Submatrix maximum queries in monge matrices are equivalent to predecessor search

Paweł Gawrychowski; Shay Mozes; Oren Weimann

doi:10.1007/978-3-662-47672-7_47

Submatrix maximum queries in monge matrices are equivalent to predecessor search

Paweł Gawrychowski, Shay Mozes, Oren Weimann

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › 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(n polylog(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-Ackerman factors.

Original language	English
Title of host publication	Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings
Editors	Magnus M. Halldorsson, Naoki Kobayashi, Bettina Speckmann, Kazuo Iwama
Publisher	Springer Verlag
Pages	580-592
Number of pages	13
ISBN (Print)	9783662476710
DOIs	https://doi.org/10.1007/978-3-662-47672-7_47
State	Published - 2015
Event	42nd International Colloquium on Automata, Languages and Programming, ICALP 2015 - Kyoto, Japan Duration: 6 Jul 2015 → 10 Jul 2015

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	9134
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	42nd International Colloquium on Automata, Languages and Programming, ICALP 2015
Country/Territory	Japan
City	Kyoto
Period	6/07/15 → 10/07/15

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2015.

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-662-47672-7_47

Cite this

Gawrychowski, P., Mozes, S., & Weimann, O. (2015). Submatrix maximum queries in monge matrices are equivalent to predecessor search. In M. M. Halldorsson, N. Kobayashi, B. Speckmann, & K. Iwama (Eds.), Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings (pp. 580-592). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9134). Springer Verlag. https://doi.org/10.1007/978-3-662-47672-7_47

Gawrychowski, Paweł ; Mozes, Shay ; Weimann, Oren. / Submatrix maximum queries in monge matrices are equivalent to predecessor search. Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings. editor / Magnus M. Halldorsson ; Naoki Kobayashi ; Bettina Speckmann ; Kazuo Iwama. Springer Verlag, 2015. pp. 580-592 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Submatrix maximum queries in 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(n polylog(n)). Our result settles the problem, improving on the O(log2 n) query-time in SODA{\textquoteright}12, and on the O(log n) query-time in ICALP{\textquoteright}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-Ackerman factors.",

author = "Pawe{\l} Gawrychowski and Shay Mozes and Oren Weimann",

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doi = "10.1007/978-3-662-47672-7_47",

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Gawrychowski, P, Mozes, S & Weimann, O 2015, Submatrix maximum queries in monge matrices are equivalent to predecessor search. in MM Halldorsson, N Kobayashi, B Speckmann & K Iwama (eds), Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9134, Springer Verlag, pp. 580-592, 42nd International Colloquium on Automata, Languages and Programming, ICALP 2015, Kyoto, Japan, 6/07/15. https://doi.org/10.1007/978-3-662-47672-7_47

Submatrix maximum queries in monge matrices are equivalent to predecessor search. / Gawrychowski, Paweł; Mozes, Shay; Weimann, Oren.
Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings. ed. / Magnus M. Halldorsson; Naoki Kobayashi; Bettina Speckmann; Kazuo Iwama. Springer Verlag, 2015. p. 580-592 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9134).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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AU - Gawrychowski, Paweł

AU - Mozes, Shay

AU - Weimann, Oren

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PY - 2015

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N2 - 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(n polylog(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-Ackerman factors.

AB - 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(n polylog(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-Ackerman factors.

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ER -

Gawrychowski P, Mozes S, Weimann O. Submatrix maximum queries in monge matrices are equivalent to predecessor search. In Halldorsson MM, Kobayashi N, Speckmann B, Iwama K, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Proceedings. Springer Verlag. 2015. p. 580-592. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-662-47672-7_47

Submatrix maximum queries in monge matrices are equivalent to predecessor search

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