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# Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves

conference contribution

posted on 2023-11-29, 18:22 authored by Karl Bringmann, Sándor Kisfaludi-Bak, Marvin Künnemann, Dániel MarxDániel Marx, André NusserThe Dynamic Time Warping (DTW) distance is a popular measure of similarity for a variety of sequence data. For comparing polygonal curves π, σ in ℝ^d, it provides a robust, outlier-insensitive alternative to the Fréchet distance. However, like the Fréchet distance, the DTW distance is not invariant under translations. Can we efficiently optimize the DTW distance of π and σ under arbitrary translations, to compare the curves' shape irrespective of their absolute location?
There are surprisingly few works in this direction, which may be due to its computational intricacy: For the Euclidean norm, this problem contains as a special case the geometric median problem, which provably admits no exact algebraic algorithm (that is, no algorithm using only addition, multiplication, and k-th roots). We thus investigate exact algorithms for non-Euclidean norms as well as approximation algorithms for the Euclidean norm.
For the L₁ norm in ℝ^d, we provide an ????(n^{2(d+1)})-time algorithm, i.e., an exact polynomial-time algorithm for constant d. Here and below, n bounds the curves' complexities. For the Euclidean norm in ℝ², we show that a simple problem-specific insight leads to a (1+ε)-approximation in time ????(n³/ε²). We then show how to obtain a subcubic ????̃(n^{2.5}/ε²) time algorithm with significant new ideas; this time comes close to the well-known quadratic time barrier for computing DTW for fixed translations. Technically, the algorithm is obtained by speeding up repeated DTW distance estimations using a dynamic data structure for maintaining shortest paths in weighted planar digraphs. Crucially, we show how to traverse a candidate set of translations using space-filling curves in a way that incurs only few updates to the data structure.
We hope that our results will facilitate the use of DTW under translation both in theory and practice, and inspire similar algorithmic approaches for related geometric optimization problems.

## History

## Preferred Citation

Karl Bringmann, Sándor Kisfaludi-Bak, Marvin Künnemann, Dániel Marx and André Nusser. Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves. In: International Symposium on Computational Geometry (SoCG). 2022.## Primary Research Area

- Algorithmic Foundations and Cryptography

## Name of Conference

International Symposium on Computational Geometry (SoCG)## Legacy Posted Date

2022-10-13## Open Access Type

- CC

## BibTeX

@inproceedings{cispa_all_3808, title = "Dynamic Time Warping Under Translation: Approximation Guided by Space-Filling Curves", author = "Bringmann, Karl and Kisfaludi-Bak, Sándor and Künnemann, Marvin and Marx, Dániel and Nusser, André", booktitle="{International Symposium on Computational Geometry (SoCG)}", year="2022", }## Usage metrics

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