A Parallel Algorithm for Constructing Projection Polyhedra

A cell decomposition C of Rd is called polytopical if it can be obtained as the projection of the boundary complex of a convex polyhedron P ⊂ Rd +1. In this paper, we present a local condition for recognizing simple polytopical cell decompositions. Based on this, we present a parallel algorithm for the CREW PRAM model to construct P if the given simple cell decomposition C is polytopical. Our algorithm runs in O(log m) time and uses O(mlogm) processors where m is the number of facets of C. This assumes the availability of a suitable data structure for navigating among the facets of C. Such a structure can be created for R2 and R3 without any added resource penalty.

[1]  Walter Whiteley,et al.  Two algorithms for polyhedral pictures , 1986, SCG '86.

[2]  P. McMullen.  Transforms, Diagrams and Representations , 1979 .

[3]  I. G. MacDonald,et al.  CONVEX POLYTOPES AND THE UPPER BOUND CONJECTURE , 1973 .

[4]  Herbert Edelsbrunner,et al.  Finding Extreme Points in Three Dimensions and Solving the Post-Office Problem in the Plane , 1985, Inf. Process. Lett..

[5]  R. Connelly,et al.  A convex 3-complex not simplicially isomorphic to a strictly convex complex , 1980, Mathematical Proceedings of the Cambridge Philosophical Society.

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

[7]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[8]  Franz Aurenhammer,et al.  A criterion for the affine equivalence of cell complexes inRd and convex polyhedra inRd+1 , 1987, Discret. Comput. Geom..

[9]  Georges Voronoi.  Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier mémoire. Sur quelques propriétés des formes quadratiques positives parfaites. , 1908 .

[10]  Atsuo Suzuki,et al.  APPROXIMATION OF A TESSELLATION OF THE PLANE BY A VORONOI DIAGRAM , 1986 .

[11]  J. Van Leeuwen,et al.  Handbook of theoretical computer science - Part A: Algorithms and complexity; Part B: Formal models and semantics , 1990 .

[12]  Carl W. Lee,et al.  Regular Triangulations of Convex Polvtopes , 1990, Applied Geometry And Discrete Mathematics.

[13]  Franz Aurenhammer,et al.  Recognising Polytopical Cell Complexes and Constructing Projection Polyhedra , 1987, J. Symb. Comput..

[14]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..