2026-07-22T17:54:33Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:9488 2024-03-06T10:43:52Z ddc:004 open_access

Improved Time and Space Bounds for Dynamic Range Mode El-Zein, Hicham He, Meng Munro, J. Ian Sandlund, Bryce dynamic data structures range query range mode range least frequent range k-frequency 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Hicham El-Zein and Meng He and J. Ian Munro and Bryce Sandlund 2018 Is Part Of LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2018.25 urn:nbn:de:0030-drops-94886 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.25 eng https://creativecommons.org/licenses/by/3.0/legalcode