Path planning in the presence of vertical obstacles

Consideration is given to the problem of finding a shortest path between two points in 3-D space with a restricted class of polyhedral obstacles (vertical buildings with a fixed number k of distinct heights). For the case when all the obstacles have equal heights, a shortest-path algorithm is presented with complexity O(n/sup 2/), i.e. the same complexity as for the 2-D case (n is the total number of corners in all the obstacles). For the general case (k distinct heights), an algorithm is presented for finding a shortest path in time O(n/sup 6k-1/). Also presented is an O(n/sup 2/) approximation algorithm that finds paths that are, at most, 8% longer than the shortest path for the case of k distinct heights when certain minimum separation requirements are satisfied, and a description is given of how the approximation algorithm can be extended to the general case (arbitrary separations). >

[1]  Chee-Keng Yap,et al.  Algorithmic and geometric aspects of robotics , 1987 .

[2]  Greg N. Frederickson,et al.  Fast Algorithms for Shortest Paths in Planar Graphs, with Applications , 1987, SIAM J. Comput..

[3]  D. T. Lee,et al.  Euclidean shortest paths in the presence of rectilinear barriers , 1984, Networks.

[4]  John F. Canny,et al.  A new algebraic method for robot motion planning and real geometry , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[5]  Kenneth L. Clarkson,et al.  Approximation algorithms for shortest path motion planning , 1987, STOC.

[6]  David M. Mount,et al.  An Output Sensitive Algorithm for Computing Visibility Graphs , 1987, FOCS.

[7]  Christos H. Papadimitriou,et al.  An Algorithm for Shortest-Path Motion in Three Dimensions , 1985, Inf. Process. Lett..

[8]  Micha Sharir,et al.  Planning, geometry, and complexity of robot motion , 1986 .

[9]  Micha Sharir,et al.  Retraction: A new approach to motion-planning , 1983, STOC.

[10]  Tomás Lozano-Pérez,et al.  An algorithm for planning collision-free paths among polyhedral obstacles , 1979, CACM.

[11]  Joseph O'Rourke,et al.  Shortest Paths on Polyhedral Surfaces , 1985, STACS.

[12]  Joseph S. B. Mitchell,et al.  The Discrete Geodesic Problem , 1987, SIAM J. Comput..

[13]  Micha Sharir.  On Shortest Paths Amidst Convex Polyhedra , 1987, SIAM J. Comput..

[14]  Micha Sharir,et al.  On Shortest Paths in Polyhedral Spaces , 1986, SIAM J. Comput..

[15]  J. Reif,et al.  Shortest Paths in Euclidean Space with Polyhedral Obstacles. , 1985 .

[16]  D. Mount.  On Finding Shortest Paths on Convex Polyhedra. , 1985 .

[17]  Emo WELZL,et al.  Constructing the Visibility Graph for n-Line Segments in O(n²) Time , 1985, Inf. Process. Lett..

[18]  Hans Rohnert,et al.  Shortest Paths in the Plane with Convex Polygonal Obstacles , 1986, Inf. Process. Lett..

[19]  John F. Canny,et al.  New lower bound techniques for robot motion planning problems , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[20]  Robert E. Tarjan,et al.  Fibonacci heaps and their uses in improved network optimization algorithms , 1987, JACM.