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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 unit-disk 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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Clustering with Neighborhoods
In the standard planar k-center clustering problem, one is given a set P of n points in the plane, and the goal is to select k center points, so as to minimize the maximum distance over points in P to their nearest center. Here we initiate the systematic study of the clustering with neighborhoods problem, which generalizes the k-center problem to allow the covered objects to be a set of general disjoint convex objects C rather than just a point set P. For this problem we first show that there is a PTAS for approximating the number of centers. Specifically, if r_opt is the optimal radius for k centers, then in n^O(1/ε²) time we can produce a set of (1+ε)k centers with radius ≤ r_opt. If instead one considers the standard goal of approximating the optimal clustering radius, while keeping k as a hard constraint, we show that the radius cannot be approximated within any factor in polynomial time unless P = NP, even when C is a set of line segments. When C is a set of unit disks we show the problem is hard to approximate within a factor of (√{13}-√3)(2-√3) ≈ 6.99. This hardness result complements our main result, where we show that when the objects are disks, of possibly differing radii, there is a (5+2√3)≈ 8.46 approximation algorithm. Additionally, for unit disks we give an O(n log k)+(k/ε)^O(k) time (1+ε)-approximation to the optimal radius, that is, an FPTAS for constant k whose running time depends only linearly on n. Finally, we show that the one dimensional version of the problem, even when intersections are allowed, can be solved exactly in O(n log n) time.
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- Award ID(s):
- 1750780
- PAR ID:
- 10344039
- Editor(s):
- Ahn, Hee-Kap; Sadakane, Kunihiko
- Date Published:
- Journal Name:
- International Symposium on Algorithms and Computation (ISAAC)
- Volume:
- 212
- Issue:
- 6
- Page Range / eLocation ID:
- 6:1--6:17
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.ISAAC.2021.6
