简单多边形内的测地线Fréchet距离
Geodesic Fréchet Distance Inside a Simple Polygon
摘要：
我们推出了一个诱人的替代参数搜索，适用于非测地线和测地线Fréchet优化问题。这种随机化方法是基于红-蓝交叉点的一种变体，与参数搜索相比，它的优雅和实用效率很有吸引力。我们提出了两个多边形曲线之间的测地线Fréchet距离的第一个算法$A$和$B$内的一个简单的边界多边形$P$。测地线Fréchet决策问题的求解速度几乎与其非测地线同胞一样快，并且在$O(k)$预处理之后需要$O(N^{2\log k)$时间和$O(k+N)$空间，其中$N$是$A$和$B$的复杂度中较大的一个，$k$是$P$的复杂度。在$O(k+N^{2\log kN\log N)$期望时间和$O(k+N^{2)$空间中用随机方法求解测地线Fréchet优化问题.这个运行时间只比标准的非测地线Fréchet算法大一个对数因子（Alt and Godau 1995）。结果还提出了测地Fréchet距离在一个多边形域的障碍和测地Hausdorff距离套点或套线段内的一个简单的多边形$P$。
英文摘要：
We unveil an alluring alternative to parametric search that applies to both the non-geodesic and geodesic Fréchet optimization problems. This randomized approach is based on a variant of red-blue intersections and is appealing due to its elegance and practical efficiency when compared to parametric search. We present the first algorithm for the geodesic Fréchet distance between two polygonal curves $A$ and $B$ inside a simple bounding polygon $P$. The geodesic Fréchet decision problem is solved almost as fast as its non-geodesic sibling and requires $O(N^{2\log k)$ time and $O(k+N)$ space after $O(k)$ preprocessing, where $N$ is the larger of the complexities of $A$ and $B$ and $k$ is the complexity of $P$. The geodesic Fréchet optimization problem is solved by a randomized approach in $O(k+N^{2\log kN\log N)$ expected time and $O(k+N^{2)$ space. This runtime is only a logarithmic factor larger than the standard non-geodesic Fréchet algorithm (Alt and Godau 1995). Results are also presented for the geodesic Fréchet distance in a polygonal domain with obstacles and the geodesic Hausdorff distance for sets of points or sets of line segments inside a simple polygon $P$.
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