On Clustering Induced Voronoi Diagrams

TL;DR

This paper introduces a generalized Voronoi diagram called clustering induced Voronoi diagram (CIVD), which incorporates influence functions over subsets of points, enabling simultaneous clustering and space partitioning with efficient approximation algorithms.

Contribution

The paper proposes a new influence-based model for Voronoi diagrams, introduces the AI decomposition technique, and develops fast algorithms for approximate CIVDs in vector and density-based cases.

Findings

Nearly linear size approximate CIVDs are achievable.

AI decomposition partitions space efficiently in $O(n ext{ log } n)$ time.

Fast algorithms for vector and density CIVDs with $O(n ext{ log }^{ ext{max}\{3,d+1 ight ext{}}} n)$ and $O(n ext{ log }^{2} n)$ complexity.

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 $(1-\epsilon)$-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.