Maximum sum submatrix

Prerequisite: Kadane's algorithm

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:

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:

15

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

Related Topic: Subarrays, Subsequences, and Subsets in Array