Given a polyline on $n$ vertices, the polyline simplification problem asks for a minimum size subsequence of these vertices defining a new polyline whose distance to the original polyline is at most a given threshold under some distance measure, usually the local Hausdorff or the local Fr\'echet distance. Here, local means that, for each line segment of the simplified polyline, only the distance to the corresponding sub-curve in the original polyline is measured. Melkman and O'Rourke [Computational Morphology '88] introduced a geometric data structure to solve polyline simplification under the local Hausdorff distance in $O(n^2 \log n)$ time, and Guibas, Hershberger, Mitchell, and Snoeyink [Int. J. Comput. Geom. Appl. '93] considered polyline simplification under the Fr\'echet distance as ordered stabbing and provided an algorithm with a running time of $O(n^2 \log^2 n)$, but they did not restrict the simplified polyline to use only vertices of the original polyline. We show that their techniques can be adjusted to solve polyline simplification under the local Fr\'echet distance in $O(n^2 \log n)$ time instead of $O(n^3)$ when applying the Imai--Iri algorithm. Our algorithm may serve as a more efficient subroutine for multiple other algorithms. We provide a simple algorithm description as well as rigorous proofs to substantiate this theorem. We also investigate the geometric data structure introduced by Melkman and O'Rourke, which we refer to as wavefront, in more detail and feature some interesting properties. As a result, we can prove that under the L$_1$ and the L$_\infty$ norm, the algorithm can be significantly simplified and then only requires a running time of $O(n^2)$. We also define a natural class of polylines where our algorithm always achieves this running time also in the Euclidean norm L$_2$.

翻译：给定一条由$n$个顶点构成的折线，折线简化问题要求从这些顶点中选取一个规模最小的子序列，使得定义的新折线在某种距离度量（通常为局部豪斯多夫距离或局部弗雷歇距离）下与原折线的距离不超过给定阈值。此处"局部"指仅测量简化折线中每个线段与原始折线中对应子曲线之间的距离。Melkman与O'Rourke [Computational Morphology '88] 引入了一种几何数据结构，可在$O(n^2 \log n)$时间内解决局部豪斯多夫距离下的折线简化问题；Guibas、Hershberger、Mitchell与Snoeyink [Int. J. Comput. Geom. Appl. '93] 将弗雷歇距离下的折线简化视为有序刺穿问题，并给出了运行时间为$O(n^2 \log^2 n)$的算法，但他们未限制简化折线仅使用原始折线的顶点。我们证明：当采用Imai-Iri算法时，通过调整上述技术，可在$O(n^2 \log n)$时间而非$O(n^3)$时间内解决局部弗雷歇距离下的折线简化问题。该算法可作为多种其他算法的更高效子程序。我们提供简洁的算法描述与严谨的证明以支撑该定理，并深入考察Melkman与O'Rourke引入的几何数据结构（称为波前），揭示其若干有趣性质。由此可证：在L$_1$与L$_\infty$范数下，该算法可显著简化，仅需$O(n^2)$运行时间；此外，我们定义了一类自然折线，在欧几里得范数L$_2$下，该算法同样始终达到此运行时间。