离散Fréchet间隙
The Discrete Fréchet Gap
摘要：
我们引入离散Fréchet间隙及其变体作为多边形曲线之间相似性的替代度量。我们相信，对于某些应用，新的度量（及其变体）可能比离散的Fréchet距离（及其变体）更好地反映我们直观的相似性概念，因为后者对彼此相对接近的点对（匹配）是无关紧要的。参考青蛙的类比，其中离散弗雷歇距离经常被描述，离散弗雷歇间隙是青蛙遍历它们的点序列所需的皮带的最长和最短位置之间的最小差。我们提出了一个优化方案，这是适合于任何单调函数定义的距离对，如间隙和比率函数。我们将此方案应用于离散Fréchet间隙的两个变体，即，单侧离散Fréchet间隙与捷径和弱离散Fréchet间隙，获得计算它们的$O(n^2 \log^2 n)$时间算法。
英文摘要：
We introduce the discrete Fréchet gap and its variants as an alternative measure of similarity between polygonal curves. We believe that for some applications the new measure (and its variants) may better reflect our intuitive notion of similarity than the discrete Fréchet distance (and its variants), since the latter measure is indifferent to (matched) pairs of points that are relatively close to each other. Referring to the frogs analogy by which the discrete Fréchet distance is often described, the discrete Fréchet gap is the minimum difference between the longest and shortest positions of the leash needed for the frogs to traverse their point sequences. We present an optimization scheme, which is suitable for any monotone function defined for pairs of distances such as the gap and ratio functions. We apply this scheme to two variants of the discrete Fréchet gap, namely, the one-sided discrete Fréchet gap with shortcuts and the weak discrete Fréchet gap, to obtain $O(n^2 \log^2 n)$-time algorithms for computing them.
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