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Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time

Authors Kevin Buchin , André Nusser , Sampson Wong

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Author Details

Kevin Buchin
  • Department of Computer Science, TU Dortmund, Germany
André Nusser
  • BARC, University of Copenhagen, Denmark
Sampson Wong
  • School of Computer Science, University of Sydney, Australia

Funding

  • Nusser, André: Part of this research was conducted while the author was at Saarbrücken Graduate School of Computer Science and Max Planck Institute for Informatics. The author is supported by the VILLUM Foundation grant 16582.

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Kevin Buchin, André Nusser, and Sampson Wong. Computing Continuous Dynamic Time Warping of Time Series in Polynomial Time. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 22:1-22:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SoCG.2022.22

Abstract

Dynamic Time Warping is arguably the most popular similarity measure for time series, where we define a time series to be a one-dimensional polygonal curve. The drawback of Dynamic Time Warping is that it is sensitive to the sampling rate of the time series. The Fréchet distance is an alternative that has gained popularity, however, its drawback is that it is sensitive to outliers.
Continuous Dynamic Time Warping (CDTW) is a recently proposed alternative that does not exhibit the aforementioned drawbacks. CDTW combines the continuous nature of the Fréchet distance with the summation of Dynamic Time Warping, resulting in a similarity measure that is robust to sampling rate and to outliers. In a recent experimental work of Brankovic et al., it was demonstrated that clustering under CDTW avoids the unwanted artifacts that appear when clustering under Dynamic Time Warping and under the Fréchet distance. Despite its advantages, the major shortcoming of CDTW is that there is no exact algorithm for computing CDTW, in polynomial time or otherwise.
In this work, we present the first exact algorithm for computing CDTW of one-dimensional curves. Our algorithm runs in time 𝒪(n⁵) for a pair of one-dimensional curves, each with complexity at most n. In our algorithm, we propagate continuous functions in the dynamic program for CDTW, where the main difficulty lies in bounding the complexity of the functions. We believe that our result is an important first step towards CDTW becoming a practical similarity measure between curves.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational Geometry
  • Curve Similarity
  • Fréchet distance
  • Dynamic Time Warping
  • Continuous Dynamic Time Warping

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