Reduce the 2D problem to 1D by fixing left and right column boundaries and compressing each row into a sum, then find the maximum subarray sum no larger than the target using a sorted list (e.g., TreeSet) for efficient prefix sum lookups. This yields O(min(m,n)^2 * max(m,n) * log(max(m,n))) time, which is optimal for this problem.
Pro tip: Mention that you can transpose the matrix to ensure the smaller dimension is squared, reducing time complexity when the matrix is highly rectangular. Also, clarify that the sorted list approach is necessary because the target constraint prevents using Kadane's algorithm directly.
Ask about matrix dimensions, value ranges, and whether the target can be negative. Discuss handling of empty matrix or no valid rectangle.
Explain that by fixing left and right columns, each row's sum between those columns can be computed, turning the problem into finding the maximum subarray sum ≤ target in a 1D array.
Use a sorted list (e.g., TreeSet) to store prefix sums. For each prefix sum, find the smallest prefix sum ≥ current - target to maximize the subarray sum without exceeding target.
If the number of rows is greater than columns, transpose the matrix to minimize the squared dimension, improving time complexity.
State time complexity O(min(m,n)^2 * max(m,n) * log(max(m,n))) and space O(max(m,n)). Walk through a small example to verify correctness.
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