Pattern matching under DTW distance

TL;DR

This paper develops efficient algorithms for pattern matching under the DTW distance, crucial for biological data analysis, providing exact and approximate solutions with improved time complexities.

Contribution

It introduces new algorithms for pattern matching under DTW with linear and polynomial time complexities, including an approximation method for general metrics.

Findings

O(m + n)-time algorithm for DTW distance at most 1

O(kmn)-time algorithm for DTW distance at most k

Derived approximation algorithm for general metrics

Abstract

In this work, we consider the problem of pattern matching under the dynamic time warping (DTW) distance motivated by potential applications in the analysis of biological data produced by the third generation sequencing. To measure the DTW distance between two strings, one must "warp" them, that is, double some letters in the strings to obtain two equal-lengths strings, and then sum the distances between the letters in the corresponding positions. When the distances between letters are integers, we show that for a pattern P with m runs and a text T with n runs: 1. There is an O(m + n)-time algorithm that computes all locations where the DTW distance from P to T is at most 1; 2. There is an O(kmn)-time algorithm that computes all locations where the DTW distance from P to T is at most k. As a corollary of the second result, we also derive an approximation algorithm for general metrics on the alphabet.