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Low space data structures for geometric range mode query. (English) Zbl 1315.68113
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 $$d = 1$$, 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.

##### MSC:
 68P05 Data structures 68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
##### Keywords:
range queries; mode; data structures; color queries
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##### References:
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