Variance-Based k-Clustering Algorithms by Voronoi Diagrams and Randomization

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概要

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• 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 buffer 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, |S_j|^<α-1>Σ__<P_i∈S_j>‖x_i-x^^-(S_j)‖^2 is considered, where ‖・‖ is the L_2 norm and x^^-(S_j)is the centroid of points in S_j, i.e., (1/|S_j|)Σ_<P_i∈S_j>x_i. The cases of α=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 α=1, 2 are treated in a unified 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 shatter function for the Voronoi diagrams. The primary shatter function is shown to be O(n^<O(kd)>), which implies that, for fixed 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 efficient randomized algorithm which, roughly speaking, finds an ε-approximate 2-clustering in O(n(1/ε)^d)time, which is quite practical and may be used to real large-scale problems such as the color quantization problem.

• 社団法人電子情報通信学会の論文
• 2000-06-25

著者

• IMAI Hiroshi

  The author is with the Department of Information Science, the University of Tokyo

• Katoh Naoki

  The Author Is With The Department Of Architecture And Architectural Systems Kyoto University

• INABA Mary

  The authors are with the Department of Information Science, University of Tokyo

• Inaba Mary

  The Authors Are With The Department Of Information Science University Of Tokyo

• Imai Hiroshi

  The Authors Are With The Department Of Information Science University Of Tokyo

• Imai Hiroshi

  The Author Is With The Department Of Information Science The University Of Tokyo

関連論文

• Minimax Geometric Fitting of Two Corresponding Sets of Points and Dynamic Furthest Voronoi Diagrams
• Variance-Based k-Clustering Algorithms by Voronoi Diagrams and Randomization

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