We present an optimal data structure for submatrix maximum queries in n× 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(log log n) time, as well as a matching lower bound, showing that O(log log n) query-time is optimal for any data structure of size O(npolylog(n)). Our result settles the problem, improving on the O(log2 n) query time in SODA'12, and on the O(log n) query-time in ICALP'14. In addition, 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-Ackermann factors.
Bibliographical noteFunding Information:
This article is based on two extended abstracts that appeared in ICALP 2014  and ICALP 2015 . S. Mozes and O. Weimann were supported in part by Israel Science Foundation grants 794/13 and 592/17. Authors’ addresses: P. Gawrychowski, Institute of Computer Science, University of Wrocław, Poland; email: firstname.lastname@example.org; S. Mozes, School of Computer Science, Interdisciplinary Center Herzliya, Israel; email: email@example.com; O. Weimann, Department of Computer Science, University of Haifa, Israel; email: firstname.lastname@example.org. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from email@example.com. © 2020 Copyright held by the owner/author(s). Publication rights licensed to ACM. 1549-6325/2020/03-ART16 $15.00 https://doi.org/10.1145/3381416
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- Monge matrix
- predecessor search
- range queries
ASJC Scopus subject areas
- Mathematics (miscellaneous)