Optimal Mesh Algorithms for the Voronoi Diagram of Line Segments and Motion Planning in the Plane

Abstract

The motion planning problem of an object with two degrees of freedom moving in the plane can be stated as follows: Given a set of polygonal obstacles in the plane, and a two-dimensional mobile object B with two degrees of freedom, determine if it is possible to move B from a start position to a final position while avoiding the obstacles. If so, plan a path for such a motion. Techniques from computational geometry have been used to develop exact algorithms for this fundamental case of motion planning. In this paper we obtain optimal mesh implementations of two different methods for planning motion in the plane. We do this by first presenting optimal mesh algorithms for some geometric problems that, in addition to being important substeps in motion planning, have numerous independent applications in computational geometry. In particular, we first show that the Voronoi diagram of a set of n nonintersecting (except possibly at endpoints) line segments in the plane can be constructed in O(√ n) time on a √ n × √ n mesh, which is optimal for the mesh. Consequently, we obtain an optimal mesh implementation of the sequential motion planning algorithm described by Ó′Dúlainy and Yap (J. Algorithms 6 (1985), 104-111); in other words, given a disc B and a polygonal obstacle set of size n, we can plan a path (if it exists) for the motion of B from a start position to a final position in O(√ n) time on a mesh of size n. We also show that the shortest path motion between a start position and a final position for a convex object B (of constant size) moving among convex polygonal obstacles of total size n can be found in O(n) time on an n × n mesh, which is worst-case optimal.