动态时间弯曲距离的高效交互计算
Towards Efficient Interactive Computation of Dynamic Time Warping Distance
摘要：
动态时间规整（DTW）是一种广泛使用的方法，它允许我们有效地比较两个速度不同的时间序列。给定长度分别为$m$和$n$的两个字符串$A$和$B$，存在计算$A$和$B$的DTW距离以及$Θ(mn)$时间和空间中的最佳比对的基本动态编程算法。在本文中，我们解决了动态字符串的DTW距离的交互式计算问题，表示为$\mathrm{D^2TW}$，其中字符式编辑操作（插入，删除，替换）可以在字符串的任意位置进行。令$M$和$N$分别为$A$和$B$的游程长度编码（RLE）的大小。我们提出了一种算法$\mathrm{D^2TW}$，占用$Θ(mN+nM)$空间，并使用$O(m+n+\#_{\mathrm{chg}}) \subseteq O(mN + nM)$时间更新一个紧凑的差分表示$\mathit{DS}$的DP表每个编辑操作，其中$\#_{\mathrm{chg}}$表示在$\mathit{DS}$中的单元格的数量，其值在编辑操作后改变。我们的方法是至少一样有效的算法最近提出的Froese等。运行在$Θ(mN + nM)$时间，更快，当$\#_{\mathrm{chg}}$小于$O(mN + nM)$，这是我们的初步实验表明，在大多数情况下可能是这种情况。
英文摘要：
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 for $A$ and $B$ together with an optimal alignment in $Θ(mn)$ time and space. In this paper, we tackle the problem of interactive computation of the DTW distance for dynamic strings, denoted $\mathrm{D^2TW}$, 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 $\mathrm{D^2TW}$ that occupies $Θ(mN+nM)$ space and uses $O(m+n+\#_{\mathrm{chg}}) \subseteq O(mN + nM)$ time to update a compact differential representation $\mathit{DS}$ of the DP table per edit operation, where $\#_{\mathrm{chg}}$ denotes the number of cells in $\mathit{DS}$ 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 $Θ(mN + nM)$ 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.
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