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 language | English |
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Title of host publication | Combinatorial Pattern Matching - 18th Annual Symposium, CPM 2007, Proceedings |
Publisher | Springer Verlag |
Pages | 286-294 |
Number of pages | 9 |
ISBN (Print) | 9783540734369 |
DOIs | |
State | Published - 2007 |
Event | 18th Annual Symposium on Combinatorial Pattern Matching, CPM 2007 - London, ON, Canada Duration: 9 Jul 2007 → 11 Jul 2007 |
Publication series
Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4580 LNCS |
ISSN (Print) | 0302-9743 |
ISSN (Electronic) | 1611-3349 |
Conference
Conference | 18th Annual Symposium on Combinatorial Pattern Matching, CPM 2007 |
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Country/Territory | Canada |
City | London, ON |
Period | 9/07/07 → 11/07/07 |
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.
Funders | Funder number |
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National Science Foundation | CNS-0335302 |
Office of Naval Research | N00014-01-C-0016 |
Division of Computer and Network Systems | 0335302 |