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

In 2015, Driemel, Krivov{s}ija and Sohler introduced the $(k,ell)$-median problem for clustering polygonal curves under the Frechet distance. Given a set of input curves, the problem asks to find $k$ median curves of at most $ell$ vertices each that minimize the sum of Frechet 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 $mathbb{R}^d$ and achieves approximation factor $(1+epsilon)$ 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 $d$, $ell$, $epsilon$ and $delta$, 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 $ell$, but of complexity at most $2ell-2$, 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.