The inherent time complexity and an efficient algorithm for subsequence matching problem

Abstract

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.

TSM-Bench: Benchmarking Time Series Database Systems for Monitoring Applications

“…Another aspect that impacts the indexing result is the similarity measure. There are a number of distance metrics for time series [15] including Euclidean Distance (ED) and its variants [8,47], Dynamic Time Warping (DTW) [9,67], Longest Common Subsequence (LCSS) [67]. Using the Euclidean Distance as a similarity measure between segment codes is sufficient for our application, as it can easily handle our highly correlated and generally time-aligned hydrological time series, where similar segments are in phase.…”