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：以美元为顶端的多线性线, 多线性简化问题要求这些顶端的最小大小子序列 。 Melkman 和 O'Rourke 的算法 88 引入了一个几何数据结构, 以在某个距离度量下, 通常在本地的Hausdorf 或本地的 Fr\\'echet 距离下, 与原始的顶端值之间的距离最多是一个给定的阈值。 这里, 本地的意思是, 对于简化的多线性能的每条线段, 只能测量原始的顶端线间距下相应的次曲线的距离 。 Melkman 和 O'Rourke 的算法 [Computeralationalalationalalalals'88] 引入了一个几何等量的数据结构, 在本地的 Ousdoralteral deal demotional as.