Largest Submatrix With Rearrangements

Approach

For Largest Submatrix With Rearrangements, begin by restating the condition that must remain true throughout the algorithm. Use histogram heights and per-row sorting; do not jump straight to code. Derive why the local state is sufficient, then walk through a small example and identify exactly when the answer is updated. Make the implementation’s ordering explicit, because most bugs here come from updating state before querying it, using an inclusive bound where a strict one is needed, or losing original indices.

Correctness and edge cases

Explain the invariant in one sentence and show why initialization, each transition, and termination preserve it. Test the empty or smallest allowed input, one-element structures, duplicates or ties, extreme values, and cases where the natural answer is absent. Use a wide numeric type when accumulated values can overflow. The intended approach runs in the standard complexity for histogram heights and per-row sorting: state the dominant scan, sort, traversal, or lookup honestly. For this formulation, target O(n log n) when sorting or balanced lookups are needed, otherwise O(n) time; auxiliary space is O(n) unless the input structure itself supplies the storage.