an:06438844
Zbl 1315.68113
Durocher, Stephane; El-Zein, Hicham; Munro, J. Ian; Thankachan, Sharma V.
Low space data structures for geometric range mode query
EN
Theor. Comput. Sci. 581, 97-101 (2015).
00344370
2015
j
68P05 68U05
range queries; mode; data structures; color queries
Summary: Let \(\mathcal{S}\) be a set of \(n\) points in \(d\) dimensions such that each point is assigned a color. Given a query range \(\mathcal{Q} = [a_1, b_1] \times [a_2, b_2] \times \ldots \times [a_d, b_d]\), the geometric range mode query problem asks to report the most frequent color (i.e., a mode) of the multiset of colors corresponding to points in \(\mathcal{S} \cap \mathcal{Q}\). When \(d = 1\), \textit{T. M. Chan} et al. [LIPICS -- Leibniz Int. Proc. Inform. 14, 290--301 (2012; Zbl 1245.68071)] gave a data structure that requires \(O(n +(n / {\Delta})^2 / w)\) words and supports range mode queries in \(O({\Delta})\) time for any \({\Delta} \geq 1\), where \(w = {\Omega}(\log n)\) is the word size. Chan et al. also proposed a data structures for higher dimensions (i.e., \(d \geq 2\)) with \(O(s_n +(n / {\Delta})^{2 d})\) words and \(O({\Delta} \cdot t_n)\) query time, where \(s_n\) and \(t_n\) denote the space and query time of a data structure that supports orthogonal range counting queries on the set \(\mathcal{S}\). In this paper we show that the space can be improved without any increase to the query time, by presenting an \(O(s_n +(n/{\Delta})^{2 d} / w)\)-word data structure that supports orthogonal range mode queries on a set of \(n\) points in \(d\) dimensions in \(O({\Delta} \cdot t_n)\) time, for any \({\Delta} \geq 1\). When \(d = 1\), these space and query time costs match those achieved by the current best known one-dimensional data structure.
Zbl 1245.68071