Improved Time and Space Bounds for Dynamic Range Mode

Authors Hicham El-Zein, Meng He, J. Ian Munro, Bryce Sandlund



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Author Details

Hicham El-Zein
  • Cheriton School of Computer Science, University of Waterloo, Canada
Meng He
  • Faculty of Computer Science, Dalhousie University, Canada
J. Ian Munro
  • Cheriton School of Computer Science, University of Waterloo, Canada
Bryce Sandlund
  • Cheriton School of Computer Science, University of Waterloo, Canada

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Hicham El-Zein, Meng He, J. Ian Munro, and Bryce Sandlund. Improved Time and Space Bounds for Dynamic Range Mode. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 25:1-25:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ESA.2018.25

Abstract

Given an array A of n elements, we wish to support queries for the most frequent and least frequent element in a subrange [l, r] of A. We also wish to support updates that change a particular element at index i or insert/ delete an element at index i. For the range mode problem, our data structure supports all operations in O(n^{2/3}) deterministic time using only O(n) space. This improves two results by Chan et al. [Timothy M. Chan et al., 2014]: a linear space data structure supporting update and query operations in O~(n^{3/4}) time and an O(n^{4/3}) space data structure supporting update and query operations in O~(n^{2/3}) time. For the range least frequent problem, we address two variations. In the first, we are allowed to answer with an element of A that may not appear in the query range, and in the second, the returned element must be present in the query range. For the first variation, we develop a data structure that supports queries in O~(n^{2/3}) time, updates in O(n^{2/3}) time, and occupies O(n) space. For the second variation, we develop a Monte Carlo data structure that supports queries in O(n^{2/3}) time, updates in O~(n^{2/3}) time, and occupies O~(n) space, but requires that updates are made independently of the results of previous queries. The Monte Carlo data structure is also capable of answering k-frequency queries; that is, the problem of finding an element of given frequency in the specified query range. Previously, no dynamic data structures were known for least frequent element or k-frequency queries.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • dynamic data structures
  • range query
  • range mode
  • range least frequent
  • range k-frequency

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