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