Clustering time series under the Fr'echet distance

The Fr'echet distance is a popular distance measure for curves. We study the problem of clustering time series under the Fr'echet distance. In particular, we give (1+varepsilon)-approximation algorithms for variations of the following problem with parameters k and ℓ. 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 ℓ, 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 varepsilon, k and ℓ. To the best of our knowledge, our algorithms are the first clustering algorithms for the Fr'echet distance which achieve an approximation factor of (1+varepsilon) or better. Keywords: time series, longitudinal data, functional data, clustering, Fr'echet distance, dynamic time warping, approximation algorithms.