On clustering induced voronoi diagrams

Danny Z. Chen, Ziyun Huang, Yangwei Liu, Jinhui Xu

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Scopus citations

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 ℝd, and determines the influence-based Voronoi cell in ℝd 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 ℝd 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 ℝd 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 logd+1 n) and O(n log2 n) time, respectively.

Original languageEnglish (US)
Title of host publicationProceedings - 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013
Pages390-399
Number of pages10
DOIs
StatePublished - 2013
Event2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013 - Berkeley, CA, United States
Duration: Oct 27 2013Oct 29 2013

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Other

Other2013 IEEE 54th Annual Symposium on Foundations of Computer Science, FOCS 2013
CountryUnited States
CityBerkeley, CA
Period10/27/1310/29/13

All Science Journal Classification (ASJC) codes

  • Computer Science(all)

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