We consider the time-convex hull problem in the presence of two orthogonal highways H. In this problem, the travelling speed on the highway is faster than off the highway, and the time-convex hull of a point set P is the closure of P with respect to the inclusion of shortest time-paths. In this paper, we provide the algorithm for constructing the time-convex hull with two orthogonal highways. We reach the optimal result of O(n log n) time for arbitrary highway speed in the L1-metric. For the L2-metric with infinite highway speed, we hit the goal of O(n log n) time as well.

中文翻译：

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L1 度量中具有两条正交公路的时凸船体的优化构造

我们考虑存在两条正交高速公路H的时间凸包问题。在这个问题中，高速公路上的行驶速度比高速公路外快，点集P的时间凸包是P的闭包关于包含最短时间路径。在本文中，我们提供了构造具有两条正交公路的时凸包的算法。对于 L1 度量中的任意高速公路速度，我们达到了 O(n log n) 时间的最佳结果。对于具有无限高速公路速度的 L2 度量，我们也达到了 O(n log n) 时间的目标。