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 ∈ P f(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 Õ_ϵ ,d(n^0.5 + c) , and the underlying data structure can be constructed in Õ_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.