Submatrix Maximum Queries in Monge Matrices are Equivalent to Predecessor Search

TL;DR

This paper introduces an optimal data structure for submatrix maximum queries in Monge matrices, establishing an equivalence with the predecessor problem and proving the optimality of the query time.

Contribution

It provides a two-way reduction showing the problem's equivalence to the predecessor search, achieving optimal query time and space bounds, and extends results to partial Monge matrices.

Findings

Achieves O(n) space and O(loglogn) query time for submatrix maximum queries.

Establishes a lower bound matching the upper bound, proving optimality.

Extends results to partial Monge matrices without additional complexity.

Abstract

We present an optimal data structure for submatrix maximum queries in n x n Monge matrices. Our result is a two-way reduction showing that the problem is equivalent to the classical predecessor problem in a universe of polynomial size. This gives a data structure of O(n) space that answers submatrix maximum queries in O(loglogn) time. It also gives a matching lower bound, showing that O(loglogn) query-time is optimal for any data structure of size O(n polylog(n)). Our result concludes a line of improvements that started in SODA'12 with O(log^2 n) query-time and continued in ICALP'14 with O(log n) query-time. Finally, we show that partial Monge matrices can be handled in the same bounds as full Monge matrices. In both previous results, partial Monge matrices incurred additional inverse-Ackerman factors.