Semialgebraic Range Reporting and Emptiness Searching with Applications

• Authors: Micha Sharir;Hayim Shaul
• Venue: SIAM Journal on Computing
• Year: 2011

In a typical range-emptiness searching (resp., reporting) problem, we are given a set $P$ of $n$ points in $\mathbb{R}^d$, and we wish to preprocess it into a data structure that supports efficient range-emptiness (resp., reporting) queries, in which we specify a range $\sigma$, which, in general, is a semialgebraic set in $\mathbb{R}^d$ of constant description complexity, and we wish to determine whether $P\cap\sigma=\emptyset$, or to report all the points in $P\cap\sigma$. Range-emptiness searching and reporting arise in many applications and have been treated by Matoušek [Comput. Geom. Theory Appl., 2 (1992), pp. 169-186] in the special case where the ranges are half-spaces bounded by hyperplanes. As shown in Matoušek's work, the two problems are closely related, and they have solutions (for the case of half-spaces) with similar performance bounds. In this paper we extend the analysis to general semialgebraic ranges and show how to adapt Matoušek's technique without the need to linearize the ranges into a higher-dimensional space. This yields more efficient solutions to several useful problems, and we demonstrate the new technique in four applications with the following results: (i) An algorithm for ray shooting amid balls in $\mathbb{R}^3$, which uses $O(n)$ storage and $O^*(n)$ preprocessing (we use the notation $O^*(n^\gamma)$ to mean an upper bound of the form $C(\varepsilon)n^{\gamma+\varepsilon}$, which holds for any $\varepsilon0$, where $C(\varepsilon)$ is a constant that depends on $\varepsilon$) and answers a query in $O^*(n^{2/3})$ time, improving the previous bound of $O^*(n^{3/4})$. (ii) An algorithm that preprocesses, in $O^*(n)$ time, a set $P$ of $n$ points in $\mathbb{R}^3$ into a data structure with $O(n)$ storage, so that, for any query line $\ell$ (or, for that matter, any simply shaped convex set), the point of $P$ farthest from $\ell$ can be computed in $O^*(n^{1/2})$ time. This in turn yields an algorithm that computes the largest-area triangle spanned by $P$ in time $O^*(n^{26/11})$, as well as nontrivial algorithms for computing the largest-perimeter or largest-height triangle spanned by $P$. (iii) An algorithm that preprocesses, in $O^*(n)$ time, a set $P$ of $n$ points in $\mathbb{R}^2$ into a data structure with $O(n)$ storage, so that, for any query $\alpha$-fat triangle $\Delta$, we can determine, in $O^*(1)$ time, whether $\Delta\cap P$ is empty. Alternatively, we can report, in $O^*(1)+O(k)$ time, the points of $\Delta\cap P$, where $k=|\Delta\cap P|$. (iv) An algorithm that preprocesses, in $O^*(n)$ time, a set $P$ of $n$ points in $\mathbb{R}^2$ into a data structure with $O(n)$ storage, so that, given any query semidisk $c$, or a circular cap larger than a semidisk, we can determine, in $O^*(1)$ time, whether $c\cap P$ is empty, or report the $k$ points in $c\cap P$ in $O^*(1)+O(k)$ time. Adapting the recent techniques of [B. Aronov and S. Har-Peled, SIAM J. Comput., 38 (2008), pp. 899-921, B. Aronov, S. Har-Peled, and M. Sharir, On approximate halfspace range counting and relative epsilon-approximations, in Proceedings of the 23rd ACM Symposium Comput. Geom., 2007, pp. 327-336, B. Aronov and M. Sharir, SIAM J. Comput., 39 (2010), pp. 2704-2725], we can turn our solutions into efficient algorithms for approximate range counting (with small relative error) for the cases mentioned above. Our technique is closely related to the notions of nearest- or farthest-neighbor generalized Voronoi diagrams and of the union or intersection of geometric objects, where sharper bounds on the combinatorial complexity of (decompositions of complements of) these structures yield faster range-emptiness searching or reporting algorithms.