## 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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Pages (from-to) | 2415-2431 |

Number of pages | 17 |

Journal | Algorithmica |

Volume | 82 |

Issue number | 9 |

DOIs | |

State | Published - Sep 1 2020 |

## All Science Journal Classification (ASJC) codes

- General Computer Science
- Computer Science Applications
- Applied Mathematics