局部Fréchet距离下二维折线的近似二次化简
Polyline Simplification under the Local Fréchet Distance has Almost-Quadratic Runtime in 2D
摘要：
给定$n$顶点上的一个顶点，顶点简化问题要求这些顶点的最小尺寸子序列定义一个新的顶点，该顶点到原始顶点的距离在某种距离度量下至多是给定的阈值，通常是局部Hausdorff或局部Fréchet距离。在这里，局部意味着对于简化曲线的每个线段，仅测量到原始曲线中的对应子曲线的距离。Melkman和O 'Rourke [Computational Morphology '88]引入了一种几何数据结构来解决$O(n^2 \log n)$时间中局部Hausdorff距离下的递归简化，以及Guibas、Hershberger、Mitchell和Snoeyink [Int. J. Comput. Geom. Appl. '93]将Fréchet距离下的顶点简化视为有序刺入，并提供了一个运行时间为$O(n^2 \log^2 n)$的算法，但他们没有限制简化的顶点仅使用原始顶点的顶点。我们表明，他们的技术可以调整，以解决在$O(n^2 \log n)$的时间，而不是$O(n^3)$时，应用Imai-Iri算法的局部Fréchet距离下的简化。我们的算法可以作为一个更有效的子程序的多个其他算法。我们提供了一个简单的算法描述以及严格的证明，以证实这一定理。我们还研究了几何数据结构介绍Melkman和O 'Rourke，我们称之为波前，在更详细的和功能的一些有趣的属性。结果证明，在L $_1$和L $_\infty$范数下，该算法可以得到明显的简化，只需要$O(n^2)$的运行时间.我们还定义了一个自然类的折线，我们的算法总是达到这个运行时间也在欧几里德范数L $_2$。
英文摘要：
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 Fréchet 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 Fréchet 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 Fréchet 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$_2$.
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