TY - JOUR

T1 - DYNAMIC TIME WARPING UNDER TRANSLATION

T2 - APPROXIMATION GUIDED BY SPACE-FILLING CURVES

AU - Bringmann, Karl

AU - Kisfaludi-Bak, Sándor

AU - Künnemann, Marvin

AU - Marx, Dániel

AU - Nusser, André

N1 - Publisher Copyright: © 2023, Carleton University. All rights reserved.

PY - 2023

Y1 - 2023

N2 - The Dynamic Time Warping (DTW) distance is a popular measure of similarity for a variety of sequence data. For comparing polygonal curves π, σ in Rd, 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 Rd, we provide an O (n2(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 Rd with (Formula Presented), we show that a simple problem-specific insight leads to a (1 + ε)-approximation in time O (n3/εd). We then show how to obtain a subcubic (Formula Presented) 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 facilitate the use of DTW under translation both in theory and practice, and inspire similar algorithmic approaches for related geometric optimization problems.

AB - The Dynamic Time Warping (DTW) distance is a popular measure of similarity for a variety of sequence data. For comparing polygonal curves π, σ in Rd, 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 Rd, we provide an O (n2(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 Rd with (Formula Presented), we show that a simple problem-specific insight leads to a (1 + ε)-approximation in time O (n3/εd). We then show how to obtain a subcubic (Formula Presented) 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 facilitate the use of DTW under translation both in theory and practice, and inspire similar algorithmic approaches for related geometric optimization problems.

U2 - 10.20382/jocg.v14i2a6

DO - 10.20382/jocg.v14i2a6

M3 - Journal article

AN - SCOPUS:85178942345

VL - 14

SP - 83

EP - 107

JO - Journal of Computational Geometry

JF - Journal of Computational Geometry

SN - 1920-180X

IS - 2

ER -