Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three

In this paper, we give an algorithm for output-sensitive construction of an j-face polytope that is defined by n halfspaces in E4. Our algorithm runs in O((n + f)log2 f) time and uses O(n + j) space. This is the first algorithm within a polylogarithmic factor of optimal O(nlogf + j) time over the whole range of j. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach also simplifies the “ultimate convex hull algorithm” of Kirkpatrick and Seidel in E2.

[1]  Jirí Matousek,et al.  Cutting hyperplane arrangements , 1990, SCG '90.

[2]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[3]  Erik Brisson,et al.  Representing geometric structures in d dimensions: topology and order , 1989, SCG '89.

[4]  Bernard Chazelle,et al.  Derandomizing an Output-sensitive Convex Hull Algorithm in Three Dimensions , 1995, Comput. Geom..

[5]  Herbert Edelsbrunner,et al.  An O(n log² h) Time Algorithm for the Three-Dimensional Convex Hull Problem , 1991, SIAM J. Comput..

[6]  GARRET SWART,et al.  Finding the Convex Hull Facet by Facet , 1985, J. Algorithms.

[7]  Ketan Mulmuley,et al.  Computational geometry : an introduction through randomized algorithms , 1993 .

[8]  Jirí Matousek,et al.  Approximations and optimal geometric divide-and-conquer , 1991, STOC '91.

[9]  Martin E. Dyer,et al.  Linear Time Algorithms for Two- and Three-Variable Linear Programs , 1984, SIAM J. Comput..

[10]  Robert E. Tarjan,et al.  Planar point location using persistent search trees , 1986, CACM.

[11]  Ray A. Jarvis,et al.  On the Identification of the Convex Hull of a Finite Set of Points in the Plane , 1973, Inf. Process. Lett..

[12]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.

[13]  John F. Canny,et al.  An efficient approach to removing geometric degeneracies , 1992, SCG '92.

[14]  Otfried Cheong.  Ray Shooting in Convex Polytopes , 1992, Symposium on Computational Geometry.

[15]  Rex A. Dwyer.  Higher-dimensional voronoi diagrams in linear expected time , 1991, Discret. Comput. Geom..

[16]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

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

[18]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[19]  Pascal Lienhardt,et al.  Topological models for boundary representation: a comparison with n-dimensional generalized maps , 1991, Comput. Aided Des..

[20]  Leonidas J. Guibas,et al.  The power of geometric duality , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[21]  Kenneth L. Clarkson,et al.  New applications of random sampling in computational geometry , 1987, Discret. Comput. Geom..

[22]  Raimund Seidel,et al.  Voronoi diagrams and arrangements , 1985, SCG '85.

[23]  Jirí Matousek,et al.  Linear optimization queries , 1992, SCG '92.

[24]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[25]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[26]  Raimund Seidel,et al.  Small-dimensional linear programming and convex hulls made easy , 1991, Discret. Comput. Geom..

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

[28]  Bernard Chazelle,et al.  Cutting hyperplanes for divide-and-conquer , 1993, Discret. Comput. Geom..

[29]  Kevin Q. Brown.  Geometric transforms for fast geometric algorithms , 1979 .

[30]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[31]  G. C. Shephard,et al.  Convex Polytopes and the Upper Bound Conjecture , 1971 .

[32]  Jirí Matousek,et al.  Efficient partition trees , 1991, SCG '91.

[33]  Raimund Seidel,et al.  Constructing higher-dimensional convex hulls at logarithmic cost per face , 1986, STOC '86.

[34]  David G. Kirkpatrick,et al.  The Ultimate Planar Convex Hull Algorithm? , 1986, SIAM J. Comput..

[35]  R. Wenger,et al.  Randomized Quick Hull , 1995 .

[36]  David P. Dobkin,et al.  Primitives for the manipulation of three-dimensional subdivisions , 1987, SCG '87.

[37]  Herbert Edelsbrunner,et al.  Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms , 1988, SCG '88.