Approximating $(k,\ell)$-Median Clustering for Polygonal Curves

In 2015, Driemel, Krivošija and Sohler introduced the (k, `)-median problem for clustering polygonal curves under the Fréchet distance. Given a set of input curves, the problem asks to find k median curves of at most ` vertices each that minimize the sum of Fréchet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input curves are restricted to lie on the real line. In this paper, we present a randomized bicriteria approximation algorithm that works for polygonal curves in R and achieves approximation factor (1 + ε) with respect to the clustering costs. The algorithm has worst-case running-time linear in the number of curves, polynomial in the maximum number of vertices per curve, i.e. their complexity, and exponential in ε and δ, i.e., the failure probability. We achieve this result through a shortcutting lemma, which guarantees the existence of a polygonal curve with similar cost as an optimal median curve of complexity `, but of complexity at most 2`− 2, and whose vertices can be computed efficiently. We combine this lemma with the superset-sampling technique by Kumar et al. to derive our clustering result. In doing so, we describe and analyze a generalization of the algorithm by Ackermann et al., which may be of independent interest. ∗Faculty of Mathematics, Ruhr-University Bochum, Germany, maike.buchin@rub.de †Hausdorff Center for Mathematics, University of Bonn, Germany, driemel@cs.uni-bonn.de ‡Faculty of Mathematics, Ruhr-University Bochum, Germany, dennis.rohde-t1b@rub.de ar X iv :2 00 9. 01 48 8v 2 [ cs .C G ] 4 S ep 2 02 0