Since its introduction to computational geometry by Alt and Godau in 1992, the Fr\'echet 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 Fr\'echet 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 $(1+\epsilon)$-approximations and show that an approximate decision can be computed in $O(cn/\epsilon)$ time for any $\epsilon > 0$.

翻译：自1992年由Alt和Godau引入计算几何领域以来，弗雷歇距离一直是曲线相似性计算算法研究的核心。研究的焦点主要集中于多边形曲线的比较，仅有的例外是Rote（2007年）提出的平面分段光滑曲线决策问题算法。我们提出了一种针对分段光滑曲线决策问题的算法，该算法在概念上更为简洁，并自然延伸至$\mathbb{R}^d$中分段光滑曲线问题的首个算法。我们假设算法接收两条连续曲线，每条曲线分别由$m$段和$n$段光滑片段组成，每个片段属于一类性质足够良好的曲线（例如有界次数的代数曲线集合）。我们将自由空间图分解为可控数量的片段，这些片段可用于类似多边形情况下的决策问题求解，时间复杂度为$O(mn)$，进而可在$O(mn\log(mn))$时间内计算弗雷歇距离。此外，我们研究了对于某个固定值$c$而言也是$c$-packed的分段光滑曲线的近似算法。我们改进了现有的$(1+\epsilon)$-近似框架，并证明对于任意$\epsilon > 0$，可以在$O(cn/\epsilon)$时间内计算出近似决策。