p67-callahan - A Decomposition Applications Potential of...

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A Decomposition of Multidimensional Point Sets with Applications to k-Nearest-Neighbors and n-Body Potential Fields PAUL B. CALLAHAN AND S. RAO Johns Hopkins University, Baltimore, Maiyland KOSARAJU Abstract. We define the notion of a well-separated pair decomposition of points in d-dimensional space. We then develop efficient sequential and parallel algorithms for computing such a decomposition. We apply the resulting decomposition to the efficient computation of k-nearest neighbors and n-body potential fields. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—geometrical problems and computations F.1.2 [Compu- tation by Abstract Devices]: Modes of Computation—parallelism and concurrency General Terms: Algorithms, Theory Additional Key Words and phrdses: Afl nearest neighbors, fast multipole method 1. Introduction We define the notion of a well-separated pair decomposition of a set P of n points in d dimensions. This consists of a bina~ tree whose leaves are points in P, with internal nodes corresponding to subsets of P in the natural way, and a list of pairs of nodes, such that the sets corresponding to each node are geometrically separated in a manner to be defined, and each distinct pair of points is “covered” by exactly one of the pairs of nodes. () We show that although there are ; pairs of points, we can always find a well-separated pair decomposition using O(n) pairs of nodes. Additionally, we show that such a decomposition can be computed in O(n log n) sequential time, which we prove is optimal, and in 0(log2n) time on a CREW PRAM using O(n) processors. Using this decomposition, we show that the k-nearest neighbors of each point can be computed in O(n) sequential time, and in O(log n) parallel time on a CREW PRAM with 0(n) processors, for any fixed k. Note that this gives The work of both authors was supported by the National Science Foundation under grant CCR 91-07293 and by NSF/DARPA under grant CCR 89-08092. Authors’ address: Computer Science Department, Johns Hopkins University, Baltimore, MD 21218. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. Q 1995 ACM 0004-5411/95/0100-0067 $03.50 Journalof the Associationfor ComputingMachinery,Vol 42,No, 1,Jmuay 1995,pp 67-90,
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68 P. B. CALLAHAN AND S. R. KOSARAJU O(logzn) parallel time for the whole problem, including the construction of the decomposition. This is the first parallel algorithm for this problem that runs in deterministic O(logcn) parallel time with O(rz ) processors for a c that is not a function of d.] The composed sequential algorithm has a strong similarity to that of Vaidya [1986; 1989].
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p67-callahan - A Decomposition Applications Potential of...

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