Topic
Smallest-circle problem
About: Smallest-circle problem is a research topic. Over the lifetime, 107 publications have been published within this topic receiving 5711 citations. The topic is also known as: minimum covering circle problem.
Papers published on a yearly basis
Papers
13 Oct 1975
TL;DR: The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space, and is used to obtain O(N log N) algorithms for most of the problems considered.
Abstract: A number of seemingly unrelated problems involving the proximity of N points in the plane are studied, such as finding a Euclidean minimum spanning tree, the smallest circle enclosing the set, k nearest and farthest neighbors, the two closest points, and a proper straight-line triangulation. For most of the problems considered a lower bound of O(N log N) is shown. For all of them the best currently-known upper bound is O(N2) or worse. The purpose of this paper is to introduce a single geometric structure, called the Voronoi diagram, which can be constructed rapidly and contains all of the relevant proximity information in only linear space. The Voronoi diagram is used to obtain O(N log N) algorithms for all of the problems.
1,244 citations
TL;DR: A linear-time algorithm is given for the classical problem of finding the smallest circle enclosing n given points in the plane, which disproves a conjecture by Shamos and Hoey that this problem requires Ω(n log n) time.
Abstract: Linear-time algorithms for linear programming in $R^2 $ and $R^3 $ are presented. The methods used are applicable for other graphic and geometric problems as well as quadratic programming. For exam...
974 citations
TL;DR: The p-center and the p-median problems relative to both the Euclidean and the rectilinear metrics are NP-hard and the reductions are from 3-satisfiability.
Abstract: 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 demand 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-median 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.
767 citations
18 Feb 2009
TL;DR: It is shown that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [6].
Abstract: In the k-means problem, we are given a finite set S of points in $\Re^m$, 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 [6].
642 citations
TL;DR: In this article, four closely related minimax location problems are considered, each involves locating a point in the plane to minimize the maximum distance (plus a possible constant) to a given finite set of points.
Abstract: Four closely related minimax location problems are considered. Each involves locating a point in the plane to minimize the maximum distance (plus a possible constant) to a given finite set of points. The distance measures considered are the Euclidean and the rectilinear. In each case efficient, finite solution procedures are given. The arguments are geometrical.
371 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2017 | 2 |
| 2016 | 6 |
| 2015 | 2 |
| 2014 | 4 |
| 2013 | 6 |
| 2012 | 6 |