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**1 - 1**of**1**### Fast Approximation of Sums of Distances

"... We show how to preprocess a set S of n points in R d (d constant) using O(kn log d 1 n) time and space so that the sum of distances of points in S to a query point q can be approximated to within a factor of O() in O(k log d 1 n) time, where is an arbitrarily small constant and k is a cons ..."

Abstract
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We show how to preprocess a set S of n points in R d (d constant) using O(kn log d 1 n) time and space so that the sum of distances of points in S to a query point q can be approximated to within a factor of O() in O(k log d 1 n) time, where is an arbitrarily small constant and k is a constant dependent only on and d. We also give applications of this technique to approximation algorithms for clustering and facility location problems. 1 Introduction Let S = fp 1 ; : : : ; p n g be a set of points in R d , with d constant. For a query point q we dene the weight of q as w(q) = n X i=1 d(q; p i ) ; (1) where d(x; y) denotes the Euclidean distance between x and y. This function appears frequently as the objective function in facility location and clustering problems [2, 3, 5, 6, 10, 18]. Unfortunately, even with preprocessing, it appears that little can be done in order to speed up the evaluation of w(q) for an arbitrary query point q, and the only known result is...