关于Fréchet距离的翻转
On Flipping the Fréchet distance
摘要：
经典的和广泛研究的两条曲线之间的Fréchet距离被定义为一个infmax，其中下确界是在曲线的所有遍历，最大值是在两个代理的所有并发位置。在这篇文章中，我们研究了一个“翻转”Fréchet测量定义的一个超级最小值-上确界是在所有遍历的曲线，最小值是在所有并发的位置的两个代理。这种测量产生了两条曲线（或一般域）之间的“社会距离”概念，其中代理人在穿越曲线的同时试图保持尽可能远的距离。我们首先研究了一维和二维两条多边形曲线之间的翻转Fréchet测度，给出了条件下界和匹配算法。然后，我们考虑多边形，它表示的最小距离，两个代理人可以保持，而限制在同一个多边形的边界上或旅行。我们调查在这种情况下的问题的几个变种，其中一些我们提供线性时间算法。最后，我们考虑这个措施的图。我们绘制我们提出的翻转Fréchet措施和现有的相关工作之间的联系在计算几何，希望我们的新措施可能会产生类似于那些执行的Fréchet距离的调查，并进一步有趣的问题出现。
英文摘要：
The classical and extensively-studied Fréchet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a "flipped" Fréchet measure defined by a sup min -- the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of "social distance" between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible. We first study the flipped Fréchet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear time algorithms. Finally, we consider this measure on graphs. We draw connections between our proposed flipped Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.
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