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 ${\mathbb{R}}^d$, a collection $\mathcal{C}=\{C_1,C_2,\dots ,C_n\}$ where each $C_i\subseteq P$, $i=1,2,\dots ,n$, 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 ${\mathbb{R}}^d$, and the goal is to construct an influence-based Voronoi diagram for $\mathcal{C}$. 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 $O(N\log N)$ in $O(T_2(N)N{\log }^{2}N+T_1(N))$ time, where N is the total cardinalities of clusters in $\mathcal{C}$, $ϵ>0$ is a small constant, and $T_1$ and $T_2$ are functions measuring how efficiently $F(C,q)$ can be evaluated.

在本文中，我们研究了Voronoi图的一般化，称为基于影响的Voronoi图（IVD）。输入包含一个点集P in${\mathbb{[R}}^d$，一个集合 $\mathcal{C}=\{C_{\mathrm{1个}}，C_2，\dots ，C_{ñ}\}$ 每个在哪里 $C_{\mathrm{一世}}\subseteq P$， $\mathrm{一世}=\mathrm{1个}，2，\dots ，ñ$是P点的聚类和影响函数$F（C，q）$测量一组点C对任意点q in的影响${\mathbb{[R}}^d$，目的是为以下人员构建基于影响的Voronoi图 $\mathcal{C}$。通过使用最近的未聚类点的称为聚类诱导的Voronoi图（CIVD）的工作，我们能够证明可以利用CIVD的空间划分能力，并将其与分治法相结合来同时求解一大类影响函数的空间分配和分配问题。这克服了CIVD在分配问题上的主要困难。我们的技术产生了$（\mathrm{1个}-ϵ）$-尺寸约为IVD $Ø（ñ\mathrm{日志}ñ）$ 在 $Ø（{Ť}_2（ñ）ñ{\mathrm{日志}}^2ñ+{Ť}_{\mathrm{1个}}（ñ））$时间，其中N是群集中基数的总基数$\mathcal{C}$， $ϵ>0$ 是一个小常数，并且 ${Ť}_{\mathrm{1个}}$ 和 ${Ť}_2$ 是衡量效率的函数 $F（C，q）$ 可以评估。