Dynamic Time Warping: Breaking the Quadratic Barrier

Dynamic Time Warping (DTW) is one of the basic similarity measures between curves or general temporal sequences (e.g., time series) that are represented as sequence of points in some metric space pX, distq. The DTW measure is massively used in many practical fields of computer science, and computing the DTW between two sequences is a classical problem in P. Despite extensive efforts to find more efficient algorithms, the best-known algorithm for computing DTW is a long-standing dynamic programming algorithm that requires quadratic runtime, which is believed to be optimal, even for the one-dimensional case X “ R, which is perhaps one of the most used in practice. In this paper, we break the nearly 50 years old quadratic time bound for computing DTW between two sequences of n points in R, by presenting a deterministic algorithm that runs in O ` n log log logn{ log logn ̆ time. Our algorithm can be extended to work also for higher dimensional spaces X “ R, for any constant d, when the underlying distance-metric dist is polyhedral (e.g., L1, L8).