A Constant-Factor Approximation for Continuous Dynamic Time Warping in 2D
This provides the first polynomial-time constant-factor approximation for 2D CDTW, a problem with no prior such guarantees, benefiting applications in shape matching and time series analysis.
The paper presents the first constant-factor approximation algorithm for Continuous Dynamic Time Warping (CDTW) in 2D, achieving a 5-approximation in O(n^5) time under the 1-norm, and a (5+ε)-approximation in O(n^5/ε^{1/2}) time under any fixed norm.
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.