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# Maximum sum submatrix

• Difficulty Level : Hard
• Last Updated : 30 Jun, 2021

Given a 2D array arr[][] of dimension N*M, the task is to find the maximum sum sub-matrix from the matrix arr[][].

Examples:

Input: arr[][] = {{0, -2, -7, 0 },  { 9, 2, -6, 2 }, { -4, 1, -4, 1 }, { -1, 8, 0, -2}}
Output: 15
Explanation: The submatrix {{9, 2}, {-4, 1}, {-1, 8}} has a sum 15, which is the maximum sum possible.

Input: arr[][] = {{1, 2}, {-5, -7}}
Output: 3

Naive Approach: The simplest approach is to generate all possible submatrices from the given matrix and calculate their sum. Finally, print the maximum sum obtained.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `// Function to find maximum sum submatrix``void` `maxSubmatrixSum(``    ``vector > matrix)``{``    ``// Stores the number of rows``    ``// and columns in the matrix``    ``int` `r = matrix.size();``    ``int` `c = matrix[0].size();` `    ``// Stores maximum submatrix sum``    ``int` `maxSubmatrix = 0;` `    ``// Take each row as starting row``    ``for` `(``int` `i = 0; i < r; i++) {` `        ``// Take each column as the``        ``// starting column``        ``for` `(``int` `j = 0; j < c; j++) {` `            ``// Take each row as the``            ``// ending row``            ``for` `(``int` `k = i; k < r; k++) {` `                ``// Take each column as``                ``// the ending column``                ``for` `(``int` `l = j; l < c; l++) {` `                    ``// Stores the sum of submatrix``                    ``// having topleft index(i, j)``                    ``// and bottom right index (k, l)``                    ``int` `sumSubmatrix = 0;` `                    ``// Iterate the submatrix``                    ``// row-wise and calculate its sum``                    ``for` `(``int` `m = i; m <= k; m++) {``                        ``for` `(``int` `n = j; n <= l; n++) {``                            ``sumSubmatrix += matrix[m][n];``                        ``}``                    ``}` `                    ``// Update the maximum sum``                    ``maxSubmatrix``                        ``= max(maxSubmatrix,``                              ``sumSubmatrix);``                ``}``            ``}``        ``}``    ``}` `    ``// Print the answer``    ``cout << maxSubmatrix;``}` `// Driver Code``int` `main()``{``    ``vector > matrix = { { 0, -2, -7, 0 },``                                    ``{ 9, 2, -6, 2 },``                                    ``{ -4, 1, -4, 1 },``                                    ``{ -1, 8, 0, -2 } };` `    ``maxSubmatrixSum(matrix);` `    ``return` `0;``}`

## Java

 `// Java program for the above approach``import` `java.util.*;``class` `GFG``{` `// Function to find maximum sum submatrix``static` `void` `maxSubmatrixSum(``int``[][] matrix)``{``  ` `    ``// Stores the number of rows``    ``// and columns in the matrix``    ``int` `r = matrix.length;``    ``int` `c = matrix[``0``].length;` `    ``// Stores maximum submatrix sum``    ``int` `maxSubmatrix = ``0``;` `    ``// Take each row as starting row``    ``for` `(``int` `i = ``0``; i < r; i++) {` `        ``// Take each column as the``        ``// starting column``        ``for` `(``int` `j = ``0``; j < c; j++) {` `            ``// Take each row as the``            ``// ending row``            ``for` `(``int` `k = i; k < r; k++) {` `                ``// Take each column as``                ``// the ending column``                ``for` `(``int` `l = j; l < c; l++) {` `                    ``// Stores the sum of submatrix``                    ``// having topleft index(i, j)``                    ``// and bottom right index (k, l)``                    ``int` `sumSubmatrix = ``0``;` `                    ``// Iterate the submatrix``                    ``// row-wise and calculate its sum``                    ``for` `(``int` `m = i; m <= k; m++) {``                        ``for` `(``int` `n = j; n <= l; n++) {``                            ``sumSubmatrix += matrix[m][n];``                        ``}``                    ``}` `                    ``// Update the maximum sum``                    ``maxSubmatrix``                        ``= Math.max(maxSubmatrix,``                              ``sumSubmatrix);``                ``}``            ``}``        ``}``    ``}` `    ``// Print the answer``    ``System.out.println(maxSubmatrix);``}``  ` `// Driver Code``public` `static` `void` `main(String[] args)``{``    ``int``[][] matrix = { { ``0``, -``2``, -``7``, ``0` `},``                        ``{ ``9``, ``2``, -``6``, ``2` `},``                        ``{ -``4``, ``1``, -``4``, ``1` `},``                        ``{ -``1``, ``8``, ``0``, -``2` `} };` `    ``maxSubmatrixSum(matrix);``}``}` `// This code is contributed by susmitakundugoaldanga.`

## Python3

 `# Python3 program for the above approach` `# Function to find maximum sum submatrix``def` `maxSubmatrixSum(matrix):``  ` `    ``# Stores the number of rows``    ``# and columns in the matrix``    ``r ``=` `len``(matrix)``    ``c ``=` `len``(matrix[``0``])` `    ``# Stores maximum submatrix sum``    ``maxSubmatrix ``=` `0` `    ``# Take each row as starting row``    ``for` `i ``in` `range``(r):` `        ``# Take each column as the``        ``# starting column``        ``for` `j ``in` `range``(c):` `            ``# Take each row as the``            ``# ending row``            ``for` `k ``in` `range``(i, r):` `                ``# Take each column as``                ``# the ending column``                ``for` `l ``in` `range``(j, c):` `                    ``# Stores the sum of submatrix``                    ``# having topleft index(i, j)``                    ``# and bottom right index (k, l)``                    ``sumSubmatrix ``=` `0` `                    ``# Iterate the submatrix``                    ``# row-wise and calculate its sum``                    ``for` `m ``in` `range``(i, k ``+` `1``):``                        ``for` `n ``in` `range``(j, l ``+` `1``):``                            ``sumSubmatrix ``+``=` `matrix[m][n]` `                    ``# Update the maximum sum``                    ``maxSubmatrix``=` `max``(maxSubmatrix, sumSubmatrix)` `    ``# Prthe answer``    ``print` `(maxSubmatrix)` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``    ``matrix ``=` `[ [ ``0``, ``-``2``, ``-``7``, ``0` `],``                ``[ ``9``, ``2``, ``-``6``, ``2` `],``                ``[ ``-``4``, ``1``, ``-``4``, ``1` `],``                ``[ ``-``1``, ``8``, ``0``, ``-``2` `] ]` `    ``maxSubmatrixSum(matrix)` `    ``# This code is contributed by mohit kumar 29.`

## C#

 `// C# program to implement``// the above approach``using` `System;``public` `class` `GFG``{``// Function to find maximum sum submatrix``static` `void` `maxSubmatrixSum(``int``[,] matrix)``{``  ` `    ``// Stores the number of rows``    ``// and columns in the matrix``    ``int` `r = matrix.GetLength(0);``    ``int` `c = matrix.GetLength(1);` `    ``// Stores maximum submatrix sum``    ``int` `maxSubmatrix = 0;` `    ``// Take each row as starting row``    ``for` `(``int` `i = 0; i < r; i++) {` `        ``// Take each column as the``        ``// starting column``        ``for` `(``int` `j = 0; j < c; j++) {` `            ``// Take each row as the``            ``// ending row``            ``for` `(``int` `k = i; k < r; k++) {` `                ``// Take each column as``                ``// the ending column``                ``for` `(``int` `l = j; l < c; l++) {` `                    ``// Stores the sum of submatrix``                    ``// having topleft index(i, j)``                    ``// and bottom right index (k, l)``                    ``int` `sumSubmatrix = 0;` `                    ``// Iterate the submatrix``                    ``// row-wise and calculate its sum``                    ``for` `(``int` `m = i; m <= k; m++) {``                        ``for` `(``int` `n = j; n <= l; n++) {``                            ``sumSubmatrix += matrix[m, n];``                        ``}``                    ``}` `                    ``// Update the maximum sum``                    ``maxSubmatrix``                        ``= Math.Max(maxSubmatrix,``                              ``sumSubmatrix);``                ``}``            ``}``        ``}``    ``}` `    ``// Print the answer``    ``Console.WriteLine(maxSubmatrix);``}` `// Driver Code``public` `static` `void` `Main(String []args)``{``    ``int``[,] matrix = { { 0, -2, -7, 0 },``                        ``{ 9, 2, -6, 2 },``                        ``{ -4, 1, -4, 1 },``                        ``{ -1, 8, 0, -2 } };` `    ``maxSubmatrixSum(matrix);``}``}` `// This code is contributed by sanjoy_62.`

## Javascript

 ``
Output:
`15`

Time Complexity: O(N6)
Auxiliary Space: O(1)

Efficient Approach using Kadane’s Algorithm: The above approach can be optimized using the following observations:

• Fix starting and ending column of the required sub-matrix say start and end respectively.
• Now, iterate each row and add row sum from starting to ending column to sumSubmatrix and insert this in an array. After iterating each row, perform Kadane’s Algorithm on this newly created array. If the sum obtained by applying Kadane’s algorithm is greater than the overall maximum sum, update the overall maximum sum.
• In the above step, the row sum from starting to ending column can be calculated in constant time by creating an auxiliary matrix of size N*M containing the prefix sum of each row.

Follow the steps below to solve the problem:

• Initialize a variable, say maxSum as INT_MIN, to store the maximum subarray sum.
• Create a matrix prefMatrix[N][M] that stores the prefix array sum of every row of the given matrix.
• Traverse the matrix row-wise using i as the row index and j as the column index and perform the following steps:
• If the value of i is 0, then set prefMatrix[i][j] = A[i][j].
• Otherwise, set prefMatrix[i][j] = prefMatrix[i][j – 1] + A[i][j].
• Now for all possible combinations of starting and ending index of the columns of submatrix over the range [0, M] perform the following steps:
• Initialize an auxiliary array A[] to stores the maximum sum for each row of the current submatrix.
• Find the sum from starting to ending column using prefMatrix as follows:
• If the value of start is positive, then store the required sum S as prefMatrix[i][end] – prefMatrix[i][start – 1].
• Otherwise, update S as prefMatrix[i][end].
• Insert S in an array arr[].
• After iterating all rows in the submatrix, perform Kadane’s algorithm on the array A[] and update the maximum sum maxSum as the maximum of maxSum and value obtained by performing the Kadane’s Algorithm in this step.
• After completing the above steps, print the value of maxSum as the result.

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach``#include ``using` `namespace` `std;` `// Function to find maximum continuous``// maximum sum in the array``int` `kadane(vector<``int``> v)``{``  ` `    ``// Stores current and maximum sum``    ``int` `currSum = 0;``    ``int` `maxSum = INT_MIN;` `    ``// Traverse the array v``    ``for` `(``int` `i = 0;``         ``i < (``int``)v.size(); i++) {` `        ``// Add the value of the``        ``// current element``        ``currSum += v[i];` `        ``// Update the maximum sum``        ``if` `(currSum > maxSum) {``            ``maxSum = currSum;``        ``}` `        ``if` `(currSum < 0) {``            ``currSum = 0;``        ``}``    ``}` `    ``// Return the maximum sum``    ``return` `maxSum;``}` `// Function to find the maximum``// submatrix sum``void` `maxSubmatrixSum(``    ``vector > A)``{``  ` `    ``// Store the rows and columns``    ``// of the matrix``    ``int` `r = A.size();``    ``int` `c = A[0].size();` `    ``// Create an auxiliary matrix``    ``int``** prefix = ``new` `int``*[r];` `    ``// Traverse the matrix, prefix``    ``// and initialize it will all 0s``    ``for` `(``int` `i = 0; i < r; i++) {` `        ``prefix[i] = ``new` `int``;``        ``for` `(``int` `j = 0; j < c; j++) {``            ``prefix[i][j] = 0;``        ``}``    ``}` `    ``// Calculate prefix sum of all``    ``// rows of matrix A[][] and``    ``// store in matrix prefix[]``    ``for` `(``int` `i = 0; i < r; i++) {` `        ``for` `(``int` `j = 0; j < c; j++) {` `            ``// Update the prefix[][]``            ``if` `(j == 0)``                ``prefix[i][j] = A[i][j];``            ``else``                ``prefix[i][j] = A[i][j]``                               ``+ prefix[i][j - 1];``        ``}``    ``}` `    ``// Store the maximum submatrix sum``    ``int` `maxSum = INT_MIN;` `    ``// Iterate for starting column``    ``for` `(``int` `i = 0; i < c; i++) {` `        ``// Iterate for last column``        ``for` `(``int` `j = i; j < c; j++) {` `            ``// To store current array``            ``// elements``            ``vector<``int``> v;` `            ``// Traverse every row``            ``for` `(``int` `k = 0; k < r; k++) {` `                ``// Store the sum of the``                ``// kth row``                ``int` `el = 0;` `                ``// Update the prefix``                ``// sum``                ``if` `(i == 0)``                    ``el = prefix[k][j];``                ``else``                    ``el = prefix[k][j]``                         ``- prefix[k][i - 1];` `                ``// Push it in a vector``                ``v.push_back(el);``            ``}` `            ``// Update the maximum``            ``// overall sum``            ``maxSum = max(maxSum, kadane(v));``        ``}``    ``}` `    ``// Print the answer``    ``cout << maxSum << ``"\n"``;``}` `// Driver Code``int` `main()``{``    ``vector > matrix = { { 0, -2, -7, 0 },``                                    ``{ 9, 2, -6, 2 },``                                    ``{ -4, 1, -4, 1 },``                                    ``{ -1, 8, 0, -2 } };` `    ``// Function Call``    ``maxSubmatrixSum(matrix);` `    ``return` `0;``}`

## Java

 `// Java program for the above approach``import` `java.util.*;``class` `GFG{` `  ``// Function to find maximum continuous``  ``// maximum sum in the array``  ``static` `int` `kadane(Vector v)``  ``{` `    ``// Stores current and maximum sum``    ``int` `currSum = ``0``;``    ``int` `maxSum = Integer.MIN_VALUE;` `    ``// Traverse the array v``    ``for` `(``int` `i = ``0``;``         ``i < (``int``)v.size(); i++)``    ``{` `      ``// Add the value of the``      ``// current element``      ``currSum += v.get(i);` `      ``// Update the maximum sum``      ``if` `(currSum > maxSum)``      ``{``        ``maxSum = currSum;``      ``}` `      ``if` `(currSum < ``0``)``      ``{``        ``currSum = ``0``;``      ``}``    ``}` `    ``// Return the maximum sum``    ``return` `maxSum;``  ``}` `  ``// Function to find the maximum``  ``// submatrix sum``  ``static` `void` `maxSubmatrixSum(``int` `[][]A)``  ``{``    ``// Store the rows and columns``    ``// of the matrix``    ``int` `r = A.length;``    ``int` `c = A[``0``].length;` `    ``// Create an auxiliary matrix``    ``int` `[][]prefix = ``new` `int``[r][];` `    ``// Traverse the matrix, prefix``    ``// and initialize it will all 0s``    ``for` `(``int` `i = ``0``; i < r; i++) {` `      ``prefix[i] = ``new` `int``;``      ``for` `(``int` `j = ``0``; j < c; j++) {``        ``prefix[i][j] = ``0``;``      ``}``    ``}` `    ``// Calculate prefix sum of all``    ``// rows of matrix A[][] and``    ``// store in matrix prefix[]``    ``for` `(``int` `i = ``0``; i < r; i++) {` `      ``for` `(``int` `j = ``0``; j < c; j++) {` `        ``// Update the prefix[][]``        ``if` `(j == ``0``)``          ``prefix[i][j] = A[i][j];``        ``else``          ``prefix[i][j] = A[i][j]``          ``+ prefix[i][j - ``1``];``      ``}``    ``}` `    ``// Store the maximum submatrix sum``    ``int` `maxSum = Integer.MIN_VALUE;` `    ``// Iterate for starting column``    ``for` `(``int` `i = ``0``; i < c; i++) {` `      ``// Iterate for last column``      ``for` `(``int` `j = i; j < c; j++) {` `        ``// To store current array``        ``// elements``        ``Vector v = ``new` `Vector();` `        ``// Traverse every row``        ``for` `(``int` `k = ``0``; k < r; k++) {` `          ``// Store the sum of the``          ``// kth row``          ``int` `el = ``0``;` `          ``// Update the prefix``          ``// sum``          ``if` `(i == ``0``)``            ``el = prefix[k][j];``          ``else``            ``el = prefix[k][j]``            ``- prefix[k][i - ``1``];` `          ``// Push it in a vector``          ``v.add(el);``        ``}` `        ``// Update the maximum``        ``// overall sum``        ``maxSum = Math.max(maxSum, kadane(v));``      ``}``    ``}` `    ``// Print the answer``    ``System.out.print(maxSum+ ``"\n"``);``  ``}` `  ``// Driver Code``  ``public` `static` `void` `main(String[] args)``  ``{``    ``int` `[][]matrix = { { ``0``, -``2``, -``7``, ``0` `},``                      ``{ ``9``, ``2``, -``6``, ``2` `},``                      ``{ -``4``, ``1``, -``4``, ``1` `},``                      ``{ -``1``, ``8``, ``0``, -``2` `} };` `    ``// Function Call``    ``maxSubmatrixSum(matrix);``  ``}``}` `// This code is contributed by 29AjayKumar`

## Python3

 `# Python3 program for the above approach``import` `sys` `# Function to find maximum continuous``# maximum sum in the array``def` `kadane(v):``    ` `    ``# Stores current and maximum sum``    ``currSum ``=` `0``    ` `    ``maxSum ``=` `-``sys.maxsize ``-` `1``    ` `    ``# Traverse the array v``    ``for` `i ``in` `range``(``len``(v)):``        ` `        ``# Add the value of the``        ``# current element``        ``currSum ``+``=` `v[i]``        ` `        ``# Update the maximum sum``        ``if` `(currSum > maxSum):``            ``maxSum ``=` `currSum``        ``if` `(currSum < ``0``):``            ``currSum ``=` `0``    ` `    ``# Return the maximum sum``    ``return` `maxSum` `# Function to find the maximum``# submatrix sum``def` `maxSubmatrixSum(A):``    ` `    ``# Store the rows and columns``    ``# of the matrix``    ``r ``=` `len``(A)``    ``c ``=` `len``(A[``0``])``    ` `    ``# Create an auxiliary matrix``    ``# Traverse the matrix, prefix``    ``# and initialize it will all 0s``    ``prefix ``=` `[[``0` `for` `i ``in` `range``(c)]``                 ``for` `j ``in` `range``(r)]``    ` `    ``# Calculate prefix sum of all``    ``# rows of matrix A[][] and``    ``# store in matrix prefix[]``    ``for` `i ``in` `range``(r):``        ``for` `j ``in` `range``(c):``            ` `            ``# Update the prefix[][]``            ``if` `(j ``=``=` `0``):``                ``prefix[i][j] ``=` `A[i][j]``            ``else``:``                ``prefix[i][j] ``=` `A[i][j] ``+` `prefix[i][j ``-` `1``]``    ` `    ``# Store the maximum submatrix sum``    ``maxSum ``=` `-``sys.maxsize ``-` `1``    ` `    ``#  Iterate for starting column``    ``for` `i ``in` `range``(c):``        ` `        ``# Iterate for last column``        ``for` `j ``in` `range``(i, c):``            ` `            ``# To store current array``            ``# elements``            ``v ``=` `[]``            ` `            ``# Traverse every row``            ``for` `k ``in` `range``(r):``                ` `                ``# Store the sum of the``                ``# kth row``                ``el ``=` `0``                ` `                ``# Update the prefix``                ``# sum``                ``if` `(i ``=``=` `0``):``                    ``el ``=` `prefix[k][j]``                ``else``:``                    ``el ``=` `prefix[k][j] ``-` `prefix[k][i ``-` `1``]``                ` `                ``# Push it in a vector``                ``v.append(el)``            ` `            ``# Update the maximum``            ``# overall sum``            ``maxSum ``=` `max``(maxSum, kadane(v))``    ` `    ``# Print the answer``    ``print``(maxSum)` `# Driver Code``matrix ``=` `[ [ ``0``, ``-``2``, ``-``7``, ``0` `],``           ``[ ``9``, ``2``, ``-``6``, ``2` `],``           ``[ ``-``4``, ``1``, ``-``4``, ``1` `],``           ``[ ``-``1``, ``8``, ``0``, ``-``2` `] ]``           ` `# Function Call``maxSubmatrixSum(matrix)` `# This code is contributed by rag2127`

## C#

 `// C# program for the above approach``using` `System;``using` `System.Collections.Generic;` `public` `class` `GFG{` `  ``// Function to find maximum continuous``  ``// maximum sum in the array``  ``static` `int` `kadane(List<``int``> v)``  ``{` `    ``// Stores current and maximum sum``    ``int` `currSum = 0;``    ``int` `maxSum = ``int``.MinValue;` `    ``// Traverse the array v``    ``for` `(``int` `i = 0;``         ``i < (``int``)v.Count; i++)``    ``{` `      ``// Add the value of the``      ``// current element``      ``currSum += v[i];` `      ``// Update the maximum sum``      ``if` `(currSum > maxSum)``      ``{``        ``maxSum = currSum;``      ``}` `      ``if` `(currSum < 0)``      ``{``        ``currSum = 0;``      ``}``    ``}` `    ``// Return the maximum sum``    ``return` `maxSum;``  ``}` `  ``// Function to find the maximum``  ``// submatrix sum``  ``static` `void` `maxSubmatrixSum(``int` `[,]A)``  ``{``    ``// Store the rows and columns``    ``// of the matrix``    ``int` `r = A.GetLength(0);``    ``int` `c = A.GetLength(1);` `    ``// Create an auxiliary matrix``    ``int` `[,]prefix = ``new` `int``[r,c];` `    ``// Traverse the matrix, prefix``    ``// and initialize it will all 0s``    ``for` `(``int` `i = 0; i < r; i++) {``      ``for` `(``int` `j = 0; j < c; j++) {``        ``prefix[i,j] = 0;``      ``}``    ``}` `    ``// Calculate prefix sum of all``    ``// rows of matrix [,]A and``    ``// store in matrix prefix[]``    ``for` `(``int` `i = 0; i < r; i++) {` `      ``for` `(``int` `j = 0; j < c; j++) {` `        ``// Update the prefix[,]``        ``if` `(j == 0)``          ``prefix[i,j] = A[i,j];``        ``else``          ``prefix[i,j] = A[i,j]``          ``+ prefix[i,j - 1];``      ``}``    ``}` `    ``// Store the maximum submatrix sum``    ``int` `maxSum = ``int``.MinValue;` `    ``// Iterate for starting column``    ``for` `(``int` `i = 0; i < c; i++) {` `      ``// Iterate for last column``      ``for` `(``int` `j = i; j < c; j++) {` `        ``// To store current array``        ``// elements``        ``List<``int``> v = ``new` `List<``int``>();` `        ``// Traverse every row``        ``for` `(``int` `k = 0; k < r; k++) {` `          ``// Store the sum of the``          ``// kth row``          ``int` `el = 0;` `          ``// Update the prefix``          ``// sum``          ``if` `(i == 0)``            ``el = prefix[k,j];``          ``else``            ``el = prefix[k,j]``            ``- prefix[k,i - 1];` `          ``// Push it in a vector``          ``v.Add(el);``        ``}` `        ``// Update the maximum``        ``// overall sum``        ``maxSum = Math.Max(maxSum, kadane(v));``      ``}``    ``}` `    ``// Print the answer``    ``Console.Write(maxSum+ ``"\n"``);``  ``}` `  ``// Driver Code``  ``public` `static` `void` `Main(String[] args)``  ``{``    ``int` `[,]matrix = { { 0, -2, -7, 0 },``                      ``{ 9, 2, -6, 2 },``                      ``{ -4, 1, -4, 1 },``                      ``{ -1, 8, 0, -2 } };` `    ``// Function Call``    ``maxSubmatrixSum(matrix);``  ``}``}` `// This code is contributed by 29AjayKumar`

## Javascript

 ``
Output:
`15`

Time Complexity: O(N3)
Auxiliary Space: O(N2)

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