Coresets for $k$-median clustering under Fr\'{e}chet and Hausdorff distances

Abstract

We give algorithms for computing coresets for $(1+\varepsilon)$-approximate $k$-median clustering of polygonal curves (under the discrete and continuous Fr\'{e}chet distance) and point sets (under the Hausdorff distance), when the cluster centers are restricted to be of low complexity. Ours is the first such result, where the size of the coreset is independent of the number of input curves/point sets to be clustered (although it still depends on the maximum complexity of each input object). Specifically, the size of the coreset is $\Theta\left(\frac{k^3lm^{\delta}d}{\varepsilon^2}\log\left( \frac{kl}{\varepsilon}\right)\right)$ for any $\delta > 0$, where $d$ is the ambient dimension, $m$ is the maximum number of points in an input curve/point set, and $l$ is the maximum number of points allowed in a cluster center. We formally characterize a general condition on the restricted space of cluster centers -- this helps us to generalize and apply the importance sampling framework, that was used by Langberg and Schulman for computing coresets for $k$-median clustering of $d$-dimensional points on normed spaces in $\mathbb{R}^d$, to the problem of clustering curves and point sets using the Fr\'{e}chet and Hausdorff metrics. Roughly, the condition places an upper bound on the number of different combinations of metric balls that the restricted space of cluster centers can hit. We also derive lower bounds on the size of the coreset, given the restriction that the coreset must be a subset of the input objects.