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

Abstract

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 
sum_{p in P} f(p,q)  for any query point q in 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_{epsilon,d}(n^{0.5 + c}), and the underlying 
data structure can be constructed in  ~O_{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.