Binary Dynamic Time Warping in Linear Time

Abstract

Dynamic time warping distance (DTW) is a widely used distance measure between time series $x,y\in {\Sigma }^n$. It was shown by Abboud, Backurs, and Williams that in the \emph{binary case}, where $|\Sigma |=2$, DTW can be computed in time $O(n^{1.87})$. We improve this running time $O(n)$. Moreover, if $x$ and $y$ are run-length encoded, then there is an algorithm running in time $\tilde{O}(k+\ell )$, where $k$ and $\ell$ are the number of runs in $x$ and $y$, respectively. This improves on the previous best bound of $O(k\ell )$ due to Dupont and Marteau.