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The Frechet distance is a popular distance measure for curves. We study the problem of clustering time series under the Frechet distance. In particular, we give (1 + e)-approximation algorithms for variations of the following problem with parameters k and e. Given n univariate time series P, each of complexity at most m, we find k time series, not necessarily from P, which we call cluster centers and which each have complexity at most e, such that (a) the maximum distance of an element of P to its nearest cluster center or (b) the sum of these distances is minimized. Our algorithms have running time near-linear in the input size for constant e, k and e. To the best of our knowledge, our algorithms are the first clustering algorithms for the Frechet distance which achieve an approximation factor of (1 + e) or better.

Clustering time series under the Fréchet distance