Tight lower bounds for Dynamic Time Warping

Dynamic Time Warping (DTW) is a popular similarity measure for aligning and comparing time series. Due to DTW’s high computation time, lower bounds are often employed to screen poor matches. Many alternative lower bounds have been proposed, providing a range of different trade-offs between tightness and computational efficiency. LB Keogh provides a useful trade-off in many applications. Two recent lower bounds, LB Improved and LB Enhanced, are substantially tighter than LB Keogh. All three have the same worst case computational complexity—linear with respect to series length and constant with respect to window size. We present four new DTW lower bounds in the same complexity class. LB Petitjean is substantially tighter than LB Improved, with only modest additional computational overhead. LB Webb is more efficient than LB Improved, while often providing a tighter bound. LB Webb is always tighter than LB Keogh. The parameter free LB Webb is usually tighter than LB Enhanced. A parameterized variant, LB Webb Enhanced, is always tighter than LB Enhanced. A further variant, LB Webb∗, is useful for some constrained distance functions. In extensive experiments, LB Webb proves to be very effective for nearest neighbor search.