An Efficient Sum Query Algorithm for Distance-Based Locally Dominating Functions

摘要

In this paper, we consider the following sum query problem: Given a point set P in R-d, and a distance-based function f(p, q) (i.e., a function of the distance between p and q) satisfying some general properties, the goal is to develop a data structure and a query algorithm for efficiently computing a (1 + epsilon)-approximate solution to the sum Sigma(p is an element of) f (p, q) for any query point q is an element of R-d and any small constant epsilon 0. Existing techniques for this problem are mainly based on some core-set techniques which often have difficulties to deal with functions with local domination property. Based on several new insights to this problem, we develop in this paper a novel technique to overcome these encountered difficulties. Our algorithm is capable of answering queries with high success probability in time no more than (O) over tilde (epsilon,d)(n(0.5+c)), and the underlying data structure can be constructed in (O) over tilde (epsilon,d) (n(1+c)) time for any c 0, where the hidden constant has only polynomial dependence on 1/epsilon and d. Our technique is simple and can be easily implemented for practical purpose.