# The One-Way Communication Complexity of Dynamic Time Warping Distance

## TL;DR

This paper determines the exact randomized one-way communication complexity of approximating the Dynamic Time Warping distance between strings, providing both upper and lower bounds, and explores implications for linear sketches.

## Contribution

It establishes tight bounds for the one-way communication complexity of DTW approximation and extends results to various metric spaces and linear sketching.

## Key findings

- Efficient one-way protocol using O(n/) bits for -approximate DTW.
- Lower bound of (n/) bits for the same problem.
- Linear sketches of DTW require (n) size.

## Abstract

We resolve the randomized one-way communication complexity of Dynamic Time Warping (DTW) distance. We show that there is an efficient one-way communication protocol using $\widetilde{O}(n/\alpha)$ bits for the problem of computing an $\alpha$-approximation for DTW between strings $x$ and $y$ of length $n$, and we prove a lower bound of $\Omega(n / \alpha)$ bits for the same problem. Our communication protocol works for strings over an arbitrary metric of polynomial size and aspect ratio, and we optimize the logarithmic factors depending on properties of the underlying metric, such as when the points are low-dimensional integer vectors equipped with various metrics or have bounded doubling dimension. We also consider linear sketches of DTW, showing that such sketches must have size $\Omega(n)$.