Approximating ( <i>k,ℓ</i> )-Median Clustering for Polygonal Curves

In 2015, Driemel, Krivošija, and Sohler introduced the k,ℓ -median clustering problem for polygonal curves under Fréchet distance. Given a set of input curves, asks to find k median at most ℓ vertices each that minimize sum distances over all their closest curve. A major shortcoming algorithm is are restricted lie on real line. this article, we present randomized bicriteria-approximation works in ℝ d achieves approximation factor (1+ɛ) with respect costs. The has worst-case running time linear number polynomial maximum per curve (i.e., complexity), exponential , ℓ, 1/ɛ 1/δ failure probability). We achieve result through shortcutting lemma, which guarantees existence similar cost as an optimal complexity but 2ℓ -2, whose can be computed efficiently. combine lemma superset sampling technique by Kumar et al. derive our result. doing so, describe analyze generalization Ackermann al., may independent interest.