On k-means for segments and polylines

05/18/2023

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by   Sergio Cabello, et al.

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We study the problem of k-means clustering in the space of straight-line segments in ℝ^2 under the Hausdorff distance. For this problem, we give a (1+ϵ)-approximation algorithm that, for an input of n segments, for any fixed k, and with constant success probability, runs in time O(n+ ϵ^-O(k) + ϵ^-O(k)·log^O(k) (ϵ^-1)). The algorithm has two main ingredients. Firstly, we express the k-means objective in our metric space as a sum of algebraic functions and use the optimization technique of Vigneron <cit.> to approximate its minimum. Secondly, we reduce the input size by computing a small size coreset using the sensitivity-based sampling framework by Feldman and Langberg <cit.>. Our results can be extended to polylines of constant complexity with a running time of O(n+ ϵ^-O(k)).