Coresets for $(k, \ell)$-Median Clustering under the Fréchet Distance

Abstract

We present an algorithm for computing ε-coresets for (k, ℓ)-median clustering of polygonal curves in R d under the Fréchet distance. This type of clustering, which has recently drawn increasing popularity due to a growing number of applications, is an adaption of Euclidean k-median clustering: we are given a set of polygonal curves in R d and want to compute k median curves such that the sum of distances from the given curves to their closest median curve is minimal. Additionally, we restrict the complexity, i.e., the number of vertices, of the median curves to be at most ℓ each, to suppress overfitting, a problem specific for sequential data. Our algorithm has running-time linear in the number of curves, sub-quartic in their maximum complexity and quadratic in ε −1 . With high probability it returns ε-coresets of size quadratic in ε −1 , logarithmic in the number of given curves and logarithmic in their maximum complexity. We achieve this result by applying the improved ε-coreset framework by Braverman et al. to a generalized k-median problem over an arbitrary metric space. Later we combine this result with the recent result by Driemel et al. on the VC dimension of metric balls under the Fréchet distance. Furthermore, our framework yields ε-coresets for any generalized k-median problem where the range space induced by the open metric balls of the underlying space has bounded VC dimension, which is of independent interest. Finally, we show that our ε-coresets can be used to improve the running-time of an existing approximation algorithm for (1, ℓ)-median clustering, taking a further step in the direction of practical (k, ℓ)-median clustering algorithms.