Given a set $P$ of $n$ points in the plane, we consider the problem of computing the number of points of $P$ in a query unit disk (i.e., all query disks have the same radius). We show that the main techniques for simplex range searching in the plane can be adapted to this problem. For example, by adapting Matoušek's results, we can build a data structure of $O(n)$ space in $O(n^{1+\delta})$ time (for any $\delta>0$) so that each query can be answered in $O(\sqrt{n})$ time; alternatively, we can build a data structure of $O(n^2/\log^2 n)$ space with $O(n^{1+\delta})$ preprocessing time (for any $\delta>0$) and $O(\log n)$ query time. Our techniques lead to improvements for several other classical problems in computational geometry.
1. Given a set of $n$ unit disks and a set of $n$ points in the plane, the batched unitdisk range counting problem is to compute for each disk the number of points in it. Previous work [Katz and Sharir, 1997] solved the problem in $O(n^{4/3}\log n)$ time. We give a new algorithm of $O(n^{4/3})$ time, which is optimal as it matches an $\Omega(n^{4/3})$time lower bound. For small $\chi$, where $\chi$ is the number of pairs of unit disks that intersect, we further improve the algorithm to $O(n^{2/3}\chi^{1/3}+n^{1+\delta})$ time, for any $\delta>0$.
2. The above result immediately leads to an $O(n^{4/3})$ time optimal algorithm for counting the intersecting pairs of circles for a set of $n$ unit circles in the plane. The previous best algorithms solve the problem in $O(n^{4/3}\log n)$ deterministic time [Katz and Sharir, 1997] or in $O(n^{4/3}\log^{2/3} n)$ expected time by a randomized algorithm [Agarwal, Pellegrini, and Sharir, 1993].
3. Given a set $P$ of $n$ points in the plane and an integer $k$, the distance selection problem is to find the $k$th smallest distance among all pairwise distances of $P$. The problem can be solved in $O(n^{4/3}\log^2 n)$ deterministic time [Katz and Sharir, 1997] or in $O(n\log n+n^{2/3}k^{1/3}\log^{5/3}n)$ expected time by a randomized algorithm [Chan, 2001]. Our new randomized algorithm runs in $O(n\log n +n^{2/3}k^{1/3}\log n)$ expected time.
4. Given a set $P$ of $n$ points in the plane, the discrete $2$center problem is to compute two smallest congruent disks whose centers are in $P$ and whose union covers $P$. An $O(n^{4/3}\log^5 n)$time algorithm was known [Agarwal, Sharir, and Welzl, 1998]. Our techniques yield a deterministic algorithm of $O(n^{4/3}\log^{10/3} n\cdot (\log\log n)^{O(1)})$ time and a randomized algorithm of $O(n^{4/3}\log^3 n\cdot (\log\log n)^{1/3})$ expected time.
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This content will become publicly available on December 17, 2024
Constructing many faces in arrangements of lines and segments
We present new algorithms for computing many faces in arrangements of lines and segments. Given a set $S$ of $n$ lines (resp., segments) and a set $P$ of $m$ points in the plane, the problem is to compute the faces of the arrangements of $S$ that contain at least one point of $P$.
For the line case, we give a deterministic algorithm of $O(m^{2/3}n^{2/3}\log^{2/3} (n/\sqrt{m})+(m+n)\log n)$ time. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of $\log^{2.22}n$ and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of $\log^{1/3}n$ in certain cases (e.g., when $m=\Theta(n)$).
For the segment case, we present a deterministic algorithm of $O(n^{2/3}m^{2/3}\log n+\tau(n\alpha^2(n)+n\log m+m)\log n)$ time, where $\tau=\min\{\log m,\log (n/\sqrt{m})\}$ and $\alpha(n)$ is the inverse Ackermann function. This improves the previously best deterministic algorithm [Agarwal, 1990] by a factor of $\log^{2.11}n$ and improves the previously best randomized algorithm [Agarwal, Matoušek, and Schwarzkopf, 1998] by a factor of $\log n$ in certain cases (e.g., when $m=\Theta(n)$). We also give a randomized algorithm of $O(m^{2/3}K^{1/3}\log n+\tau(n\alpha(n)+n\log m+m)\log n\log K)$ expected time, where $K$ is the number of intersections of all segments of $S$.
In addition, we consider the query version of the problem, that is, preprocess $S$ to compute the face of the arrangement of $S$ that contains any given query point. We present new results that improve the previous work for both the line and the segment cases. In particulary, for the line case, we build a data structure of $O(n\log n)$ space in $O(n\log n)$ randomized time, so that the face containing the query point can be obtained in $O(\sqrt{n\log n})$ time with high probability (more specifically, the query returns a binary search tree representing the face so that standard binarysearchbased queries on the face can be handled in $O(\log n)$ time each and the face itself can be output explicitly in time linear in its size).
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 Award ID(s):
 2300356
 NSFPAR ID:
 10546351
 Publisher / Repository:
 Journal of Computational Geometry
 Date Published:
 Journal Name:
 Journal of computational geometry
 Volume:
 14
 Issue:
 1
 ISSN:
 1920180X
 Format(s):
 Medium: X
 Right(s):
 Creative Commons Attribution 4.0 International
 Sponsoring Org:
 National Science Foundation
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