Minimum Absolute Difference in Sliding Submatrix - Solution & Explanation

Problem Overview: You are given a matrix and a fixed submatrix size k × k. For every possible sliding submatrix, compute the minimum absolute difference between any two elements inside that window. The result for each position depends only on the values contained in that specific k × k region.

Approach 1: Brute Force Pair Comparison (O((m−k+1)(n−k+1) · k⁴) time, O(1) space)

For each valid top‑left corner of a k × k window, enumerate all elements in that submatrix. Compare every pair of values and track the smallest absolute difference. A k × k window contains k² numbers, so pairwise comparison requires O(k⁴) operations per window. This approach uses constant extra space but becomes slow when k grows because the number of pair comparisons increases quadratically. It mainly serves as a conceptual baseline that proves the problem can be solved by direct enumeration.

Approach 2: Enumeration + Sorting (O((m−k+1)(n−k+1) · k² log(k²)) time, O(k²) space)

A more efficient observation: the minimum absolute difference among numbers always appears between two adjacent elements after sorting. For each sliding k × k region, first collect its k² values into a temporary list. Sort the list using a standard comparison sort. Then iterate once through the sorted values and compute abs(a[i] - a[i-1]) to find the smallest gap. Sorting costs O(k² log(k²)), and the linear scan adds O(k²). Because there are (m−k+1)(n−k+1) windows, this becomes the dominant complexity.

The algorithm relies heavily on sequential traversal of the matrix (a common pattern in matrix problems) and temporary containers for each window. Sorting simplifies the difference calculation since only adjacent values must be checked instead of every pair. Most implementations store the window values in an array or vector, apply a built‑in sort, then scan once to compute the minimum gap.

This method fits naturally with problems involving value comparisons in arrays and ordering via sorting. It trades extra memory and sorting cost for a dramatic reduction in pair comparisons.

Recommended for interviews: Start by describing the brute‑force pair comparison to demonstrate understanding of the requirement. Then move to the sorted enumeration approach. Interviewers expect the key insight that the minimum absolute difference must appear between adjacent elements after sorting. Implementing that idea clearly shows comfort with matrix traversal, temporary buffers, and algorithmic optimization.