  We present algorithms for the computation of $\varepsilon$-coresets for
$k$-median clustering of point sequences in $\mathbb{R}^d$ under the
$p$-dynamic time warping (DTW) distance. Coresets under DTW have not been
investigated before, and the analysis is not directly accessible to existing
methods as DTW is not a metric. The three main ingredients that allow our
construction of coresets are the adaptation of the $\varepsilon$-coreset
framework of sensitivity sampling, bounds on the VC dimension of approximations
to the range spaces of balls under DTW, and new approximation algorithms for
the $k$-median problem under DTW. We achieve our results by investigating
approximations of DTW that provide a trade-off between the provided accuracy
and amenability to known techniques. In particular, we observe that given $n$
curves under DTW, one can directly construct a metric that approximates DTW on
this set, permitting the use of the wealth of results on metric spaces for
clustering purposes. The resulting approximations are the first with polynomial
running time and achieve a very similar approximation factor as
state-of-the-art techniques. We apply our results to produce a practical
algorithm approximating $(k,\ell)$-median clustering under DTW.
