eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-08-14
25:1
25:13
10.4230/LIPIcs.ESA.2018.25
article
Improved Time and Space Bounds for Dynamic Range Mode
El-Zein, Hicham
1
He, Meng
2
Munro, J. Ian
1
Sandlund, Bryce
1
Cheriton School of Computer Science, University of Waterloo, Canada
Faculty of Computer Science, Dalhousie University, Canada
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol112-esa2018/LIPIcs.ESA.2018.25/LIPIcs.ESA.2018.25.pdf
dynamic data structures
range query
range mode
range least frequent
range k-frequency