In this paper, we study a generalization of the classical Voronoi diagram, called clustering induced Voronoi diagram (CIVD). Different from the traditional model, CIVD takes as its sites the power set $U$ of an input set $P$ of objects. For each subset $C$ of $P$, CIVD uses an influence function $F(C,q)$ to measure the total (or joint) influence of all objects in $C$ on an arbitrary point $q$ in the space $\mathbb{R}^d$, and determines the influence-based Voronoi cell in $\mathbb{R}^d$ for $C$. This generalized model offers a number of new features (e.g., simultaneous clustering and space partition) to Voronoi diagram which are useful in various new applications. We investigate the general conditions for the influence function which ensure the existence of a small-size (e.g., nearly linear) approximate CIVD for a set $P$ of $n$ points in $\mathbb{R}^d$ for some fixed $d$. To construct CIVD, we first present a standalone new technique, called approximate influence (AI) decomposition, for the general CIVD problem. With only $O(n\log n)$ time, the AI decomposition partitions the space $\mathbb{R}^{d}$ into a nearly linear number of cells so that all points in each cell receive their approximate maximum influence from the same (possibly unknown) site (i.e., a subset of $P$). Based on this technique, we develop assignment algorithms to determine a proper site for each cell in the decomposition and form various $(1-\epsilon)$-approximate CIVDs for some small fixed $\epsilon>0$. Particularly, we consider two representative CIVD problems, vector CIVD and density-based CIVD, and show that both of them admit fast assignment algorithms; consequently, their $(1-\epsilon)$-approximate CIVDs can be built in $O(n \log^{\max\{3,d+1\}}n)$ and $O(n \log^{2} n)$ time, respectively.

翻译：本文研究经典沃罗诺伊图的一种推广形式——聚类诱导沃罗诺伊图（CIVD）。与传统模型不同，CIVD以输入对象集合P的幂集U作为源点集。对于P的每个子集C，CIVD使用影响函数F(C,q)度量C中所有对象对空间ℝ^d中任意点q的联合影响，并据此确定C在ℝ^d中基于影响的沃罗诺伊单元。该推广模型为沃罗诺伊图提供了聚类与空间划分同步进行等新特性，适用于多种新兴应用场景。本文系统研究了影响函数需满足的通用条件，以确保在固定维度d下，对ℝ^d中n个点构成的集合P存在规模近似线性的近似CIVD。为构建CIVD，我们首先提出通用CIVD问题的独立新技术——近似影响分解。该分解仅需O(n log n)时间即可将ℝ^d划分为近线性数量的单元，使得同一单元内所有点都能从同一（可能未知）源点集（即P的子集）获取近似最大影响值。基于该技术，我们开发分配算法为每个单元确定恰当的源点集，并针对固定小参数ε>0构建多种(1-ε)近似CIVD。特别地，通过研究向量CIVD与基于密度的CIVD两类代表性问题，证明两者均存在快速分配算法；其(1-ε)近似CIVD的构建时间复杂度分别为O(n log^{max{3,d+1}} n)和O(n log² n)。