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

2026-05-07

AI Summary

This work addresses the absence of efficient constant-factor approximation algorithms for two-dimensional continuous dynamic time warping (CDTW) by proposing the first polynomial-time constant-factor approximation algorithm. Specifically, the authors design an O(n⁵)-time 5-approximation algorithm under the ℓ¹ norm and extend this result to any fixed norm—including the Euclidean norm—via polygonal norm approximation techniques, achieving a (5+ε)-approximation in O(n⁵/ε¹/²) time. This study constitutes the first demonstration that constant-factor approximations for two-dimensional CDTW are attainable within polynomial time, thereby filling a notable theoretical gap in the field.

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.

Problem

• Continuous Dynamic Time Warping
• approximation algorithm
• polygonal curves
• algorithmic complexity
• 2D

Innovation

• Continuous Dynamic Time Warping
• constant-factor approximation
• polygonal curves
• computational geometry
• normed spaces