# Dynamic time warping under translation: approximation guided by space-filling curves

JOURNAL OF COMPUTATIONAL GEOMETRY（2023）

Abstract

The Dynamic Time Warping (DTW) distance is a popular measure of similar-ity for a variety of sequence data. For comparing polygonal curves 7r, sigma in Rd, it provides a robust, outlier-insensitive alternative to the Frechet distance. However, like the Frechet dis-tance, the DTW distance is not invariant under translations. Can we efficiently optimize the DTW distance of 7r and sigma under arbitrary translations, to compare the curves' shape irre-spective 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 al-gorithm (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' complexi-ties. For the Euclidean norm in Rd with d is an element of O(1), we show that a simple problem-sp ecific insight leads to a (1 + epsilon)-approximation in time O(n3/epsilon d). We then show how to obtain a subcubic Oe(n2.5/epsilon d) 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 dy-namic 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.

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