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)

(i+1, j)
(i+1, j-1)
(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

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

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.

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 ; 
  
    // creat 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 column into 
    // 'dp' first column 
    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.*; 
  
class GFG { 
      
    static int N = 4; 
  
    // function find maximum sum path 
    static int MaximumPath(int Mat[][]) 
    { 
        int result = 0; 
  
        // creat 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 column into 
        // 'dp' first column 
        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 column into 
    # dp first column 
    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;  
  
        // creat 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 column into  
        // 'dp' first column  
        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; 
  
    // creat 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 column  
    // into 'dp' first column 
    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 
?> 

Output:

Time complexity : O(N²)

This article is contributed by Nishant Singh . If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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My Personal Notes

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

C++

Java

Python3

C#

PHP

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