Rectilinear Path Problems among Rectilinear Obstacles Revisited

We present eecient algorithms for nding rectilinear collision-free paths between two given points among a set of rectilinear obstacles. Our results improve the time complexity of previous results for nding the shortest rectilinear path, the minimum-bend shortest rectilinear path, the shortest minimum-bend rectilinear path and the minimum-cost rectilinear path. For nding the shortest rectilinear path, we use graph-theoretic approach and obtain an algorithm with O(m logt + t log 3=2 t) running time where t is the number of extreme edges of given obstacles, and m is the number of obstacleedges. Based on this result we also obtain an O(N logN+(m+N)logt+(t+N)log 2 (t+N)) running time algorithm for computing the L 1 minimum spanning tree of given N terminals among rectilinear obstacles. For nding the minimum-bend shortest path, the shortest minimum-bend rectilinear path and the minimum-cost rectilinear path, we devise a new dynamic-searching approach and derive algorithms that run in O(m log 2 m) time using O(m logm) space or run in O(m log 3=2 m) time and space.