Efficient planar two-center algorithms

Let P be a set of points in the plane. The goal is to place two unit disks in the plane such that the number of points from P covered by the disks is maximized. In addition, the distance between the centers of the two disks should not exceed a specified constant R c ≥ 0. We propose two algorithms to solve this problem. The first algorithm is a simple exhaustive algorithm which runs in O(n 4) time. We then improve this algorithm by a constructing connectivity region and building a segment tree to compute two optimal disks. The resulting algorithm has O(n 3 log n) time complexity.

Proceedings of the …, 2006

SIAM Journal on Computing, 2011

Let P be a simple polygon, and let Q be a set of points in P . We present an almost-linear time algorithm for computing a minimum cover of Q by disks that are contained in P . We generalize the algorithm above, so that it can compute a minimum cover of Q by homothets of a fixed compact convex set of constant description complexity O that are contained in P . This improves previous results of Katz and Morgenstern . We also consider the disk-cover problem when Q is contained in a (not too wide) annulus, and present an O(|Q| log |Q|) algorithm for this case.

2009

In the k-means problem, we are given a finite set S of points in ℜ m, and integer k ≥ 1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [7].

2004

Computational Geometry, 2000

Given a set P of n points in the plane, we seek two squares such that their center points belong to P , their union contains P , and the area of the larger square is minimal. We present efficient algorithms for three variants of this problem: in the first the squares are axis parallel, in the second they are free to rotate but must remain parallel to each other, and in the third they are free to rotate independently.

SIAM Journal on Computing, 1984

Given n demand points in the plane, the p-center problem is to find p supply points (anywhere in the plane) so as to minimize the maximum distance from a demo& point to its respective nearest supply point. The p-median problem is to minimize the sum of distances from demand points to their respective nearest supply points. We prove that the p-center and the p-media problems relative to both the Euclidean and the rectilinear metrics are NP-hard. In fact, we prove that it is NP-hard even to approximate the p-center problems sufficiently closely. The reductions are from 3-satisfiability.

Information Processing Letters, 2005

We describe a simple and fast O(n log 2 1 ε )-time algorithm for finding a (1 + ε)-approximation of the smallest enclosing disk of a planar set of n points or disks. Experimental results of a readily available implementation are presented.

Journal of Mathematics Research, 2016

Geometric Disks Covering (GDC) is one of the most typical and well studied problems in computational geometry. Geometric disks are well known 2-D objects which have surface area with circular boundaries but differ from polygons whose surfaces area are bounded by straight line segments. Unlike polygons covering with disks is a rigorous task because of the circular boundaries that do not tessellate. In this paper, we investigate an area approximate polygon to disks that facilitate tiling as a guide to disks covering with least overlap difference. Our study uses geometry of tessellable regular polygons to show that hexagonal tiling is the most efficient way to tessellate the plane in terms of the total perimeter per area coverage.

Cccg, 2006

In this paper, we propose an algorithm for computing the farthest-segment Voronoi diagram for the edges of a convex polygon and apply this to obtain an O(n log n) algorithm for the following proximity problem: Given a set P of n(> 2) points in the plane, we have O(n 2) implicitly defined segments on pairs of points. For each point p ∈ P , find a segment from this set of implicitly defined segments that is farthest from p. We improve the previously known time bound of O(nh + n log n) for this problem, where h is the number of vertices on the convex hull of P .