A Constant-Factor Approximation for Continuous Dynamic Time Warping in 2D

TL;DR

This paper introduces the first polynomial-time constant-factor approximation algorithm for 2D Continuous Dynamic Time Warping (CDTW), extending to all polygonal norms and achieving near-optimal approximation ratios.

Contribution

It presents the first polynomial-time constant-factor approximation algorithm for 2D CDTW and extends it to all polygonal norms, including Euclidean.

Findings

First 5-approximation algorithm for 2D CDTW with O(n^5) time

Extension to all polygonal norms with (5+ε)-approximation

Applicable to Euclidean 2-norm with near-optimal approximation

Abstract

Continuous Dynamic Time Warping (CDTW) is a robust similarity measure for polygonal curves that has recently found a variety of applications. Despite its practical use, not much is known about the algorithmic complexity of computing it in 2D, especially when one requires either an exact solution or strong approximation guarantees. We fill this gap by introducing a $5$-approximation algorithm with running time $O(n^5)$ under the 1-norm. This is the first constant-factor approximation for 2D CDTW with polynomial running time. We extend our algorithm to all polygonal norms on $\mathbb{R}^2$, which we subsequently use in order to achieve a $(5+\varepsilon)$-approximation with time complexity $O(n^5 / \varepsilon^{1/2})$ for CDTW in 2D under any fixed norm. The latter result in particular includes the usual Euclidean 2-norm.