Rectilinear Path Problems among Rectilinear Obstacles Revisited

Abstract

Efficient algorithms are presented for finding rectilinear collision-free paths between two given points among a set of rectilinear obstacles. The results improve the time complexity of previous results for finding the shortest rectilinear path the minimum-bend shortest rectilinear path, the shortest minimum-bend rectilinear path and the minimum-cost rectilinear path. For finding the shortest rectilinear path, a graph-theoretic approach is used and an algorithm is obtained with $O(m \log t + t \log^{3/2}t)$ running time, where t is the number of extreme edges of given obstacles and m is the number of obstacle edges. Based on this result an $O(N \log N + (m + N) \log t + (t+N) \log^{2} (t + N))$ running time algorithm for computing the $L_{1}$ minimum spanning tree of given N terminals among rectilinear obstacles is obtained. For finding 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 $O(m \log m)$ space or run in $O(m \log^{3/2} m)$ time and space.