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. In this paper, we improve the long-standing running time bound for the simplification of polylines under the local Fr\'echet distance. The best algorithm known so far is by Imai and Iri and has a cubic running time in $n$. We present an algorithm with a running time of $O(n^2)$ under the $L_1$ and $L_\infty$ norm, and $O(n^2 \log n)$ under $L_{p \in (1,\infty)}$ (including the Euclidean norm $L_2$). Our approach is based on the ideas of Chan and Chin, who showed that under the local Hausdorff distance, the Imai-Iri algorithm can be improved to run in quadratic time for $L_1$, $L_2$, and $L_\infty$. However, the Hausdorff distance does not take the order of the points along the polyline into account. The Fr\'echet distance, which is sensitive to the course of the polylines, is hence often deemed the superior distance measure for polyline similarity but it also more intricate to compute. So far, the significantly faster simplification algorithms for the Hausdorff distance made them preferable for practical application. But our new algorithm for simplification under the Fr\'echet distance matches the running time bounds for the Hausdorff distance up to logarithmic factors and thus allows the usage of this more suitable distance measure.

翻译：在以美元为顶端的多线性线上, 多线性简化问题要求这些顶端的最小规模子序列为最小规模的子序列, 定义一个新的多线性, 其距离与原多线的距离在某种距离测量下最多为给定阈值。 在本文中, 我们改进了本地Fr\\'echet 距离下多线性简化的长期运行时间。 目前已知的最佳算法是Imai和Iri, 其运行时间为美元。 我们展示了一个运行时间为O (n) 2美元 的算法, 其运行时间在$1美元和$$L_ intyalfty 标准下为O(n_ 2\log n) 。 与原始多线性多线性算值相比, 远线性算得更远。 光线性计算速度( halpilus) 的计算方法基于Chan 和 Chin的理念, 其中显示在本地的距离下, Imai- Irial 算算法可以改进到新时间 时间为$1美元, 美元, $L_2rial liver liental dal deal deal deal deal deal deal deal demout the demout the demout thesocial deal deal deal deal demout.