A faster reduction of the dynamic time warping distance to the longest increasing subsequence length

TL;DR

This paper introduces a novel reduction of the dynamic time warping (DTW) distance to the longest increasing subsequence (LIS) problem, enabling the application of efficient LIS algorithms to time series similarity analysis.

Contribution

It presents a method to represent DTW distance as an LIS problem, allowing faster algorithms for DTW-related computations using existing LIS techniques.

Findings

DTW distance can be represented by LIS length of an integer sequence.

The integer sequence length is O(c n^2), with c being the maximum dissimilarity.

Time-efficient algorithms for DTW problems are developed using LIS techniques.

Abstract

The similarity between a pair of time series, i.e., sequences of indexed values in time order, is often estimated by the dynamic time warping (DTW) distance, instead of any in the well-studied family of measures including the longest common subsequence (LCS) length and the edit distance. Although it may seem as if the DTW and the LCS(-like) measures are essentially different, we reveal that the DTW distance can be represented by the longest increasing subsequence (LIS) length of a sequence of integers, which is the LCS length between the integer sequence and itself sorted. For a given pair of time series of length $n$ such that the dissimilarity between any elements is an integer between zero and $c$, we propose an integer sequence that represents any substring-substring DTW distance as its band-substring LIS length. The length of the produced integer sequence is $O(c n^2)$, which can be translated to $O(n^2)$ for constant dissimilarity functions. To demonstrate that techniques developed under the LCS(-like) measures are directly applicable to analysis of time series via our reduction of DTW to LIS, we present time-efficient algorithms for DTW-related problems utilizing the semi-local sequence comparison technique developed for LCS-related problems.