We give the first approximate-near-neighbor data structure for time series under the continuous Fr\'echet distance. Given a parameter \epsilon \in (0,1], the data structure can be used to preprocess n curves in R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fr\'echet distance 2+\epsilon, or answering that there exists no curve in the input within distance 1. In both cases, the answer is correct. Our data structure uses space in n\cdot O\left({\epsilon^{-1}}\right)^{k} + O(nm) and query time in O\left(k\right). We show that under some conditions the approximation factor achieved by our data structure is optimal in the cell-probe model of computation. Concretely, we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by O (L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}. Our data structure uses only a constant number of probes per query and does not have any dependency on L. In particular, this shows that our solution is optimal if only a constant number of probes is allowed. Our second positive result is a probabilistic data structure based on locality-sensitive hashing, which achieves space in O (nm) and query time in O (k), and which answers queries with an approximation factor in O (k). Both of our data structures make use of the concept of signatures, which were originally introduced for the problem of clustering time series under the Fr\'echet distance.
We give the first approximate-near-neighbor data structure for time series under the continuous Fr\'echet distance. Given a parameter \epsilon \in (0,1], the data structure can be used to preprocess n curves in R (aka time series), each of complexity m, to answer queries with a curve of complexity k by either returning a curve that lies within Fr\'echet distance 2+\epsilon, or answering that there exists no curve in the input within distance 1. In both cases, the answer is correct. Our data structure uses space in n\cdot O\left({\epsilon^{-1}}\right)^{k} + O(nm) and query time in O\left(k\right). We show that under some conditions the approximation factor achieved by our data structure is optimal in the cell-probe model of computation. Concretely, we show that for any data structure which achieves an approximation factor less than 2 and which supports curves of arclength at most L, uses a word size bounded by \mathcal{O (L^{1-\epsilon}) for some constant \epsilon>0, and answers the query using only a constant number of probes, the number of words used to store the data structure must be at least L^{\Omega(k)}. Our data structure uses only a constant number of probes per query and does not have any dependency on L. In particular, this shows that our solution is optimal if only a constant number of probes is allowed. Our second positive result is a probabilistic data structure based on locality-sensitive hashing, which achieves space in \mathcal{O (nm) and query time in \mathcal{O (k), and which answers queries with an approximation factor in \mathcal{O (k). Both of our data structures make use of the concept of signatures, which were originally introduced for the problem of clustering time series under the Fr\'echet distance.
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