用于短Fréchet查询的次线性数据结构
Sublinear data structures for short Fréchet queries
摘要：
我们研究度量数据结构曲线加倍空间，如运动物体的轨迹在欧几里德$\mathbb{R}^d$，其中两条曲线之间的距离是使用离散Fréchet距离测量。我们在\emph{asymmetric}设置中设计数据结构，其中输入是一条曲线（或一组$n$曲线），每条曲线的复杂性为$m$，查询具有复杂性曲线$k\ll m$。我们发现，存在近似的数据结构，是独立的输入大小$N = d \cdot n \cdot m$，我们研究如何保持它们的动态，如果输入流中给出。具体而言，我们研究了两种类型的数据结构：（i）距离预言机，其中的任务是存储输入曲线的压缩版本，它可以用于回答查询曲线到输入曲线的距离的查询，以及（ii）最近邻数据结构，其中的任务是预处理一组输入曲线，以回答最接近查询曲线的输入曲线的查询。在这两种情况下，我们感兴趣的是近似。对于嵌入在具有常数$d$的Euclidean $\mathbb{R}^d$中的曲线，我们的距离预言机使用$\mathcal{O}((k \log(ε^{-1}) ε^{-d})^k)$中的空间（$ε$是精度参数）。oracle在$\mathcal{O}(k^2)$中及时执行$(1+ε)$ -近似查询，并且是确定性的。我们将展示如何使用多对数附加内存在流中保持这种距离Oracle。在流中，我们可以在$\mathcal{O}(k^4 \log^2 m)$时间内动态地回答到目前为止看到的流部分的距离查询。我们将我们的技术应用于第二个问题，近似近邻（ANN）的数据结构，并实现了指数级的改善，在依赖于输入曲线的复杂性相比，现有技术。
英文摘要：
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 Fréchet 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 $\mathcal{O}((k \log(ε^{-1}) ε^{-d})^k)$ ($ε$ is the precision parameter). The oracle performs $(1+ε)$-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.
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