On Clustering Induced Voronoi Diagrams
Danny Z. Chen, Ziyun Huang, Yangwei Liu, Jinhui Xu
TL;DR
The paper generalizes Voronoi diagrams to Clustering Induced Voronoi Diagrams (CIVD), where sites are all subsets of a point set $P$ and the joint influence $F(C,q)$ governs region assignment, enabling simultaneous clustering and space partitioning. It introduces Approximate Influence (AI) Decomposition, a framework that partitions space into $O(n\log n)$ cells using a box-clustering tree guided by a distance-tree from well-separated pair decompositions, and proves conditions under which a small CIVD exists. Leveraging AI, the authors develop assignment algorithms for two CIVD variants—vector CIVD and density-based CIVD—achieving $(1-\epsilon)$-approximate CIVDs in near-linear to near-quadratic time depending on dimension, specifically $O\left(n\log^{\max\{3,d+1\}}n\right)$ and $O\left(n\log^{2}n\right)$ respectively. This work provides a general, practical approach to constructing CIVDs efficiently and demonstrates potential applications in data mining and social networks by enabling joint clustering and space partitioning with principled approximation guarantees.
Abstract
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 $(1-ε)$-approximate CIVDs for some small fixed $ε>0$. 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 $(1-ε)$-approximate CIVDs can be built in $O(n \log^{\max\{3,d+1\}}n)$ and $O(n \log^{2} n)$ time, respectively.