We study data structures for storing a set of polygonal curves in Rd
such that, given a query curve, we can efficiently retrieve similar curves from
the set, where similarity is measured using the discrete Fr\'echet distance or
the dynamic time warping distance. To this end we devise the first
locality-sensitive hashing schemes for these distance measures. A major
challenge is posed by the fact that these distance measures internally optimize
the alignment between the curves. We give solutions for different types of
alignments including constrained and unconstrained versions. For unconstrained
alignments, we improve over a result by Indyk from 2002 for short curves. Let
n be the number of input curves and let m be the maximum complexity of a
curve in the input. In the particular case where m≤4dαlogn, for some fixed α>0, our solutions imply an approximate near-neighbor
data structure for the discrete Fr\'echet distance that uses space in
O(n1+αlogn) and achieves query time in O(nαlog2n) and
constant approximation factor. Furthermore, our solutions provide a trade-off
between approximation quality and computational performance: for any parameter
k∈[m], we can give a data structure that uses space in O(22kmk−1nlogn+nm), answers queries in O(22kmklogn) time and achieves
approximation factor in O(m/k).Comment: Proc. of 33rd International Symposium on Computational Geometry
(SoCG), 201