TY - JOUR

T1 - Influence-based Voronoi diagrams of clusters

AU - Huang, Ziyun

AU - Chen, Danny Z.

AU - Xu, Jinhui

N1 - Funding Information:
The research of Ziyun Huang was supported in part by NSF Grant CCF-1422324 and IIS-1422591 . The research of D. Z. Chen was supported in part by NSF Grant CCF-1617735 . The research of Jinhui Xu was partially supported by NSF through grants CCF-1422324 , IIS-1422591 , CCF-1716400 , and IIS-1910492 .
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/6

Y1 - 2021/6

N2 - In this paper, we study a generalization of Voronoi diagram, called the Influence-based Voronoi Diagram (IVD). The input consists of a point set P in Rd, a collection C={C1,C2,…,Cn} where each Ci⊆P, i=1,2,…,n, is a cluster of points of P, and an influence function F(C,q) measuring the influence from a set C of points to any point q in Rd, and the goal is to construct an influence-based Voronoi diagram for C. By making use of a recent work called the Clustering Induced Voronoi Diagram (CIVD) for unclustered points, we are able to show that it is possible to utilize CIVD's space-partition ability and combine it with a divide-and-conquer algorithm to simultaneously resolve the space partition and assignment problems for a large class of influence functions. This overcomes a major difficulty of CIVD on the assignment problem. Our technique yields a (1−ϵ)-approximate IVD with size O(NlogN) in O(T2(N)Nlog2N+T1(N)) time, where N is the total cardinalities of clusters in C, ϵ>0 is a small constant, and T1 and T2 are functions measuring how efficiently F(C,q) can be evaluated.

AB - In this paper, we study a generalization of Voronoi diagram, called the Influence-based Voronoi Diagram (IVD). The input consists of a point set P in Rd, a collection C={C1,C2,…,Cn} where each Ci⊆P, i=1,2,…,n, is a cluster of points of P, and an influence function F(C,q) measuring the influence from a set C of points to any point q in Rd, and the goal is to construct an influence-based Voronoi diagram for C. By making use of a recent work called the Clustering Induced Voronoi Diagram (CIVD) for unclustered points, we are able to show that it is possible to utilize CIVD's space-partition ability and combine it with a divide-and-conquer algorithm to simultaneously resolve the space partition and assignment problems for a large class of influence functions. This overcomes a major difficulty of CIVD on the assignment problem. Our technique yields a (1−ϵ)-approximate IVD with size O(NlogN) in O(T2(N)Nlog2N+T1(N)) time, where N is the total cardinalities of clusters in C, ϵ>0 is a small constant, and T1 and T2 are functions measuring how efficiently F(C,q) can be evaluated.

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U2 - 10.1016/j.comgeo.2021.101746

DO - 10.1016/j.comgeo.2021.101746

M3 - Article

AN - SCOPUS:85100606820

SN - 0925-7721

VL - 96

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

M1 - 101746

ER -