Sublinear data structures for short Fr\'echet queries

TL;DR

This paper develops sublinear, approximate data structures for short curve queries under the discrete Fréchet distance in doubling spaces, enabling efficient streaming and dynamic updates.

Contribution

It introduces novel approximate data structures for short curve queries that are independent of input size and can be maintained dynamically in streaming settings.

Findings

Distance oracle uses space independent of total input size.

Queries are answered with a (1+ε) approximation in O(k^2) time.

Achieves exponential improvement over previous methods for nearest-neighbor queries.

Abstract

We study metric data structures for curves in doubling spaces, such as trajectories of moving objects in Euclidean $\mathbb{R}^d$, where the distance between two curves is measured using the discrete Fr\'echet distance. We design data structures in an \emph{asymmetric} setting where the input is a curve (or a set of $n$ curves) each of complexity $m$ and the queries are with curves of complexity $k\ll m$. We show that there exist approximate data structures that are independent of the input size $N = d \cdot n \cdot m$ and we study how to maintain them dynamically if the input is given in the stream. Concretely, we study two types of data structures: (i) distance oracles, where the task is to store a compressed version of the input curve, which can be used to answer queries for the distance of a query curve to the input curve, and (ii) nearest-neighbor data structures, where the task is to preprocess a set of input curves to answer queries for the input curve closest to the query curve. In both cases we are interested in approximation. For curves embedded in Euclidean $\mathbb{R}^d$ with constant $d$, our distance oracle uses space in $\mathcal{O}((k \log(\epsilon^{-1}) \epsilon^{-d})^k)$ ($\epsilon$ is the precision parameter). The oracle performs $(1+\epsilon)$-approximate queries in time in $\mathcal{O}(k^2)$ and is deterministic. We show how to maintain this distance oracle in the stream using polylogarithmic additional memory. In the stream, we can dynamically answer distance queries to the portion of the stream seen so far in $\mathcal{O}(k^4 \log^2 m)$ time. We apply our techniques to the second problem, approximate near neighbor (ANN) data structures, and achieve an exponential improvement in the dependency on the complexity of the input curves compared to the state of the art.