Number Of Submatrices That Sum To Target - Leetcode Solution

The Number Of Submatrices That Sum To Target problem on LeetCode is a difficult problem that combines several different concepts. In this problem, we are given a matrix of integers and a target sum, and we have to find the number of non-empty submatrices whose sum is equal to the target.

To solve this problem, we can use a two-dimensional prefix sum array to efficiently compute the sum of any submatrix in O(1) time. The idea is to precompute the sum of all submatrices that start at (0,0) and end at each (i,j) element in the matrix. This can be done by the following formula:

prefix[i][j] = matrix[i][j] + prefix[i-1][j] + prefix[i][j-1] - prefix[i-1][j-1]

where prefix[i][j] represents the sum of all elements in the submatrix that starts at (0,0) and ends at (i,j).

Once we have precomputed the prefix sum array, we can iterate over all possible submatrices in the matrix using two nested loops. For each submatrix, we can compute its sum in O(1) time using the prefix sum array. If the sum of the submatrix is equal to the target, we can increment a counter that keeps track of the number of submatrices that satisfy the condition.

The time complexity of this solution is O(N^2*M^2), where N and M are the dimensions of the matrix. However, this solution can be optimized using a hashmap to store the count of submatrices with a particular sum. The idea is to fix the top and bottom rows of the submatrix, and then iterate over all possible combinations of left and right column indices. For each combination, we can compute the sum of the submatrix in O(1) time using the prefix sum array, and then increment the corresponding count in the hashmap. Finally, we can iterate over all possible submatrices again and check if there is a submatrix with a sum equal to the target by looking up the count in the hashmap.

The time complexity of this optimized solution is O(N^2*M), where N and M are the dimensions of the matrix. This is much faster than the naive solution and can solve the problem within the given time limit on LeetCode.

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