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 Rd, 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)-Approximate solution to the sum P p2P f(p, q) for any query point q 2 Rd and any small constant 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 Od(n0.5+c), and the underlying data structure can be constructed in d(n1+c) time for any c > 0, where the hidden constant has only polynomial dependence on and d. Our technique is simple and can be easily implemented for practical purpose.