Two-dimensional range minimum queries

Amihood Amir, Johannes Fischer, Moshe Lewenstein

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

29 Scopus citations

Abstract

We consider the two-dimensional Range Minimum Query problem: for a static (m x n)-matrix of size N = mn which may be preprocessed, answer on-line queries of the form "where is the position of a minimum element in an axis-parallel rectangle?". Unlike the one-dimensional version of this problem which can be solved in provably optimal time and space, the higher-dimensional case has received much less attention. The only result we are aware of is due to Gabow, Bentley and Tarjan [1], who solve the problem in O(N log N) preprocessing time and space and O(log N) query time. We present a class of algorithms which can solve the 2-dimensional RMQ-problem with O(kN) additional space, O(N log [k+1] N) preprocessing time and O(1) query time for any k > 1, where log[k+1] denotes the iterated application of k + 1 logarithms. The solution converges towards an algorithm with O(N log* N) preprocessing time and space and O(1) query time. All these algorithms are significant improvements over the previous results: query time is optimal, preprocessing time is quasi-linear in the input size, and space is linear. While this paper is of theoretical nature, we believe that our algorithms will turn out to have applications in different fields of computer science, e.g., in computational biology.

Original languageEnglish
Title of host publicationCombinatorial Pattern Matching - 18th Annual Symposium, CPM 2007, Proceedings
PublisherSpringer Verlag
Pages286-294
Number of pages9
ISBN (Print)9783540734369
DOIs
StatePublished - 2007
Event18th Annual Symposium on Combinatorial Pattern Matching, CPM 2007 - London, ON, Canada
Duration: 9 Jul 200711 Jul 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4580 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference18th Annual Symposium on Combinatorial Pattern Matching, CPM 2007
Country/TerritoryCanada
CityLondon, ON
Period9/07/0711/07/07

Bibliographical note

Funding Information:
Wireless Networks; MANET; routing; Geographic Routing Protocol; Location System; mobility; scalability This work was supported in part by the ONR "MINUTEMAN project under contract N00014-01-C-0016, and in part by NSF "WHYNET" project under contract CNS-0335302.

Funding Information:
This work was supported in part by the ONR “MINUTEMAN” project under contract N00014-01-C_0016, and in part by NSF “WHYNET” project under contract CNS-0335302.

Funding

Wireless Networks; MANET; routing; Geographic Routing Protocol; Location System; mobility; scalability This work was supported in part by the ONR "MINUTEMAN project under contract N00014-01-C-0016, and in part by NSF "WHYNET" project under contract CNS-0335302. This work was supported in part by the ONR “MINUTEMAN” project under contract N00014-01-C_0016, and in part by NSF “WHYNET” project under contract CNS-0335302.

FundersFunder number
National Science FoundationCNS-0335302
Office of Naval ResearchN00014-01-C-0016
Division of Computer and Network Systems0335302

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