We give the first subquadratic-time approximation schemes for dynamic time
warping (DTW) and edit distance (ED) of several natural families of point
sequences in Rd, for any fixed d≥1. In particular, our
algorithms compute (1+ε)-approximations of DTW and ED in time
near-linear for point sequences drawn from k-packed or k-bounded curves, and
subquadratic for backbone sequences. Roughly speaking, a curve is
κ-packed if the length of its intersection with any ball of radius r
is at most κ⋅r, and a curve is κ-bounded if the sub-curve
between two curve points does not go too far from the two points compared to
the distance between the two points. In backbone sequences, consecutive points
are spaced at approximately equal distances apart, and no two points lie very
close together. Recent results suggest that a subquadratic algorithm for DTW or
ED is unlikely for an arbitrary pair of point sequences even for d=1. Our
algorithms work by constructing a small set of rectangular regions that cover
the entries of the dynamic programming table commonly used for these distance
measures. The weights of entries inside each rectangle are roughly the same, so
we are able to use efficient procedures to approximately compute the cheapest
paths through these rectangles