TY - JOUR

T1 - An Efficient Sum Query Algorithm for Distance-Based Locally Dominating Functions

AU - Huang, Ziyun

AU - Xu, Jinhui

N1 - Funding Information:
This research was supported in part by NSF through Grants IIS-1422591, CCF-1422324, CNS-1547167, CCF-1716400, and IIS-1910492. A preliminary version of this paper has appeared in the Proceedings of the 28th International Symposium on Algorithms and Computation (ISAAC 2017).

PY - 2020/9/1

Y1 - 2020/9/1

N2 - In this paper, we consider the following sum query problem: Given a point set P in Rd, 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∈Pf(p, q) for any query point q∈ Rd 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(n0.5+c) , and the underlying data structure can be constructed in O~ ϵ,d(n1+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.

AB - In this paper, we consider the following sum query problem: Given a point set P in Rd, 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∈Pf(p, q) for any query point q∈ Rd 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(n0.5+c) , and the underlying data structure can be constructed in O~ ϵ,d(n1+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.

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U2 - 10.1007/s00453-020-00691-w

DO - 10.1007/s00453-020-00691-w

M3 - Article

AN - SCOPUS:85081579951

VL - 82

SP - 2415

EP - 2431

JO - Algorithmica

JF - Algorithmica

SN - 0178-4617

IS - 9

ER -