On clustering induced voronoi diagrams

In this paper, we study a generalization of the classical Voronoi diagram, called the 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 in uence function F(C; q) to measure the total (or joint) infiuence of all objects in C on an arbitrary point q in the space R^d and determines the infiuence-based Voronoi cell in R^d for C. This generalized model offers a number of new features (e.g., simultaneous clustering and space partition) to the Voronoi diagram which are useful in various new applications. We investigate the general conditions for the infiuence function which ensure the existence of a small-size (e.g., nearly linear) approximate CIVD for a set P of n points in R^d for some fixed d. To construct CIVD, we first present a stand-alone new technique, called approximate infiuence (AI) decomposition, for the general CIVD problem. With only O(n log n) time, the AI decomposition partitions the space Rd into a nearly linear number of cells so that all points in each cell receive their approximate maximum infiuence 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-)- approximate CIVDs for some small fixed > 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-)-approximate CIVDs can be built in O(n log^max{3,d+1} n) and O(n log² n) time, respectively.