近似离散Fréchet距离的数据结构
Data Structures for Approximate Discrete Fréchet Distance
AI总结：
本文针对欧几里得空间和一般测地度量空间中的c-packed曲线，提出了近线性时间的近似连续Fréchet距离静态算法，并构建了首个支持快速查询离散Fréchet距离的数据结构及多种扩展。
摘要：
Fréchet距离是衡量曲线P和Q之间距离的一种常用度量。条件性下界禁止在强次二次时间内进行(1+ε)近似Fréchet距离计算，即使在使用任意多项式级时间和空间对P进行预处理的情况下也是如此。因此，人们已经在现实输入假设下研究了Fréchet距离，例如假设两条曲线都是c-packed的。在本文中，我们研究了欧几里得空间R^d和一般测地度量空间X中的c-packed曲线。在R^d中，我们提供了一种近线性时间的静态算法，用于计算c-packed曲线之间的(1+ε)近似连续Fréchet距离。我们的算法对维度d具有线性依赖关系，而先前的算法对d具有指数依赖关系。在一般测地度量空间X中，先前知之甚少。我们提供了该模型下的第一个数据结构，从而也是第一个算法。给定一个具有n个顶点的c-packed输入曲线P，我们在O(n log n)时间内对其进行预处理，使得给定包含常数ε和具有m个顶点的曲线Q的查询时，我们可以在n中多对数时间、m、1/ε和现实主义参数c中线性时间内返回P和Q之间离散Fréchet距离的(1+ε)近似。最后，我们展示了数据结构的若干扩展；支持对P的动态扩展/截断更新，回答地图匹配查询，以及回答Hausdorff距离查询。
英文摘要：
The Fréchet distance is a popular distance measure between curves $P$ and $Q$. Conditional lower bounds prohibit $(1 + \varepsilon)$-approximate Fréchet distance computations in strongly subquadratic time, even when preprocessing $P$ using any polynomial amount of time and space. As a consequence, the Fréchet distance has been studied under realistic input assumptions, for example, assuming both curves are $c$-packed. In this paper, we study $c$-packed curves in Euclidean space $\mathbb R^d$ and in general geodesic metrics $\mathcal X$. In $\mathbb R^d$, we provide a nearly-linear time static algorithm for computing the $(1+\varepsilon)$-approximate continuous Fréchet distance between $c$-packed curves. Our algorithm has a linear dependence on the dimension $d$, as opposed to previous algorithms which have an exponential dependence on $d$. In general geodesic metric spaces $\mathcal X$, little was previously known. We provide the first data structure, and thereby the first algorithm, under this model. Given a $c$-packed input curve $P$ with $n$ vertices, we preprocess it in $O(n \log n)$ time, so that given a query containing a constant $\varepsilon$ and a curve $Q$ with $m$ vertices, we can return a $(1+\varepsilon)$-approximation of the discrete Fréchet distance between $P$ and $Q$ in time polylogarithmic in $n$ and linear in $m$, $1/\varepsilon$, and the realism parameter $c$. Finally, we show several extensions to our data structure; to support dynamic extend/truncate updates on $P$, to answer map matching queries, and to answer Hausdorff distance queries.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
