重新审视分段光滑曲线之间的Fréchet距离
Revisiting the Fréchet distance between piecewise smooth curves
AI总结：
本文提出了一种概念更简单且可扩展至$\mathbb{R}^d$的分段光滑曲线Fréchet距离决策算法，并给出了$c$-packed曲线的近似算法。
摘要：
自1992年Alt和Godau将其引入计算几何以来，Fréchet距离一直是曲线相似性计算算法研究的主要支柱。研究的重点一直是比较折线，一个显著的例外是Rote（2007）提出的针对平面分段光滑曲线决策问题的算法。我们提出了一种针对分段光滑曲线决策问题的算法，该算法在概念上更简单，并且自然地扩展为$\mathbb{R}^d$中分段光滑曲线该问题的首个算法。我们假设算法给定两条连续曲线，分别由$m$和$n$个光滑片段组成，其中每个片段属于足够良好表现的曲线类，例如有界次数的代数曲线集合。我们引入了对自由空间图的分解，将其分解为数量可控的片段，可用于类似于折线情况求解决策问题，时间复杂度为$O(mn)$，从而使Fréchet距离的计算在$O(mn\log(mn))$时间内完成。此外，我们研究了分段光滑曲线的近似算法，这些曲线对于某个固定值$c$也是$c$-packed的。我们调整了现有的$(1+ε)$-近似框架，并证明对于任何$ε> 0$，可以在$O(cn/ε)$时间内计算出近似决策。
英文摘要：
Since its introduction to computational geometry by Alt and Godau in 1992, the Fréchet distance has been a mainstay of algorithmic research on curve similarity computations. The focus of the research has been on comparing polygonal curves, with the notable exception of an algorithm for the decision problem for planar piecewise smooth curves due to Rote (2007). We present an algorithm for the decision problem for piecewise smooth curves that is both conceptually simpler and naturally extends to the first algorithm for the problem for piecewise smooth curves in $\mathbb{R}^d$. We assume that the algorithm is given two continuous curves, each consisting of a sequence of $m$, resp.\ $n$, smooth pieces, where each piece belongs to a sufficiently well-behaved class of curves, such as the set of algebraic curves of bounded degree. We introduce a decomposition of the free space diagram into a controlled number of pieces that can be used to solve the decision problem similarly to the polygonal case, in $O(mn)$ time, leading to a computation of the Fréchet distance that runs in $O(mn\log(mn))$ time. Furthermore, we study approximation algorithms for piecewise smooth curves that are also $c$-packed for some fixed value $c$. We adapt the existing framework for $(1+ε)$-approximations and show that an approximate decision can be computed in $O(cn/ε)$ time for any $ε> 0$.
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