We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for $$k \le h$$ k ≤ h . Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s , we show how to construct a map, called a shortest k-path map , so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of $$\varTheta (kn)$$ Θ ( k n ) on the size of this map, and show that it can be computed in $$O(k^2n \log n)$$ O ( k 2 n log n ) time, where n is the total number of obstacle vertices.

中文翻译：

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平面上有障碍物违规的最短路径

我们研究了在 h 个凸障碍物之间寻找平面中最短路径的问题，其中允许路径通过（违反）最多 k 个障碍物，因为 $$k \le h$$ k ≤ h 。等效地，问题是找到最短路径，如果从输入中移除 k 个障碍物，则该路径成为无障碍物。给定一个固定的源点 s ，我们展示了如何构造一个地图，称为最短 k 路径地图，以便地图同一区域中的所有目的地都具有相同的组合最短路径，最多可通过 k 个障碍物。我们证明了 $$\varTheta (kn)$$ Θ ( kn ) 在这张地图的大小上的紧界，并表明它可以在 $$O(k^2n \log n)$$ O ( k 2 n log n ) 时间，其中 n 是障碍物顶点的总数。