A Simple Efficient Approximation Algorithm for Dynamic Time Warping

Dynamic time warping (DTW) is a widely used curve similarity measure. We present a simple and efficient (1 + is an element of)approximation algorithm for DTW between a pair of point sequences, say, P and Q, each of which is sampled from a curve. We prove that the running time of the algorithm is O (k(2) /is an element of n log sigma) for a pair of k-packed curves with a total of n points, assuming that the spreads of P and Q are bounded by sigma. The spread of a point set is the ratio of the maximum to the minimum pairwise distance, and a curve is called k-packed if the length of its intersection with any disk of radius r is at most kr. Although an algorithm with similar asymptotic time complexity was presented in [1], our algorithm is considerably simpler and more e ffi cient in practice. We have implemented our algorithm. Our experiments on both synthetic and real_world data sets show that it is an order of magnitude faster than the standard exact DP algorithm on point sequences of length 5,000 or more while keeping the approximation error within 5-10%. We demonstrate the e ffi cacy of our algorithm by using it in two applications | computing the k most similar trajectories to a query trajectory, and running the iterative closest point method for a pair of trajectories. We show that we can achieve 8-12 times speedup using our algorithm as a subroutine in these applications, without compromising much in accuracy.