Submatrix Maximum Queries in Monge and Partial Monge Matrices Are Equivalent to Predecessor Search

Paweł Gawrychowski; Shay Mozes; Oren Weimann

doi:10.1145/3381416

Submatrix Maximum Queries in Monge and Partial Monge Matrices Are Equivalent to Predecessor Search

Paweł Gawrychowski, Shay Mozes, Oren Weimann

Department of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Article number	16
Journal	ACM Transactions on Algorithms
Volume	16
Issue number	2
DOIs	https://doi.org/10.1145/3381416
State	Published - Apr 2020

Bibliographical note

Funding Information:
This article is based on two extended abstracts that appeared in ICALP 2014 [17] and ICALP 2015 [18]. 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: gawry@cs.uni.wroc.pl; S. Mozes, School of Computer Science, Interdisciplinary Center Herzliya, Israel; email: smozes@idc.ac.il; O. Weimann, Department of Computer Science, University of Haifa, Israel; email: oren@cs.haifa.ac.il. 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 permissions@acm.org. © 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

Publisher Copyright:
© 2020 ACM.

Keywords

Monge matrix
predecessor search
range queries

ASJC Scopus subject areas

Mathematics (miscellaneous)

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10.1145/3381416

Cite this

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title = "Submatrix Maximum Queries in Monge and Partial Monge Matrices Are Equivalent to Predecessor Search",

abstract = "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.",

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note = "Funding Information: This article is based on two extended abstracts that appeared in ICALP 2014 [17] and ICALP 2015 [18]. S. Mozes and O. Weimann were supported in part by Israel Science Foundation grants 794/13 and 592/17. Authors{\textquoteright} addresses: P. Gawrychowski, Institute of Computer Science, University of Wroc{\l}aw, Poland; email: gawry@cs.uni.wroc.pl; S. Mozes, School of Computer Science, Interdisciplinary Center Herzliya, Israel; email: smozes@idc.ac.il; O. Weimann, Department of Computer Science, University of Haifa, Israel; email: oren@cs.haifa.ac.il. 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 permissions@acm.org. {\textcopyright} 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 Publisher Copyright: {\textcopyright} 2020 ACM.",

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Submatrix Maximum Queries in Monge and Partial Monge Matrices Are Equivalent to Predecessor Search

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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