Space-Efficient Preprocessing Schemes for Range Minimum Queries on Static Arrays

• Published in: SIAM journal on computing Vol. 40; no. 2; pp. 465 - 492
• Main Authors: Fischer, Johannes, Heun, Volker
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Applied sciences; Arrays; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Construction; Data processing. List processing. Character string processing; Data structures; Entropy; Exact sciences and technology; Graph theory; Information retrieval; Mathematical analysis; Mathematical functions; Mathematics; Memory organisation. Data processing; Miscellaneous; On-line systems; Optimization algorithms; Preprocessing; Queries; Sciences and techniques of general use; Software; Studies; Theoretical computing; 68P05; On line; Query; 68R05; 68W32; range queries; Linear time; Entropy; trees; N order; lowest common ancestors; Integer; Array; Input; Tree; arrays; Data structure; 05C05
• ISSN: 0097-5397; 1095-7111
• DOI: 10.1137/090779759

Given a static array of n totally ordered objects, the range minimum query problem is to build a data structure that allows to answer efficiently subsequent on-line queries of the form of the position of a minimum element in the subarray ranging from i to j. The authors focus on two settings, where (1) the input array is available at query time, and (2) the input array is available only at construction time. In setting (1), the authors show new data structures (a) of size ... bits and query time O(c(n)) for any positive integer function ... for an arbitrary constant 0 < ε < 1, or (b) with O(nH(k)) + o(n) bits and O(1) query time, where H(k) denotes the empirical entropy of kth order of the input array. In setting (2), the authors give a data structure of size 2n+o(n) bits and query time O(1). All data structures can be constructed in linear time and almost in-place. (ProQuest: ... denotes formulae/symbols omitted.)