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 ${\mathbb {R}}^d$, 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 $q \in {\mathbb {R}}^d$ 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 ${\tilde{O}}_{\epsilon ,d}(n^{0.5 + c})$, and the underlying data structure can be constructed in ${\tilde{O}}_{\epsilon ,d}(n^{1+c})$ 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.