Approximating shortest paths on a convex polytope in three dimensions

We present an approximation algorithm that, given a convex polytope P with n faces in lR3, points s, t c 8P, and a parameter O < & <1, constructs a path on t3P from s to twhose length is at most (1 +E)dP(S, t), where dp (s, t) is the length of the shortest path between s and t on 8P. The algorithm runs in O(n . min {1/s’5, logn} + l/e4’5 log(l/&)) time. and is relatively simple to implement. We also present an extension of the algorithm that computes approximate shortest paths from a given source point on 8P to all vertices of P.

[1]  A. V. Pogorelov.  Extrinsic geometry of convex surfaces , 1973 .

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

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

[4]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[5]  Micha Sharir,et al.  On the shortest paths between two convex polyhedra , 2018, JACM.

[6]  James A. Storer,et al.  A single-exponential upper bound for finding shortest paths in three dimensions , 1994, JACM.

[7]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[8]  C. Bajaj.  The Algebraic Complexity of Shortest Paths in Polyhedral Spaces , 1985 .

[9]  Yijie Han,et al.  Shortest paths on a polyhedron , 1990, SCG '90.

[10]  Subhash Suri,et al.  Practical methods for approximating shortest paths on a convex polytope in R3 , 1995, SODA '95.

[11]  Imre Bárány,et al.  On the number of convex lattice polytopes , 1992 .

[12]  Micha Sharir,et al.  On shortest paths amidst convex polyhedra , 1987, SCG '86.

[13]  David G. Kirkpatrick,et al.  A Linear Algorithm for Determining the Separation of Convex Polyhedra , 1985, J. Algorithms.

[14]  George E. Andrews,et al.  A LOWER BOUND FOR THE VOLUME OF STRICTLY CONVEX BODIES WITH MANY BOUNDARY LATTICE POINTS , 1963 .

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

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

[17]  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).

[18]  Micha Sharir,et al.  On shortest paths in polyhedral spaces , 1986, STOC '84.