$(2+\epsilon)$-ANN for 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 $\mathbb{R}$ (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 $n\cdot \mathcal{O}\left({\epsilon^{-1}}\right)^{k} + \mathcal{O}(nm)$ and has query time in $\mathcal{O}\left(k\right)$. Our second data structure achieves a $(2+\epsilon)$ approximation factor, uses space in $n\cdot \mathcal{O}\left(\frac{m}{k\epsilon}\right)^{k} + \mathcal{O}(nm)$ and has query time in $\mathcal{O}\left(k\cdot 2^k\right)$. 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 $L^{\Omega(k)}$.