We give the first approximate-near-neighbor data structures for time series under the continuous Fr\'echet distance. For an attainable approximation factor c>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 c , or answering that there exists no curve in the input within distance 1. In both cases, the answer is correct. Our first data structure achieves a (5+\epsilon) approximation factor, uses space in n\cdot O\left({\epsilon^{-1}}\right)^{k} + O(nm) and has query time in O\left(k\right). Our second data structure achieves a (2+\epsilon) approximation factor, uses space in n\cdot O\left(m{k\epsilon}\right)^{k} + O(nm) and has query time in O\left(k\cdot 2^k\right). As an indicator of hardness, 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 third 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). All 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 study approximate-near-neighbor data structures for time series under the continuous Fr\'echet distance. For an attainable approximation factor c>1 and a query radius r, an approximate-near-neighbor 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 cr , or answering that there exists no curve in the input within distance r. In both cases, the answer is correct. Our first data structure achieves a (5+\epsilon) approximation factor, uses space in n\cdot O\left({\epsilon^{-1}}\right)^{k} + O(nm) and has query time in O\left(k\right). Our second data structure achieves a (2+\epsilon) approximation factor, uses space in n\cdot O\left(m{k\epsilon}\right)^{k} + O(nm) and has query time in O\left(k\cdot 2^k\right). (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 third positive result is a probabilistic data structure based on locality-sensitive hashing, which achieves space in \mathcal{O}( n\log n+ nm) and query time in \mathcal{O}(k \log n ), and which answers queries with an approximation factor in \mathcal{O}(k). All 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 . In addition, we show lower bounds for this problem. Consider any data structure which achieves an approximation factor less than 2 and which supports curves of arclength up to L and answers the query using only a constant number of probes. We show that under reasonable assumptions on the word size any such data structure needs space in L^{\Omega(k) .
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