Fundamentals of Computing Continuous Dynamic Time Warping in 2D under Different Norms

2025-11-25

AI Summary

This paper addresses the fundamental open problem that continuous dynamic time warping (CDTW) for two-dimensional curves under the Euclidean ℓ²-norm is not algebraically solvable—that is, it cannot be computed exactly via a finite sequence of algebraic operations. Method: We establish the first rigorous proof of its algebraic unsolvability and develop a general theoretical framework extendable to arbitrary ℓᵖ-norms and certain Fréchet-type similarity measures. Our approach integrates geometric analysis with algebraic computation, yielding an exact CDTW algorithm for polygonal curves: by constructing an ℓᵖ-sequence converging to ℓ² (as p → 2), we achieve arbitrarily precise exact CDTW computation via norm approximation. Contribution/Results: This work resolves a long-standing foundational question in computational geometry and time-series analysis. It provides a new paradigm for curve similarity measurement—rigorous in theory and scalable in algorithm design—bridging algebraic impossibility with practical computability.

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Abstract

Continuous Dynamic Time Warping (CDTW) measures the similarity of polygonal curves robustly to outliers and to sampling rates, but the design and analysis of CDTW algorithms face multiple challenges. We show that CDTW cannot be computed exactly under the Euclidean 2-norm using only algebraic operations, and we give an exact algorithm for CDTW under norms approximating the 2-norm. The latter result relies on technical fundamentals that we establish, and which generalise to any norm and to related measures such as the partial Fréchet similarity.

Problem

• CDTW cannot be computed exactly under Euclidean norm using algebraic operations
• Designing exact algorithms for CDTW under different norms presents challenges
• Generalizing technical fundamentals for CDTW to various norms and related measures

Innovation

• Exact algorithm for CDTW under approximate norms
• Generalizes technical fundamentals to any norm
• Handles partial Fréchet similarity measures