Binary Dynamic Time Warping in Linear Time

TL;DR

This paper presents a new linear-time algorithm for computing binary dynamic time warping (DTW) distances, significantly improving efficiency over previous methods, especially for run-length encoded data.

Contribution

It introduces a linear-time algorithm for binary DTW and an efficient method for run-length encoded sequences, surpassing prior bounds.

Findings

Binary DTW can be computed in linear time.

Run-length encoded sequences allow for near-linear computation.

Improves previous quadratic bounds for specific cases.

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.