Applications of Weighted Voronoi Diagrams and Randomization to Variance-based K-clustering

Abstract

In this paper we consider the k-clustering problem for a set S of n points p i = (x i) in the d-dimensional space with variance-based errors as clustering criteria, motivated from the color quantization problem of computing a color lookup table for frame buuer display. As the inter-cluster criterion to minimize, the sum of intra-cluster errors over every cluster is used, and as the intra-cluster criterion of a cluster S j , jS j j ?1 X pi2Sj kx i ? x(S j)k 2 is considered, where k k is the L 2 norm and x(S j) is the centroid of points in S j , i.e., (1=jS j j) P pi2Sj x i. The cases of = 1; 2 correspond to the sum of squared errors and the all-pairs sum of squared errors, respectively. The k-clustering problem under the criterion with = 1; 2 are treated in a uniied manner by characterizing the optimum solution to the k-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 O(n O(kd)), which implies that, for xed k, this clustering problem can be solved in a polynomial time. For the problem with the most typical intra-cluster criterion of the sum of squared errors, we also present an eecient randomized algorithm which, roughly speaking, nds an-approximate 2-clustering in O(n(1==) d) time, which is quite practical and may be used to real large-scale problems such as the color quantization problem.

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@inproceedings{Inaba2009ApplicationsOW, title={Applications of Weighted Voronoi Diagrams and Randomization to Variance-based K-clustering}, author={Mary Inaba and Naoki Katohy and Hiroshi Imai}, year={2009} }