Maximum sum path in a Matrix

Given an n*m matrix, the task is to find the maximum sum of elements of cells starting from the cell (0, 0) to cell (n-1, m-1). 
However, the allowed moves are right, downwards or diagonally right, i.e, from location (i, j) next move can be (i+1, j), or, (i, j+1), or (i+1, j+1). Find the maximum sum of elements satisfying the allowed moves.
Examples:

Input:
mat[][] = {{100, -350, -200},
           {-100, -300, 700}}
Output: 500
Explanation: 
Path followed is 100 -> -300 -> 700

Input:
mat[][] = {{500, 100, 230},
           {1000, 300, 100},
           {200, 1000, 200}}
Explanation:
Path followed is 500 -> 1000 -> 300 -> 1000 -> 200

Naive Approach: Recursion

Going through the Naive approach by traversing every possible path. But, this is costly. So, use Dynamic Programming here in order to reduce the time complexity.

C++

#include <bits/stdc++.h>
using namespace std;
#define N 100

// No of rows and columns
int n, m;

// Declaring the matrix of maximum
// 100 rows and 100 columns
int a[N][N];

// For storing current sum
int current_sum = 0;

// For continuous update of
// maximum sum required
int total_sum = 0;

// Function to Input the matrix of size n*m
void inputMatrix()
{
    n = 3;
    m = 3;
    a[0][0] = 500;
    a[0][1] = 100;
    a[0][2] = 230;
    a[1][0] = 1000;
    a[1][1] = 300;
    a[1][2] = 100;
    a[2][0] = 200;
    a[2][1] = 1000;
    a[2][2] = 200;
}

// Function to calculate maximum sum of path
int maximum_sum_path(int i, int j)
{
    // Checking boundary condition
    if (i == n - 1 && j == m - 1)
        return a[i][j];

    // Checking whether the position hasn't
    // visited the last row or the last column.
    // Making recursive call for all the possible
    // moves from the current cell and then adding the
    // maximum returned by the calls and updating it.
    if (i < n - 1 & j < m - 1) {
        int current_sum = max(maximum_sum_path(i, j + 1),
                              max(
                                  maximum_sum_path(i + 1, j + 1),
                                  maximum_sum_path(i + 1, j)));
        total_sum = a[i][j] + current_sum;
    }

    // Checking whether
    // position has reached last row
    else if (i == n - 1)
        total_sum = a[i][j]
                    + maximum_sum_path(i, j + 1);

    // If the position is in the last column
    else
        total_sum = a[i][j]
                    + maximum_sum_path(i + 1, j);

    // Returning the updated maximum value
    return total_sum;
}

// Driver Code
int main()
{
    inputMatrix();

    // Calling the implemented function
    int maximum_sum = maximum_sum_path(0, 0);

    cout << maximum_sum;
    return 0;
}

Java

// Java program for
// Recursive implementation of
// Max sum path
import java.io.*;

class GFG {

    // Function for finding maximum sum
    public static int maxPathSum(int matrix[][], int i, int j)
    {

        if (i<0 || j<0) {
          return -100_000_000;
        }
        if (i==0 && j==0) {
            return matrix[i][j];
        }

          int up = matrix[i][j] + maxPathSum(matrix, i - 1, j);
        int right = matrix[i][j] + maxPathSum(matrix, i, j - 1);
        int up_left_diagonal = matrix[i][j] + maxPathSum(matrix, i - 1, j - 1);
        return  Math.max(up , Math.max( right, up_left_diagonal));
    }

    /* Driver program to test above functions */
    public static void main(String[] args)
    {
        int matrix [][] ={{100, -350, -200}, {-100, -300, 700}};
        System.out.print(maxPathSum(matrix, 1, 2));
    }
}

// This code is contributed by Rohini Chandra

Python3

# No of rows and columns
n = 0
m = 0
# Declaring the matrix of maximum
# 100 rows and 100 columns
rows, cols = (100, 100)
a = [[0 for i in range(cols)] for j in range(rows)]

# For storing current sum
current_sum = 0

# For continuous update of
# maximum sum required
total_sum = 0

# Function to Input the matrix of size n*m


def inputMatrix():
  global n, m, a
  n = 3
  m = 3
  a[0][0] = 500
  a[0][1] = 100
  a[0][2] = 230
  a[1][0] = 1000
  a[1][1] = 300
  a[1][2] = 100
  a[2][0] = 200
  a[2][1] = 1000
  a[2][2] = 200


# Function to calculate maximum sum of path
def maximum_sum_path(i, j):
  global n, m, a, total_sum, current_sum

  # Checking boundary condition
  if i == n - 1 and j == m - 1:
      return a[i][j]

  # Checking whether the position hasn't
  # visited the last row or the last column.
  # Making recursive call for all the possible
  # moves from the current cell and then adding the
  # maximum returned by the calls and updating it.
  if (i < (n - 1)) & (j < (m - 1)):
      current_sum = max(maximum_sum_path(
          i, j + 1), max(maximum_sum_path(i + 1, j + 1),
                     maximum_sum_path(i + 1, j)))
      total_sum = a[i][j] + current_sum
  # Checking whether
  # position has reached last row
  elif i == n - 1:
      total_sum = a[i][j] + maximum_sum_path(i, j + 1)
  # If the position is in the last column
  else:
      total_sum = a[i][j] + maximum_sum_path(i + 1, j)

  # Returning the updated maximum value
  return total_sum


# Driver program
if __name__ == "__main__":
  inputMatrix()

  # Calling the implemented function
  maximum_sum = maximum_sum_path(0, 0)

  print(maximum_sum)

Javascript

<script>

const N = 100

// No of rows and columns
let n, m;

// Declaring the matrix of maximum
// 100 rows and 100 columns
let a = new Array(N);
for(let i=0;i<N;i++){
    a[i] = new Array(N);
}

// For storing current sum
let current_sum = 0;

// For continuous update of
// maximum sum required
let total_sum = 0;

// Function to Input the matrix of size n*m
function inputMatrix()
{
    n = 3;
    m = 3;
    a[0][0] = 500;
    a[0][1] = 100;
    a[0][2] = 230;
    a[1][0] = 1000;
    a[1][1] = 300;
    a[1][2] = 100;
    a[2][0] = 200;
    a[2][1] = 1000;
    a[2][2] = 200;
}

// Function to calculate maximum sum of path
function maximum_sum_path(i, j)
{

    // Checking boundary condition
    if (i == n - 1 && j == m - 1)
        return a[i][j];

    // Checking whether the position hasn't
    // visited the last row or the last column.
    // Making recursive call for all the possible
    // moves from the current cell and then adding the
    // maximum returned by the calls and updating it.
    if (i < n - 1 & j < m - 1) {
        let current_sum = Math.max(maximum_sum_path(i, j + 1),
                            Math.max(
                                maximum_sum_path(i + 1, j + 1),
                                maximum_sum_path(i + 1, j)));
        total_sum = a[i][j] + current_sum;
    }

    // Checking whether
    // position has reached last row
    else if (i == n - 1)
        total_sum = a[i][j]
                    + maximum_sum_path(i, j + 1);

    // If the position is in the last column
    else
        total_sum = a[i][j]
                    + maximum_sum_path(i + 1, j);

    // Returning the updated maximum value
    return total_sum;
}

// Driver Code
inputMatrix();

// Calling the implemented function
let maximum_sum = maximum_sum_path(0, 0);

document.write(maximum_sum,"</br>");

// This code is contributed by shinjanpatra

</script>

Output

3000

Time Complexity: O(2^N*M) 

Auxiliary Space: O(N*M)

2. Efficient Approach(Memoization) : Dynamic programming is used to solve the above problem in a recursive way.  

1. Allot a position at the beginning of the matrix at (0, 0).
2. Check each next allowed position from the current position and select the path with maximum sum.
3. Take care of the boundaries of the matrix, i.e, if the position reaches the last row or last column then the only possible choice will be right or downwards respectively.
4. Use a map to store track of all the visiting positions and before visiting any (i, j), check whether or not the position is visited before.
5. Update the maximum of all possible paths returned by each recursive calls made.
6. Go till the position reaches the destination cell, i.e, (n-1.m-1).

Below is the implementation of the above approach: 

C++

#include <bits/stdc++.h>
using namespace std;
#define N 100

// No of rows and columns
int n, m;

// Declaring the matrix of maximum
// 100 rows and 100 columns
int a[N][N];

// Variable visited is used to keep
// track of all the visited positions
// Variable dp is used to store
// maximum sum till current position
vector<vector<int> > dp(N, vector<int>(N)),
    visited(N, vector<int>(N));

// For storing current sum
int current_sum = 0;

// For continuous update of
// maximum sum required
int total_sum = 0;

// Function to Input the matrix of size n*m
void inputMatrix()
{
    n = 3;
    m = 3;
    a[0][0] = 500;
    a[0][1] = 100;
    a[0][2] = 230;
    a[1][0] = 1000;
    a[1][1] = 300;
    a[1][2] = 100;
    a[2][0] = 200;
    a[2][1] = 1000;
    a[2][2] = 200;
}

// Function to calculate maximum sum of path
int maximum_sum_path(int i, int j)
{
    // Checking boundary condition
    if (i == n - 1 && j == m - 1)
        return a[i][j];

    // Checking whether or not (i, j) is visited
    if (visited[i][j])
        return dp[i][j];

    // Marking (i, j) is visited
    visited[i][j] = 1;

    // Updating the maximum sum till
    // the current position in the dp
    int& total_sum = dp[i][j];

    // Checking whether the position hasn't
    // visited the last row or the last column.
    // Making recursive call for all the possible
    // moves from the current cell and then adding the
    // maximum returned by the calls and updating it.
    if (i < n - 1 & j < m - 1) {
        int current_sum = max(maximum_sum_path(i, j + 1),
                              max(
                                  maximum_sum_path(i + 1, j + 1),
                                  maximum_sum_path(i + 1, j)));
        total_sum = a[i][j] + current_sum;
    }

    // Checking whether
    // position has reached last row
    else if (i == n - 1)
        total_sum = a[i][j]
                    + maximum_sum_path(i, j + 1);

    // If the position is in the last column
    else
        total_sum = a[i][j]
                    + maximum_sum_path(i + 1, j);

    // Returning the updated maximum value
    return dp[i][j] = total_sum;
}

// Driver Code
int main()
{
    inputMatrix();

    // Calling the implemented function
    int maximum_sum = maximum_sum_path(0, 0);

    cout << maximum_sum;
    return 0;
}

Java

class GFG{

static final int N = 100;

// No of rows and columns
static int n, m;

// Declaring the matrix of maximum
// 100 rows and 100 columns
static int a[][] = new int[N][N];

// Variable visited is used to keep
// track of all the visited positions
// Variable dp is used to store
// maximum sum till current position
static int dp[][] = new int[N][N];
static int visited[][] = new int[N][N];

// For storing current sum
static int current_sum = 0;

// For continuous update of
// maximum sum required
static int total_sum = 0;

// Function to Input the matrix
// of size n*m
static void inputMatrix()
{
    n = 3;
    m = 3;
    a[0][0] = 500;
    a[0][1] = 100;
    a[0][2] = 230;
    a[1][0] = 1000;
    a[1][1] = 300;
    a[1][2] = 100;
    a[2][0] = 200;
    a[2][1] = 1000;
    a[2][2] = 200;
}

// Function to calculate maximum sum of path
static int maximum_sum_path(int i, int j)
{

    // Checking boundary condition
    if (i == n - 1 && j == m - 1)
        return a[i][j];

    // Checking whether or not
    // (i, j) is visited
    if (visited[i][j] != 0)
        return dp[i][j];

    // Marking (i, j) is visited
    visited[i][j] = 1;

    int total_sum = 0;

    // Checking whether the position hasn't
    // visited the last row or the last column.
    // Making recursive call for all the possible
    // moves from the current cell and then adding the
    // maximum returned by the calls and updating it.
    if (i < n - 1 & j < m - 1)
    {
        int current_sum = Math.max(
                          maximum_sum_path(i, j + 1),
                          Math.max(
                          maximum_sum_path(i + 1, j + 1),
                          maximum_sum_path(i + 1, j)));
        total_sum = a[i][j] + current_sum;
    }

    // Checking whether position
    // has reached last row
    else if (i == n - 1)
        total_sum = a[i][j] +
                    maximum_sum_path(i, j + 1);

    // If the position is in the last column
    else
        total_sum = a[i][j] +
                    maximum_sum_path(i + 1, j);

    // Updating the maximum sum till
    // the current position in the dp
    dp[i][j] = total_sum;

    // Returning the updated maximum value
    return total_sum;
}

// Driver Code
public static void main(String[] args)
{
    inputMatrix();

    // Calling the implemented function
    int maximum_sum = maximum_sum_path(0, 0);

    System.out.println(maximum_sum);
}
}

// This code is contributed by jrishabh99

C#

// C# program to implement
// the above approach
using System;
class GFG{

static readonly int N = 100;

// No of rows and columns
static int n, m;

// Declaring the matrix of maximum
// 100 rows and 100 columns
static int[,]a = new int[N, N];

// Variable visited is used to keep
// track of all the visited positions
// Variable dp is used to store
// maximum sum till current position
static int [,]dp = new int[N, N];
static int [,]visited = new int[N, N];

// For storing current sum
static int current_sum = 0;

// For continuous update of
// maximum sum required
static int total_sum = 0;

// Function to Input the matrix
// of size n*m
static void inputMatrix()
{
  n = 3;
  m = 3;
  a[0, 0] = 500;
  a[0, 1] = 100;
  a[0, 2] = 230;
  a[1, 0] = 1000;
  a[1, 1] = 300;
  a[1, 2] = 100;
  a[2, 0] = 200;
  a[2, 1] = 1000;
  a[2, 2] = 200;
}

// Function to calculate maximum
// sum of path
static int maximum_sum_path(int i,
                            int j)
{
  // Checking boundary condition
  if (i == n - 1 && j == m - 1)
    return a[i, j];

  // Checking whether or not
  // (i, j) is visited
  if (visited[i, j] != 0)
    return dp[i, j];

  // Marking (i, j) is visited
  visited[i, j] = 1;

  int total_sum = 0;

  // Checking whether the position
  // hasn't visited the last row
  // or the last column.
  // Making recursive call for all
  // the possible moves from the
  // current cell and then adding the
  // maximum returned by the calls
  // and updating it.
  if (i < n - 1 & j < m - 1)
  {
    int current_sum = Math.Max(maximum_sum_path(i, j + 1),
                      Math.Max(maximum_sum_path(i + 1,
                                                j + 1),
                               maximum_sum_path(i + 1, j)));
    total_sum = a[i, j] + current_sum;
  }

  // Checking whether position
  // has reached last row
  else if (i == n - 1)
    total_sum = a[i, j] +
                maximum_sum_path(i, j + 1);

  // If the position is
  // in the last column
  else
    total_sum = a[i, j] +
                maximum_sum_path(i + 1, j);

  // Updating the maximum
  // sum till the current
  // position in the dp
  dp[i, j] = total_sum;

  // Returning the updated
  // maximum value
  return total_sum;
}

// Driver Code
public static void Main(String[] args)
{
  inputMatrix();

  // Calling the implemented function
  int maximum_sum = maximum_sum_path(0, 0);

  Console.WriteLine(maximum_sum);
}
}

// This code is contributed by shikhasingrajput

Javascript

<script>
    // Javascript program to implement the above approach

    let N = 100;

    // No of rows and columns
    let n, m;

    // Declaring the matrix of maximum
    // 100 rows and 100 columns
    let a = new Array(N);

    // Variable visited is used to keep
    // track of all the visited positions
    // Variable dp is used to store
    // maximum sum till current position
    let dp = new Array(N);
    let visited = new Array(N);
    for(let i = 0; i < N; i++)
    {
        a[i] = new Array(N);
        dp[i] = new Array(N);
        visited[i] = new Array(N);
        for(let j = 0; j < N; j++)
        {
            a[i][j] = 0;
            dp[i][j] = 0;
            visited[i][j] = 0;
        }
    }

    // For storing current sum
    let current_sum = 0;

    // For continuous update of
    // maximum sum required
    let total_sum = 0;

    // Function to Input the matrix
    // of size n*m
    function inputMatrix()
    {
        n = 3;
        m = 3;
        a[0][0] = 500;
        a[0][1] = 100;
        a[0][2] = 230;
        a[1][0] = 1000;
        a[1][1] = 300;
        a[1][2] = 100;
        a[2][0] = 200;
        a[2][1] = 1000;
        a[2][2] = 200;
    }

    // Function to calculate maximum sum of path
    function maximum_sum_path(i, j)
    {

        // Checking boundary condition
        if (i == n - 1 && j == m - 1)
            return a[i][j];

        // Checking whether or not
        // (i, j) is visited
        if (visited[i][j] != 0)
            return dp[i][j];

        // Marking (i, j) is visited
        visited[i][j] = 1;

        let total_sum = 0;

        // Checking whether the position hasn't
        // visited the last row or the last column.
        // Making recursive call for all the possible
        // moves from the current cell and then adding the
        // maximum returned by the calls and updating it.
        if (i < n - 1 & j < m - 1)
        {
            let current_sum = Math.max(
                              maximum_sum_path(i, j + 1),
                              Math.max(
                              maximum_sum_path(i + 1, j + 1),
                              maximum_sum_path(i + 1, j)));
            total_sum = a[i][j] + current_sum;
        }

        // Checking whether position
        // has reached last row
        else if (i == n - 1)
            total_sum = a[i][j] + maximum_sum_path(i, j + 1);

        // If the position is in the last column
        else
            total_sum = a[i][j] + maximum_sum_path(i + 1, j);

        // Updating the maximum sum till
        // the current position in the dp
        dp[i][j] = total_sum;

        // Returning the updated maximum value
        return total_sum;
    }

    inputMatrix();

    // Calling the implemented function
    let maximum_sum = maximum_sum_path(0, 0);

    document.write(maximum_sum);

// This code is contributed by suresh07.
</script>

Python3

N=100

# No of rows and columns
n, m=-1,-1

# Declaring the matrix of maximum
# 100 rows and 100 columns
a=[[-1]*N for _ in range(N)]

# Variable visited is used to keep
# track of all the visited positions
# Variable dp is used to store
# maximum sum till current position
dp,visited=[[-1]*N for _ in range(N)],[[False]*N for _ in range(N)]

# For storing current sum
current_sum = 0

# For continuous update of
# maximum sum required
total_sum = 0

# Function to Input the matrix of size n*m
def inputMatrix():
    global n, m
    n = 3
    m = 3
    a[0][0] = 500
    a[0][1] = 100
    a[0][2] = 230
    a[1][0] = 1000
    a[1][1] = 300
    a[1][2] = 100
    a[2][0] = 200
    a[2][1] = 1000
    a[2][2] = 200


# Function to calculate maximum sum of path
def maximum_sum_path(i, j):
    global total_sum
    # Checking boundary condition
    if (i == n - 1 and j == m - 1):
        return a[i][j]

    # Checking whether or not (i, j) is visited
    if (visited[i][j]):
        return dp[i][j]

    # Marking (i, j) is visited
    visited[i][j] = True

    # Updating the maximum sum till
    # the current position in the dp
    total_sum = dp[i][j]

    # Checking whether the position hasn't
    # visited the last row or the last column.
    # Making recursive call for all the possible
    # moves from the current cell and then adding the
    # maximum returned by the calls and updating it.
    if (i < n - 1 and j < m - 1) :
        current_sum = max(maximum_sum_path(i, j + 1),
                              max(
                                  maximum_sum_path(i + 1, j + 1),
                                  maximum_sum_path(i + 1, j)))
        total_sum = a[i][j] + current_sum


    # Checking whether
    # position has reached last row
    elif (i == n - 1):
        total_sum = a[i][j] + maximum_sum_path(i, j + 1)

    # If the position is in the last column
    else:
        total_sum = a[i][j] + maximum_sum_path(i + 1, j)

    # Returning the updated maximum value
    dp[i][j] = total_sum
    return total_sum


# Driver Code
if __name__ == '__main__':
    inputMatrix()

    # Calling the implemented function
    maximum_sum = maximum_sum_path(0, 0)

    print(maximum_sum)

Output

3000

Time Complexity: O(N*M)
Auxiliary Space: O(N*M) + O(N+M) -> O(NxM) is for the declared matrix of size nxm and O(N+M) is for the stack space (maximum recursive recursion calls).

3. Tabulation approach: This approach will remove the extra space complexity i.e O(N+M) caused by recursion in the above approach. 

1. Declare an array of size NxM – dp[N][M].
2. Traverse through the created array row-wise and start filling the values in it.
3. First row of the `dp` array i.e. index (0,j) ,where j represents column, will contain the values same as the first row of the given `matrix` array.
4. Otherwise, we will calculate the values for up (i-1,j), right (i,j-1) and up_left_diagonal (i-1,j-1) for the current cell and the maximum value of them will be the value for the current cell ie dp[i][j].
5. Finally, the maximum path sum of the matrix will be at dp[n-1][m-1] where n=no. of rows and m=no. of columns.

Java

// Java program for
// Recursive implementation of
// Max sum path
import java.io.*;

class GFG {

    /* Driver program to test above functions */
    public static void main(String[] args)
    {
        int matrix [][] = {{100, -350, -200}, {-100, -300, 700}};
        int n = matrix.length;
        int m = matrix[0].length;
        int dp[][] = new int [n][m], up, right, up_left_diagonal;
        for (int i=0; i<n; i++){
            for (int j=0; j<m; j++){
                 if (i==0) {
                    dp[i][j] = matrix[i][j];
                 }
                 else {

                    up = matrix[i][j];
                    if(i>0) up += dp[i-1][j];
                    else up -= (int)Math.pow(10,9);

                    right = matrix[i][j];
                    if(j>0) right += dp[i][j-1];
                    else right -= (int)Math.pow(10,9);

                    up_left_diagonal = matrix[i][j];
                    if(i>0 && j>0) up_left_diagonal += dp[i-1][j-1];
                    else up_left_diagonal -= (int)Math.pow(10,9);

                    dp[i][j] = Math.max(up , Math.max( right, up_left_diagonal));
                 }
            }
        }
        System.out.println(dp[n-1][m-1]);
    }
}

// This code is contributed by Rohini Chandra

C#

// C# program for
// Recursive implementation of
// Max sum path
using System;

class GFG
{
    // Function for finding maximum sum
    public static int maxPathSum(int[,] matrix, int i, int j)
    {

        if (i<0 || j<0) {
          return -1000000000;
        }
        if (i==0 && j==0) {
            return matrix[i,j];
        }

          int up = matrix[i,j] + maxPathSum(matrix, i - 1, j);
        int right = matrix[i,j] + maxPathSum(matrix, i, j - 1);
        int up_left_diagonal = matrix[i,j] + maxPathSum(matrix, i - 1, j - 1);
        return  Math.Max(up , Math.Max( right, up_left_diagonal));
    }

    /* Driver program to test above functions */
    public static void Main(string[] args)
    {
        int[,] matrix = {{100, -350, -200}, {-100, -300, 700}};
        Console.Write(maxPathSum(matrix, 1, 2));
    }
}

Output

500

Time complexity: O(NxM)

Auxiliary Space: O(NxM)