n" ;

// This code is contributed by ita_c
?>

Javascript

<script>

// Javascript program to find Maximum path sum
// start any column in row '0' and ends
// up to any column in row 'n-1'

    let N = 4;

    // function find maximum sum path
    function MaximumPath(Mat)
    {
        let result = 0;

        // create 2D matrix to store the sum
        // of the path
        let dp = new Array(N);

        // initialize all dp matrix as '0'
        for(let i=0;i<N;i++)
        {
            dp[i]=new Array(N+2);
            for(let j=0;j<N+2;j++)
            {
                dp[i][j]=0;
            }
        }


        // copy all element of first row into
        // 'dp' first row
        for (let i = 0; i < N; i++)
            dp[0][i + 1] = Mat[0][i];

        for (let i = 1; i < N; i++)
            for (let j = 1; j <= N; j++)
                dp[i][j] = Math.max(dp[i - 1][j - 1],
                                    Math.max(dp[i - 1][j],
                                    dp[i - 1][j + 1])) +
                                    Mat[i][j - 1];

        // Find maximum path sum that end ups
        // at any column of last row 'N-1'
        for (let i = 0; i <= N; i++)
            result = Math.max(result, dp[N - 1][i]);

        // return maximum sum path
        return result;
    }

    // driver code
    let Mat = [[4, 2, 3, 4],
       [2, 9, 1, 10],
       [15, 1, 3, 0],
       [16, 92, 41, 44]]
    document.write(MaximumPath(Mat))

    // This code is contributed by rag2127

</script>

Output

120

Time complexity : O(N²)
Auxiliary space: O(N²)

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