Decomposition algorithm for moving a ladder among rectangular obstacles

In this paper we consider the problem of moving a ladder amidst n rectangular obstacles. The problem of moving the ladder between two different placements is solved approximately, by decomposing it into several "local motion planning" problems We give an O(n log n) time algorithm to construct and solve all the local problems. A weighted graph MG, called the motion graph with O(n) vertices and O(n) edges, is constructed from the solutions of the various local sub-problems. The vertices of MG correspond to the placements of the ladder and edges correspond to motions between these placements. The weight on its edges represents the length of the longest ladder moveable between the two placements corresponding to the two end vertices. We give an O(n) algorithm to construct collision-free paths for a given ladder between a pair of free placements by searching the graph MG. Furthermore, using an algorithm like Dijkstra\´s shortest path algorithm we estimate the length of the longest ladder moveable between any two free placements, in O(n log n) time.