The Inherent Time Complexity and An Efficient Algorithm for Subsequence Matching Problem

Subsequence matching is an important and fundamental problem on time series data. This paper studies the inherent time complexity of the subsequence matching problem and designs a more efficient algorithm for solving the problem. Firstly, it is proved that the subsequence matching problem is incomputable in time O (n^1−δ) even allowing polynomial time preprocessing if the hypothesis SETH is true, where n is the size of the input time series and 0 ≤ δ < 1, i.e., the inherent complexity of the subsequence matching problem is ω (n^{1− δ}). Secondly, an efficient algorithm for subsequence matching problem is proposed. In order to improve the efficiency of the algorithm, we design a new summarization method as well as a novel index for series data. The proposed algorithm supports both Euclidean Distance and DTW distance with or without z-normalization. Experimental results show that the proposed algorithm is up to about 3 ∼ 10 times faster than the state of art algorithm on the constrained z-normalized Euclidean Distance and DTW distance, and is up to 7 ∼ 12 times faster on Euclidean Distance.