Towards Efficient Interactive Computation of Dynamic Time Warping Distance

Abstract

The dynamic time warping (DTW) is a widely-used method that allows us to efficiently compare two time series that can vary in speed. Given two strings A and B of respective lengths m and n, there is a fundamental dynamic programming algorithm that computes the DTW distance $\mathsf {dtw}(A,B)$ for A and B together with an optimal alignment in $\varTheta (mn)$ time and space. In this paper, we tackle the problem of interactive computation of the DTW distance for dynamic strings, denoted $\mathbf {D^2TW}$, where character-wise edit operation (insertion, deletion, substitution) can be performed at an arbitrary position of the strings. Let M and N be the sizes of the run-length encoding (RLE) of A and B, respectively. We present an algorithm for $\mathbf {D^2TW}$ that occupies $\varTheta (mN+nM)$ space and uses $O(m+n+\#_{\mathrm {chg}}) \subseteq O(mN + nM)$ time to update a compact differential representation $DS$ of the DP table per edit operation, where $\#_{\mathrm {chg}}$ denotes the number of cells in $DS$ whose values change after the edit operation. Our method is at least as efficient as the algorithm recently proposed by Froese et al. running in $\varTheta (mN + nM)$ time, and is faster when $\#_{\mathrm {chg}}$ is smaller than $O(mN + nM)$ which, as our preliminary experiments suggest, is likely to be the case in the majority of instances.