Abstract
We consider the two-dimensional Range Minimum Query problem: for a static (m ×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(NlogN) preprocessing time and space and O(logN) query time. We present a class of algorithms which can solve the 2-dimensional RMQ-problem with O(kN) additional space, 
preprocessing time and O(1) query time for any k > 1, where 
denotes the iterated application of k + 1 logarithms. The solution converges towards an algorithm with 
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.
The second author was partially funded by the German Research Foundation (DFG, Bioinformatics Initiative).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Gabow, H.N., Bentley, J.L., Tarjan, R.E.: Scaling and related techniques for geometry problems. In: Proc. of the ACM Symp. on Theory of Computing, pp. 135–143. ACM Press, New York (1984)
Harel, D., Tarjan, R.E.: Fast algorithms for finding nearest common ancestors. SIAM J. Comput. 13(2), 338–355 (1984)
Schieber, B., Vishkin, U.: On finding lowest common ancestors: Simplification and parallelization. SIAM J. Comput. 17(6), 1253–1262 (1988)
Berkman, O., Breslauer, D., Galil, Z., Schieber, B., Vishkin, U.: Highly parallelizable problems. In: Proc. of the ACM Symp. on Theory of Computing, pp. 309–319. ACM Press, New York (1989)
Bender, M.A., Farach-Colton, M., Pemmasani, G., Skiena, S., Sumazin, P.: Lowest common ancestors in trees and directed acyclic graphs. J. Algorithms 57(2), 75–94 (2005)
Fischer, J., Heun, V.: A new succinct representation of RMQ-information and improvements in the enhanced suffix array. In: Proc. ESCAPE. LNCS (to appear)
Fischer, J., Heun, V.: Theoretical and practical improvements on the RMQ-problem, with applications to LCA and LCE. In: Lewenstein, M., Valiente, G. (eds.) CPM 2006. LNCS, vol. 4009, pp. 36–48. Springer, Heidelberg (2006)
Gusfield, D.: Algorithms on Strings, Trees, and Sequences. Cambridge University Press, Cambridge (1997)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Amir, A., Fischer, J., Lewenstein, M. (2007). Two-Dimensional Range Minimum Queries. In: Ma, B., Zhang, K. (eds) Combinatorial Pattern Matching. CPM 2007. Lecture Notes in Computer Science, vol 4580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73437-6_29
Download citation
DOI: https://doi.org/10.1007/978-3-540-73437-6_29
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73436-9
Online ISBN: 978-3-540-73437-6
eBook Packages: Computer ScienceComputer Science (R0)