Two dimensional range minimum queries and fibonacci lattices

Authors:
Gerth Stølting Brodal;Pooya Davoodi;Moshe Lewenstein;Rajeev Raman;Satti Srinivasa Rao
Affiliations:
MADALGO, Aarhus University, Denmark;Polytechnic Institute of New York University, United States;Bar-Ilan University, Israel;University of Leicester, UK;Seoul National University, S. Korea
Venue:
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
Year:
2012

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Abstract

Given a matrix of size N, two dimensional range minimum queries (2D-RMQs) ask for the position of the minimum element in a rectangular range within the matrix. We study trade-offs between the query time and the additional space used by indexing data structures that support 2D-RMQs. Using a novel technique--the discrepancy properties of Fibonacci lattices--we give an indexing data structure for 2D-RMQs that uses O(N/c) bits additional space with O(clogc(loglogc)2) query time, for any parameter c, 4≤c≤N. Also, when the entries of the input matrix are from {0,1}, we show that the query time can be improved to O(clogc) with the same space usage.