四个苏联人遛狗--计算Fréchet距离的改进边界
Four Soviets Walk the Dog-Improved Bounds for Computing the Fréchet Distance
摘要：
给定平面上的两条多边形曲线，有许多方法可以定义它们之间的相似性。一个常用的测量方法是Fréchet距离。自1992年Alt和Godau提出以来，已有许多变体和扩展被研究。尽管如此，即使在20多年后，Alt和Godau用于计算Fréchet距离的原始$O(n^2 \log n)$算法仍然是最先进的（这里，$n$表示每条曲线上的边数）。这导致Helmut Alt猜想相关的决策问题是3SUM-困难的。在最近的工作中，Agarwal等人展示了如何打破Fréchet距离离散版本的二次障碍，其中考虑点序列而不是多边形曲线。在他们工作的基础上，我们给出了一个在指针机器上计算两条多边形曲线在时间上的Fréchet距离的随机算法$O(n^2 \sqrt{\log n}(\log\log n)^{3/2})$和在字RAM上计算两条多边形曲线在时间上的Fréchet距离的随机算法$O(n^2(\log\log n)^2)$.此外，我们证明了存在一个代数决策树的决策问题的深度$O(n^{2-\varepsilon})$，一些$\varepsilon > 0$。我们认为，这揭示了这个研究得很好的问题的一个有趣的新方面。最后，我们展示了如何获得第一次次二次算法计算弱Fréchet距离的字RAM。
英文摘要：
Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original $O(n^2 \log n)$ algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here, $n$ denotes the number of edges on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time $O(n^2 \sqrt{\log n}(\log\log n)^{3/2})$ on a pointer machine and in time $O(n^2(\log\log n)^2)$ on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth $O(n^{2-\varepsilon})$, for some $\varepsilon > 0$. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fréchet distance on a word RAM.
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