Largest rectangular sub-matrix whose sum is 0
Given a 2D matrix, find the largest rectangular sub-matrix whose sum is 0. for example consider the following N x M input matrix
Input : 1, 2, 3 -3, -2, -1 1, 7, 5 Output : 1, 2, 3 -3, -2, -1 Input : 9, 7, 16, 5 1, -6, -7, 3 1, 8, 7, 9 7, -2, 0, 10 Output :-6, -7 8, 7 -2, 0
The naive solution for this problem is to check every possible rectangle in given 2D array. This solution requires 4 nested loops and time complexity of this solution would be O(n^4).
The solution is based on Maximum sum rectangle in a 2D matrix. The idea is to reduce the problem to 1 D array. We can use Hashing to find maximum length of sub-array in 1-D array in O(n) time. We fix the left and right columns one by one and find the largest sub-array with 0 sum contiguous rows for every left and right column pair. We basically find top and bottom row numbers (which have sum is zero) for every fixed left and right column pair.
To find the top and bottom row numbers, calculate sum of elements in every row from left to right and store these sums in an array say temp. So temp[i] indicates sum of elements from left to right in row i. If we find largest subarray with 0 sum on temp, and no. of elements is greater than previous no. of elements then update the values of final row_up, final row_down, final col_left, final col_right.
-6 -7 8 7 -2 0
Time Complexity: O(n3)
Auxiliary Space: O(1)
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