AbstractA collection of

nballs inddimensions forms ak-ply system if no point in the space is covered by more thankballs. We show that for everyk-ply systemG, there is a sphereSthat intersects at mostO(k^{1/d}n^{1 - 1/d}) balls ofGand divides the remainder ofGinto two parts: those in the interior and those in the exterior of the sphereS, respectively, so that the larger part contains at most (1 - 1/(d+ 2))nballs. This bound ofO(k^{1/d}n^{1 - 1/d}) is the best possible in bothnandk. We also present a simple randomized algorithm to find such a sphere inO(n) time. Our result implies that everyk-nearest neighbor graphs ofnpoints inddimensions has a separator of sizeO(k^{1/d}n^{1 - 1/d}). In conjunction with a result of Koebe that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, our result not only gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions. The separator algorithm can be used for point location and geometric divide and conquer in a fixed dimensional space.The abstract is also available as a LaTeX file, a DVI file, or a PostScript file.

Categories and Subject Descriptors: E.1 [Data Structures] --graphs,trees; F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problems --computation of transforms; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems --computaions on discrete structures,geometrical problems and computations,sorting and searching; G.2.1 [Discrete Mathematics]: Combinatorics; G.2.2 [Discrete Mathematics]: Graph Theory --graph algorithms,trees; G.3 [Probability and Statistics] --probabilistic algorithms,random number generation; G.4 [Mathematical Software] --algorithm analysis,efficiency

General Terms: Algorithms, Theory

Additional Key Words and Phrases: Centerpoint, computational geometry, graph algorithms, graph separators, partitioning, probabilistic method, point location, randomized algorithm, sphere-preserving mapping

Selected references

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