ANN for Time Series Under the Fréchet Distance

Abstract

We study approximate-near-neighbor data structures for time series under the continuous Fréchet 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 Fréchet 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 Fréchet 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 arc length 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 $L^{\varOmega (k)}$.

A full version of this paper can be found on arXiv [9]. We thank Karl Bringmann and André Nusser for useful discussions on the topic of this paper. Special thanks go to the anonymous reviewer who pointed out an error in an earlier version of the manuscript, and to Andrea Cremer for careful reading.