### 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 + 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 O~ _{ϵ} _{,} _{d}(n^{0.5} ^{+} ^{c}) , and the underlying data structure can be constructed in O~ _{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.

Original language | English (US) |
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Journal | Algorithmica |

DOIs | |

State | Accepted/In press - Jan 1 2020 |

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### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Computer Science Applications
- Applied Mathematics

### Cite this

*Algorithmica*. https://doi.org/10.1007/s00453-020-00691-w