再论曲线的Fréchet距离
Fréchet Distance for Curves, Revisited
摘要：
$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\eps}{\varepsilon} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$我们重新讨论了在$L_1$、$L_2$和$L_\infty$范数下计算多边形曲线之间的Fréchet距离的问题，重点是离散Fréchet距离，其中只考虑顶点之间的距离。我们开发了两个自然类的曲线的有效算法。特别地，给定两条各自具有$n$顶点的多边形曲线，如果其中一条曲线是\emph{$κ$-bounded}，则可以在三维中在大致$O(nκ^3\log n/\eps^3)$时间内计算它们的离散Fréchet距离的$\eps$ -近似。以前，只有$κ$近似算法是已知的。如果这两条曲线都是所谓的\emph{\backbone~curves}，在分子生物学中被广泛用于蛋白质骨架的建模，我们可以在二维中近似线性时间的$\eps$ -近似它们的弗雷歇距离，在三维中近似$O(n^{4/3}\log nm)$时间。在第二部分中，我们提出了一个精确计算Fréchet距离的伪输出敏感算法。算法的复杂度是一个我们称为\emph{\bwnumber{}}的量的函数，在最坏的情况下是二次的，但在实践中往往要小得多。
英文摘要：
$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\eps}{\varepsilon} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)}$ We revisit the problem of computing Fréchet distance between polygonal curves under $L_1$, $L_2$, and $L_\infty$ norms, focusing on discrete Fréchet distance, where only distance between vertices is considered. We develop efficient algorithms for two natural classes of curves. In particular, given two polygonal curves of $n$ vertices each, a $\eps$-approximation of their discrete Fréchet distance can be computed in roughly $O(nκ^3\log n/\eps^3)$ time in three dimensions, if one of the curves is \emph{$κ$-bounded}. Previously, only a $κ$-approximation algorithm was known. If both curves are the so-called \emph{\backbone~curves}, which are widely used to model protein backbones in molecular biology, we can $\eps$-approximate their Fréchet distance in near linear time in two dimensions, and in roughly $O(n^{4/3}\log nm)$ time in three dimensions. In the second part, we propose a pseudo--output-sensitive algorithm for computing Fréchet distance exactly. The complexity of the algorithm is a function of a quantity we call the \emph{\bwnumber{}}, which is quadratic in the worst case, but tends to be much smaller in practice.
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