TY - JOUR

T1 - On clustering induced voronoi diagrams

AU - Chen, Danny Z.

AU - Huang, Ziyun

AU - Liu, Yangwei

AU - Xu, Jinhui

N1 - Funding Information:
The research of the first author was supported in part by NSF grant CCF-1217906, and the research of the last three authors was supported in part by NSF grants CCF-1716400, IIS-1422591, and CCF-1422324.
Funding Information:
∗Received by the editors October 22, 2015; accepted for publication (in revised form) May 15, 2017; published electronically November 7, 2017. A preliminary version of this paper appeared in the Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, 2013. http://www.siam.org/journals/sicomp/46-6/M104487.html Funding: The research of the first author was supported in part by NSF grant CCF-1217906, and the research of the last three authors was supported in part by NSF grants CCF-1716400, IIS-1422591, and CCF-1422324. †Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN 40556 (dchen@cse.nd.edu). ‡Corresponding authors. Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY 14260 (ziyunhua@buffalo.edu, jinhui@buffalo.edu). §Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY 14260 (yangweil@buffalo.edu).
Publisher Copyright:
Copyright © by SIAM.

PY - 2017

Y1 - 2017

N2 - 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 Rd and determines the infiuence-based Voronoi cell in Rd 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 Rd 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 logmax{3,d+1} n) and O(n log2 n) time, respectively.

AB - 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 Rd and determines the infiuence-based Voronoi cell in Rd 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 Rd 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 logmax{3,d+1} n) and O(n log2 n) time, respectively.

UR - http://www.scopus.com/inward/record.url?scp=85039956476&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85039956476&partnerID=8YFLogxK

U2 - 10.1137/15M1044874

DO - 10.1137/15M1044874

M3 - Article

AN - SCOPUS:85039956476

SN - 0097-5397

VL - 46

SP - 1679

EP - 1711

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 6

ER -