Shortest paths among transient obstacles

We present an optimal algorithm for determining a time-minimal rectilinear path among transient rectilinear obstacles. An obstacle is transient if it exists only for a specific time interval, i.e., it appears and then disappears at specific times. Given a point robot moving with bounded speed among non-intersecting transient rectilinear obstacles and a pair of points (s, d), we determine a time-minimal, obstacle-avoiding path from s to d. Our algorithm runs in $$\varTheta (n \log n)$$ time, where n is the total number of vertices in the obstacle polygons. The main challenge in solving this problem arises when the robot may be required to wait for an obstacle to disappear, before it can continue moving towards its destination. Our algorithm builds on the continuous Dijkstra paradigm, which simulates propagating a wavefront from the source point. We also present an $$O(n^2 \log n)$$ time algorithm for computing the Euclidean shortest path map among transient polygonal obstacles in the plane. This decreases the time complexity of the existing algorithm (Fujimura, in: Proceedings 1992 IEEE international conference on robotics and automation, vol 2, pp 1488–1493, 1992. https://doi.org/10.1109/ROBOT.1992.220041 ) by a factor of n. The shortest path map can be preprocessed for point location, after which a shortest path query from s to any point d can be answered in time $$O(\log n + k)$$ where, k is the number of edges in the path.

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