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

TL;DR

This paper introduces a randomized approximation algorithm for the $(k, ext{ell})$-median clustering problem of polygonal curves in higher dimensions, improving applicability beyond line-restricted curves with near-optimal cost guarantees.

Contribution

It extends previous work by developing a bicriteria-approximation algorithm for polygonal curves in $ extbf{R}^d$, with a novel shortcutting lemma and analysis.

Findings

Achieves $(1+ ext{epsilon})$ approximation factor for clustering cost.

Runs in linear time in the number of curves, polynomial in complexity, exponential in dimension and parameters.

Provides a generalized algorithm potentially useful for broader clustering problems.

Abstract

In 2015, Driemel, Krivo\v{s}ija and Sohler introduced the $(k,\ell)$-median problem for clustering polygonal curves under the Fr\'echet 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 Fr\'echet 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 $2\ell-2$, 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.