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

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 + ϵ) -approximate solution to the sum ∑ _p∈Pf(p, q) for any query point q∈ R^d and any small constant ϵ> 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~ _ϵ,d(n^0.5+c) , and the underlying data structure can be constructed in O~ _ϵ,d(n^1+c) time for any c> 0 , where the hidden constant has only polynomial dependence on 1 / ϵ and d. Our technique is simple and can be easily implemented for practical purpose.