A Faster Reduction of the Dynamic Time Warping Distance to the Longest Increasing Subsequence Length

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(cn(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 DTWto LIS, we present time-efficient algorithms for DTW-related problems utilizing the semi-local sequence comparison technique developed for LCS-related problems.