Skip to content
Related Articles

Related Articles

Maximum path sum in a triangle.

View Discussion
Improve Article
Save Article
Like Article
  • Difficulty Level : Medium
  • Last Updated : 16 Jun, 2022

We have given numbers in form of a triangle, by starting at the top of the triangle and moving to adjacent numbers on the row below, find the maximum total from top to bottom. 
Examples : 

Input : 
   3
  7 4
 2 4 6
8 5 9 3
Output : 23
Explanation : 3 + 7 + 4 + 9 = 23 

Input :
   8
 -4 4
 2 2 6
1 1 1 1
Output : 19
Explanation : 8 + 4 + 6 + 1 = 19

Method1: We can go through the brute force by checking every possible path but that is much time taking so we should try to solve this problem with the help of dynamic programming which reduces the time complexity. 

Implementation of Recursive Approach:

C++




// C++ program for
// Recursive implementation of
// Max sum problem in a triangle
#include<bits/stdc++.h>
using namespace std;
#define N 3
 
//  Function for finding maximum sum
int maxPathSum(int tri[][N], int i, int j, int row, int col){
     if(j == col ){
         return 0;
     }
   
     if(i == row-1 ){
         return tri[i][j] ;
     }
   
     return tri[i][j] + max(maxPathSum(tri, i+1, j, row, col),
                            maxPathSum(tri, i+1, j+1, row, col)) ;
}
 
/* Driver program to test above functions */
int main()
{
   int tri[N][N] = {  {1, 0, 0},
                      {4, 8, 0},
                      {1, 5, 3} };
   cout << maxPathSum(tri, 0, 0, 3, 3);
   return 0;
}

Java



// Java program for
// Recursive implementation of
// Max sum problem in a triangle
import java.io.*;
 
class GFG {
    static int N = 3;
 
    //  Function for finding maximum sum
    public static int maxPathSum(int tri[][], int i, int j,
                                 int row, int col)
    {
        if (j == col) {
            return 0;
        }
 
        if (i == row - 1) {
            return tri[i][j];
        }
 
        return tri[i][j]
            + Math.max(
                maxPathSum(tri, i + 1, j, row, col),
                maxPathSum(tri, i + 1, j + 1, row, col));
    }
 
    /* Driver program to test above functions */
    public static void main(String[] args)
    {
        int tri[][]
            = { { 1, 0, 0 }, { 4, 8, 0 }, { 1, 5, 3 } };
        System.out.print(maxPathSum(tri, 0, 0, 3, 3));
    }
}
 
// This code is contributed by Rohit Pradhan
Python3



# Python program for
# Recursive implementation of
# Max sum problem in a triangle
N = 3
 
#  Function for finding maximum sum
def maxPathSum(tri, i, j, row, col):
     if(j == col ):
         return 0
   
     if(i == row-1 ):
         return tri[i][j]
   
     return tri[i][j] + max(maxPathSum(tri, i+1, j, row, col),
                            maxPathSum(tri, i+1, j+1, row, col))
 
# Driver program to test above functions
tri = [  [1, 0, 0],[4, 8, 0],[1, 5, 3] ]
print(maxPathSum(tri, 0, 0, 3, 3))
 
# This code is contributed by shinjanpatra
Javascript



<script>
 
// JavaScript program for
// Recursive implementation of
// Max sum problem in a triangle
 
const N = 3
 
//  Function for finding maximum sum
function maxPathSum(tri, i, j, row, col){
     if(j == col ){
         return 0;
     }
   
     if(i == row-1 ){
         return tri[i][j] ;
     }
   
     return tri[i][j] + Math.max(maxPathSum(tri, i+1, j, row, col),
                            maxPathSum(tri, i+1, j+1, row, col)) ;
}
 
/* Driver program to test above functions */
let tri = [  [1, 0, 0],[4, 8, 0],[1, 5, 3] ];
document.write(maxPathSum(tri, 0, 0, 3, 3));
 
// This code is contributed by shinjanpatra
 
</script>
Output
14
Complexity Analysis:
  • Time Complexity: O(2N*N)
  • Space Complexity:  O(N)
If we should left shift every element and put 0 at each empty position to make it a regular matrix, then our problem looks like minimum cost path. 
So, after converting our input triangle elements into a regular matrix we should apply the dynamic programming concept to find the maximum path sum. 
Method 2: DP Top-Down
Since there are overlapping subproblems, we can avoid the repeated work done in method 1 by storing the min-cost path calculated so far using top-down approach
C++



// C++ program for Dynamic
// Programming implementation (Top-Down) of
// Max sum problem in a triangle
#include<bits/stdc++.h>
using namespace std;
#define N 3
 
//  Function for finding maximum sum
int maxPathSum(int tri[][N], int i, int j, int row, int col, vector<vector<int>> &dp){
     if(j == col ){
         return 0;
     }
   
     if(i == row-1 ){
         return tri[i][j] ;
     }
   
     if(dp[i][j] != -1){
         return dp[i][j] ;
     }
   
     return dp[i][j] = tri[i][j] + max(maxPathSum(tri, i+1, j, row, col, dp),
                                   maxPathSum(tri, i+1, j+1, row, col, dp)) ;
}
 
/* Driver program to test above functions */
int main()
{
   int tri[N][N] = {  {1, 0, 0},
                      {4, 8, 0},
                      {1, 5, 3} };
   vector<vector<int>> dp(N, vector<int>(N, -1) ) ;
   cout << maxPathSum(tri, 0, 0, N, N, dp);
   return 0;
}
Python3



# C++ program for Dynamic
# Programming implementation (Top-Down) of
# Max sum problem in a triangle
N = 3
 
#  Function for finding maximum sum
def maxPathSum(tri, i, j, row, col, dp):
     if(j == col):
         return 0
   
     if(i == row-1):
         return tri[i][j]
   
     if(dp[i][j] != -1):
         return -1
   
     dp[i][j] = tri[i][j] + max(maxPathSum(tri, i+1, j, row, col, dp),
                                   maxPathSum(tri, i+1, j+1, row, col, dp))
     return dp[i][j]
 
# Driver program to test above functions
tri = [ [1, 0, 0],
        [4, 8, 0],
        [1, 5, 3] ]
dp = [[-1 for i in range(N)]for j in range(N)]
print(maxPathSum(tri, 0, 0, N, N, dp))
 
 
# This code is contributed by shinjanpatra
Javascript



<script>
 
b// JavaScript program for Dynamic
// Programming implementation (Top-Down) of
// Max sum problem in a triangle
const N = 3
 
//  Function for finding maximum sum
function maxPathSum(tri, i, j, row, col, dp){
     if(j == col)
         return 0
   
     if(i == row-1)
         return tri[i][j]
   
     if(dp[i][j] != -1)
         return -1
   
     dp[i][j] = tri[i][j] + Math.max(maxPathSum(tri, i+1, j, row, col, dp),
                                   maxPathSum(tri, i+1, j+1, row, col, dp))
     return dp[i][j]
}
 
// Driver program to test above functions
let tri = [ [1, 0, 0],
        [4, 8, 0],
        [1, 5, 3] ]
let dp = new Array(N).fill(-1).map(()=>new Array(N).fill(-1))
document.write(maxPathSum(tri, 0, 0, N, N, dp),"</br>")
 
// This code is contributed by shinjanpatra
 
</script>
Output
14
Complexity Analysis:
  • Time Complexity:  O(m*n) where m = no of rows and n = no of columns
  • Space Complexity: O(n2)
Method 3: DP(Bottom – UP)
Since there are overlapping subproblems, we can avoid the repeated work done in method 1 by storing the min-cost path calculated so far using the bottom-up approach thus reducing stack space
C++



// C++ program for Dynamic
// Programming implementation of
// Max sum problem in a triangle
#include<bits/stdc++.h>
using namespace std;
#define N 3
 
//  Function for finding maximum sum
int maxPathSum(int tri[][N], int n, vector<vector<int>> &dp)
{
     // loop for bottom-up calculation
     for(int j = 0; j < n; j++ ){
            dp[n-1][j] = tri[n-1][j] ;
        }
         
        for(int i = n-2; i >= 0; i--){
            for(int j = i; j >= 0; j-- ){
                dp[i][j] = tri[i][j] + max(dp[i+1][j] , dp[i+1][j+1]) ;
            }
        }
         
        return dp[0][0] ;
      
}
 
/* Driver program to test above functions */
int main()
{
   int tri[N][N] = {  {1, 0, 0},
                      {4, 8, 0},
                      {1, 5, 3} };
   vector<vector<int>> dp(N, vector<int>(N, -1) ) ;
   cout << maxPathSum(tri, N, dp);
   return 0;
}
Output
14
Complexity Analysis:
  • Time Complexity: O(m*n) where m = no of rows and n = no of columns
  • Space Complexity: O(n2)
Method 4: Space Optimization (Without Changning input matrix)
We do not need a 2d matrix we only need a 1d array that stores the minimum of the immediate next column  and thus we can reduce space
C++



// C++ program for Dynamic
// Programming implementation of
// Max sum problem in a triangle
#include<bits/stdc++.h>
using namespace std;
#define N 3
 
//  Function for finding maximum sum
int maxPathSum(int tri[][N], int n, vector<vector<int>> &dp)
{
        vector<int> front(n, -1) , curr(n, -1) ;
        
        for(int j = 0; j < n; j++ ){
            front[j] = tri[n-1][j] ;
        }
         
        for(int i = n-2; i >= 0; i--){
            for(int j = i; j >= 0; j-- ){
                curr[j] = tri[i][j] + max(front[j] , front[j+1]) ;
            }
            front = curr ;
        }
         
        return front[0] ;
      
}
 
/* Driver program to test above functions */
int main()
{
   int tri[N][N] = {  {1, 0, 0},
                      {4, 8, 0},
                      {1, 5, 3} };
   vector<vector<int>> dp(N, vector<int>(N, -1) ) ;
   cout << maxPathSum(tri, N, dp);
   return 0;
}
Output
14
Complexity Analysis:
  • Time Complexity: O(m*n) where m = no of rows and n = no of columns
  • Space Complexity: O(n)
Method 5: Space Optimization (Changing input matrix)
Applying, DP in bottom-up manner we should solve our problem as: 
Example: 
   3
  7 4
 2 4 6
8 5 9 3

Step 1 :
3 0 0 0
7 4 0 0
2 4 6 0
8 5 9 3

Step 2 :
3  0  0  0
7  4  0  0
10 13 15 0

Step 3 :
3  0  0  0
20 19 0  0

Step 4:
23 0 0 0

output : 23
C++



// C++ program for Dynamic
// Programming implementation of
// Max sum problem in a triangle
#include<bits/stdc++.h>
using namespace std;
#define N 3
 
//  Function for finding maximum sum
int maxPathSum(int tri[][N], int m, int n)
{
     // loop for bottom-up calculation
     for (int i=m-1; i>=0; i--)
     {
        for (int j=0; j<=i; j++)
        {
            // for each element, check both
            // elements just below the number
            // and below right to the number
            // add the maximum of them to it
            if (tri[i+1][j] > tri[i+1][j+1])
                tri[i][j] += tri[i+1][j];
            else
                tri[i][j] += tri[i+1][j+1];
        }
     }
 
     // return the top element
     // which stores the maximum sum
     return tri[0][0];
}
 
/* Driver program to test above functions */
int main()
{
   int tri[N][N] = {  {1, 0, 0},
                      {4, 8, 0},
                      {1, 5, 3} };
   cout << maxPathSum(tri, 2, 2);
   return 0;
}
Java



// Java Program for Dynamic
// Programming implementation of
// Max sum problem in a triangle
import java.io.*;
 
class GFG {
         
    static int N = 3;
     
    // Function for finding maximum sum
    static int maxPathSum(int tri[][], int m, int n)
    {
        // loop for bottom-up calculation
        for (int i = m - 1; i >= 0; i--)
        {
            for (int j = 0; j <= i; j++)
            {
                // for each element, check both
                // elements just below the number
                // and below right to the number
                // add the maximum of them to it
                if (tri[i + 1][j] > tri[i + 1][j + 1])
                    tri[i][j] += tri[i + 1][j];
                else
                    tri[i][j] += tri[i + 1][j + 1];
            }
        }
     
        // return the top element
        // which stores the maximum sum
        return tri[0][0];
    }
     
    /* Driver program to test above functions */
    public static void main (String[] args)
    {
        int tri[][] = { {1, 0, 0},
                        {4, 8, 0},
                        {1, 5, 3} };
        System.out.println ( maxPathSum(tri, 2, 2));
    }
}
 
// This code is contributed by vt_m
Python3



# Python program for
# Dynamic Programming
# implementation of Max
# sum problem in a
# triangle
 
N = 3
 
# Function for finding maximum sum
def maxPathSum(tri, m, n):
 
    # loop for bottom-up calculation
    for i in range(m-1, -1, -1):
        for j in range(i+1):
 
            # for each element, check both
            # elements just below the number
            # and below right to the number
            # add the maximum of them to it
            if (tri[i+1][j] > tri[i+1][j+1]):
                tri[i][j] += tri[i+1][j]
            else:
                tri[i][j] += tri[i+1][j+1]
 
    # return the top element
    # which stores the maximum sum
    return tri[0][0]
 
# Driver program to test above function
 
tri = [[1, 0, 0],
       [4, 8, 0],
       [1, 5, 3]]
print(maxPathSum(tri, 2, 2))
 
# This code is contributed
# by Soumen Ghosh.
C#



// C# Program for Dynamic Programming
// implementation of Max sum problem
// in a triangle
using System;
 
class GFG {
     
    // Function for finding maximum sum
    static int maxPathSum(int [,]tri,
                          int m, int n)
    {
        // loop for bottom-up calculation
        for (int i = m - 1; i >= 0; i--)
        {
            for (int j = 0; j <= i; j++)
            {
                // for each element,
                // check both elements
                // just below the number
                // and below right to
                // the number add the
                // maximum of them to it
                if (tri[i + 1,j] >
                       tri[i + 1,j + 1])
                    tri[i,j] +=
                           tri[i + 1,j];
                else
                    tri[i,j] +=
                       tri[i + 1,j + 1];
            }
        }
     
        // return the top element
        // which stores the maximum sum
        return tri[0,0];
    }
     
    /* Driver program to test above
    functions */
    public static void Main ()
    {
        int [,]tri = { {1, 0, 0},
                        {4, 8, 0},
                        {1, 5, 3} };
                         
        Console.Write (
             maxPathSum(tri, 2, 2));
    }
}
 
// This code is contributed by nitin mittal.
PHP



<?php
// PHP program for Dynamic
// Programming implementation of
// Max sum problem in a triangle
 
// Function for finding
// maximum sum
function maxPathSum($tri, $m, $n)
{
    // loop for bottom-up
    // calculation
    for ( $i = $m - 1; $i >= 0; $i--)
    {
        for ($j = 0; $j <= $i; $j++)
        {
            // for each element, check
            // both elements just below
            // the number and below right
            // to the number add the maximum
            // of them to it
            if ($tri[$i + 1][$j] > $tri[$i + 1]
                                       [$j + 1])
                $tri[$i][$j] += $tri[$i + 1][$j];
            else
                $tri[$i][$j] += $tri[$i + 1]
                                    [$j + 1];
        }
    }
 
    // return the top element
    // which stores the maximum sum
    return $tri[0][0];
}
 
// Driver Code
$tri= array(array(1, 0, 0),
            array(4, 8, 0),
            array(1, 5, 3));
echo maxPathSum($tri, 2, 2);
 
// This code is contributed by ajit
?>
Javascript



<script>
// Javascript Program for Dynamic
// Programming implementation of
// Max sum problem in a triangle
    let N = 3;
       
    // Function for finding maximum sum
    function maxPathSum(tri, m, n)
    {
     
        // loop for bottom-up calculation
        for (let i = m - 1; i >= 0; i--)
        {
            for (let j = 0; j <= i; j++)
            {
             
                // for each element, check both
                // elements just below the number
                // and below right to the number
                // add the maximum of them to it
                if (tri[i + 1][j] > tri[i + 1][j + 1])
                    tri[i][j] += tri[i + 1][j];
                else
                    tri[i][j] += tri[i + 1][j + 1];
            }
        }
       
        // return the top element
        // which stores the maximum sum
        return tri[0][0];
    }
         
// Driver code   
    let tri = [[1, 0, 0],
                  [4, 8, 0],
                  [1, 5, 3]];
       document.write( maxPathSum(tri, 2, 2));
               
// This code is contributed by susmitakundugoaldanga.           
</script>
Output
14
Complexity Analysis:
  • Time Complexity: O(m*n) where m = no of rows and n = no of columns
  • Space Complexity: O(1)
This article is contributed by Shivam Pradhan (anuj_charm). If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

My Personal Notes
arrow_drop_up
Previous
Maximum games played by winner
Next

Minimum Sum Path in a Triangle
Recommended Articles
Page :
Maximum sum of a path in a Right Number Triangle
16, Apr 17
Maximum path sum in an Inverted triangle | SET 2
15, Jan 19
Maximum size of subset of given array such that a triangle can be formed by any three integers as the sides of the triangle
27, Sep 21
Minimum Sum Path in a Triangle
26, Mar 18
Minimum length of the shortest path of a triangle
03, Oct 18
Maximum path sum for each position with jumps under divisibility condition
05, Jan 17
Maximum sum path in a matrix from top to bottom and back
07, May 19
Paths from entry to exit in matrix and maximum path sum
28, May 19
Find maximum path sum in a 2D matrix when exactly two left moves are allowed
28, May 19
Maximum path sum from top left to bottom right of a matrix passing through one of the given cells
15, Jun 21
Maximum path sum when at most K elements can be picked from a row
22, Mar 22
Maximum sum path in a matrix from top to bottom
02, Dec 17
Maximum path sum in the given arrays with at most K jumps
24, Apr 20
Maximum path sum that starting with any cell of 0-th row and ending with any cell of (N-1)-th row
25, Feb 17
Maximum sum path in a Matrix
26, Jan 20
Maximum sum path in a matrix from top-left to bottom-right
10, Sep 20
Maximum area of triangle having different vertex colors
11, Feb 17
Maximum Perimeter Triangle from array
26, Mar 18
Maximum count of Equilateral Triangles that can be formed within given Equilateral Triangle
15, Aug 20
Maximum height when coins are arranged in a triangle
12, Nov 16
Maximum number of squares that can fit in a right angle isosceles triangle
20, Sep 17
Maximum number of 2x2 squares that can be fit inside a right isosceles triangle
08, Apr 17
Sum of numbers in the Kth level of a Fibonacci triangle
17, Apr 20
Angle Sum Property of a Triangle
18, Jan 21
Article Contributed By :
https://media.geeksforgeeks.org/auth/avatar.png
GeeksforGeeks
Vote for difficulty
Current difficulty :
Medium





Improved By :
Article Tags :
Practice Tags :
Report Issue

We use cookies to ensure you have the best browsing experience on our website. By using our site, you
acknowledge that you have read and understood our
Cookie Policy &
        Privacy Policy

Lightbox

Start Your Coding Journey Now!