用可伸缩皮带计算Fréchet距离
Computing the Fréchet Distance with a Retractable Leash
摘要：
所有已知的曲线间Fréchet距离的算法都分为两个步骤：首先，它们为决策版本构造一个有效的预言机;其次，它们使用这个预言机从一组有限的临界值中找到最优值。我们提出了一种新的方法，避免了决策版本的弯路。这给出了在多面体距离函数下$R^d$中多边形曲线之间Fréchet距离的第一个二次时间算法（例如，$L_1$和$L_\infty$）。我们还得到了一个$(1+\varepsilon)$ -近似的Fréchet距离下的欧几里德度量，在二次时间为任何固定的$\varepsilon > 0$。对于确切的欧几里德的情况下，我们的框架目前产生一个算法的运行时间$O(n^2 \log^2 n)$。然而，我们推测，它可能最终导致一个更快的精确算法。
英文摘要：
All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet distance between polygonal curves in $R^d$ under polyhedral distance functions (e.g., $L_1$ and $L_\infty$). We also get a $(1+\varepsilon)$-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed $\varepsilon > 0$. For the exact Euclidean case, our framework currently yields an algorithm with running time $O(n^2 \log^2 n)$. However, we conjecture that it may eventually lead to a faster exact algorithm.
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