Fast approximations for sums of distances, clustering and the Fermat--Weber problem

Authors:
Prosenjit Bose;Anil Maheshwari;Pat Morin
Affiliations:
School of Computer Science, Carleton University, 1125 Colonel By Dr., Ottawa, ON, Canada K1S 5B6;School of Computer Science, Carleton University, 1125 Colonel By Dr., Ottawa, ON, Canada K1S 5B6;School of Computer Science, Carleton University, 1125 Colonel By Dr., Ottawa, ON, Canada K1S 5B6
Venue:
Computational Geometry: Theory and Applications
Year:
2003

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Abstract

We describe two data structures that preprocess a set S of n points in Rd (d constant) so that the sum of Euclidean distances of points in S to a query point q can be quickly approximated to within a factor of ε. This preprocessing technique has several applications in clustering and facility location. Using it, we derive an O(n log n) time deterministic and O(n) time randomized ε-approximation algorithm for the so called Fermat-Weber problem in any fixed dimension.