Applications of Weighted Voronoi Diagrams and Randomization to Variance-Based k-Clustering (Extended Abstract)


In this paper we consider the<italic>k</italic>-clustering problem for a set <italic>S</italic> of <italic>n</italic> points <inline-equation><f><inf>i</inf>=<fen lp="par"><b>x<inf>i</inf></b><rp post="par"> </fen></f> </inline-equation> in the<italic>d</italic>-dimensional space with variance-based errors as clustering criteria, motivated from the color quantization problem of computing a color lookup table for frame buffer display. As the inter-cluster criterion to minimize, the sum on intra-cluster errors over every cluster is used, and as the intra-cluster criterion of a cluster<inline-equation><f>S<inf>j</inf></f> </inline-equation>, <display-equation><fd><fl><fen lp="vb"><g>a </g>-1<sum align="c"><ll>p<inf>i</inf>&#8712;S<inf>j</inf></ll></sum><fen lp="vb" style="d"><b>x<inf>i</inf>-<ovl>x</ovl></b><fen lp="par"> S<inf>j</inf><rp post="par"></fen><rp post="vb"style="d"></fen> <sup>2</sup><rp post="vb"></fen></fl></fd></display-equation> is considered, where <inline-equation><f><fen lp="vb" style="d">&dot;<rp post="vb" style="d"></fen></f> </inline-equation> is the <inline-equation><f>L<inf>2</inf></f></inline-equation> norm and <inline-equation><f><b><ovl>x</ovl><fen lp="par">S<inf>j</inf><rp post="par"></fen></b></f></inline-equation> is the centroid of points in <inline-equation><f>S<inf>j</inf></f></inline-equation>, i.e., <inline-equation><f><fen lp="par">1/<fen lp="vb">S<inf>j</inf><rp post="vb"></fen><rp post="par"></fen><sum align="c"><ll>p<inf>i</inf>&#8712;S<inf>j</inf></ll></sum><b>x<inf>i</inf></b></f></inline-equation>. The cases of <inline-equation><f><g>a</g>=1,2</f></inline-equation> correspond to the sum of squared errors and the all-pairs sum of squared errors, respectively. The <italic>k</italic>-clustering problem under the criterion with <inline-equation><f><g>a</g>=1,2</f></inline-equation> are treated in a unified manner by characterizing the optimum solution to the <italic>k</italic>clustering problem by the ordinary Euclidean Voronoi diagram and the weighted Voronoi diagram with both multiplicative and additive weights. With this framework, the problem is related to the generalized primary shutter function for the Voronoi diagrams. The primary shutter function is shown to be <inline-equation><f>O<fen lp="par">n<sup>O<fen lp="par">kd<rp post="par"></fen></sup><rp post="par"></fen></f></inline-equation>, which implies that, for fixed <italic>k</italic>, this clustering problem can be solved in a polynomial time. For the problem with the most typicalintra-cluster criterion of the sum of squared errors, we also present anefficient randomized algorithm which, roughly speaking, finds an <inline-equation><f>&#8712;</f></inline-equation>&#8211;approximate 2&#8211;clustering in<inline-equation><f>O<fen lp="par">n<fen lp="par">1/&#8712;<rp post="par"></fen><sup>d</sup><rp post="par"></fen></f></inline-equation> time, which is quite practical and may be used to real large-scale problems such as the color quantization problem.

DOI: 10.1145/177424.178042
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@inproceedings{Inaba1994ApplicationsOW, title={Applications of Weighted Voronoi Diagrams and Randomization to Variance-Based k-Clustering (Extended Abstract)}, author={Mary Inaba and Naoki Katoh and Hiroshi Imai}, booktitle={Symposium on Computational Geometry}, year={1994} }