An Amortized O(1) Lower Bound for Dynamic Time Warping in Motif Discovery

Motif discovery is a critical operation for analyzing series data in many applications. Recent works demonstrate the importance of finding motifs with Dynamic Time Warping. However, existing algorithms spend most of their time in computing lower bounds of Dynamic Time Warping to filter out the unpromising candidates. Specifically, the time complexity for computing these lower bounds is $O(L)$ for each pair of subsequences, where $L$ is the length of the motif (subsequences). This paper proposes two new lower bounds, called $LB_{f}$ and $LB_{M}$, both of them only cost amortized $O(1)$ time for each pair of subsequences. On real datasets, the proposed lower bounds are at least one magnitude faster than the state-of-the-art lower bounds used in motif discovery while still keeping satisfying effectiveness. Based on these faster lower bounds, this paper designs an efficient motif discovery algorithm that significantly reduces the cost of lower bounds. The experiments conducted on real datasets show the proposed algorithm is 5.6 times faster than the state-of-the-art algorithms on average.