Almost tight upper bounds for vertical decompositions in four dimensions

We show that the complexity of the vertical decomposition of an arrangement of n fixed-degree algebraic surfaces or surface patches in four dimensions is O(n/sup 4+/spl epsi//) for any /spl epsi/ > 0. This improves the best previously known upper bound for this problem by a near-linear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O (n/sup 2d-4+/spl epsi//), for any /spl epsi/ > 0, on the complexity of vertical decompositions in dimensions d /spl ges/ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems.

[1]  Micha Sharir,et al.  New bounds for lower envelopes in three dimensions, with applications to visibility in terrains , 1993, SCG '93.

[2]  Bernard Chazelle,et al.  An Algorithm for Generalized Point Location and its Applications , 1990, J. Symb. Comput..

[3]  Micha Sharir,et al.  Efficient randomized algorithms for some geometric optimization problems , 1996, Discret. Comput. Geom..

[4]  Micha Sharir,et al.  The overlay of lower envelopes and its applications , 1996, Discret. Comput. Geom..

[5]  Micha Sharir,et al.  Ray Shooting Amidst Spheres in Three Dimensions and Related Problems , 1997, SIAM J. Comput..

[6]  Leonidas J. Guibas,et al.  Diameter, width, closest line pair, and parametric searching , 1992, SCG '92.

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

[8]  Micha Sharir,et al.  Computing envelopes in four dimensions with applications , 1994, SCG '94.

[9]  Leonidas J. Guibas,et al.  Vertical decomposition of arrangements of hyperplanes in four dimensions , 1993, Discret. Comput. Geom..

[10]  Micha Sharir,et al.  Voronoi diagrams of lines in 3-space under polyhedral convex distance functions , 1995, SODA '95.

[11]  Leonidas J. Guibas,et al.  A Singly-Expenential Stratification Scheme for Real Semi-Algebraic Varieties and Its Applications , 1989, ICALP.

[12]  George E. Collins,et al.  Quantifier elimination for real closed fields by cylindrical algebraic decomposition , 1975 .

[13]  Boaz Tagansky,et al.  A new technique for analyzing substructures in arrangements of piecewise linear surfaces , 1996, Discret. Comput. Geom..

[14]  Sivan Toledo,et al.  Applications of parametric searching in geometric optimization , 1992, SODA '92.

[15]  Micha Sharir,et al.  Vertical Decomposition of Shallow Levels in 3-Dimensional Arrangements and Its Applications , 1999, SIAM J. Comput..

[16]  Leonidas J. Guibas,et al.  Vertical decompositions for triangles in 3-space , 1994, SCG '94.

[17]  Micha Sharir,et al.  Vertical Decomposition of a Single Cell in a Three-Dimensional Arrangement of Surfaces , 1997, Discret. Comput. Geom..

[18]  Micha Sharir,et al.  Vertical decomposition of a single cell in a three-dimensional arrangement of surfaces and its applications , 1996, SCG '96.

[19]  P. Agarwal,et al.  Randomized Algorithms for Geometric Optimization Problems , 2001 .

[20]  Micha Sharir,et al.  Arrangements and Their Applications , 2000, Handbook of Computational Geometry.

[21]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.

[22]  Herbert Edelsbrunner,et al.  The upper envelope of piecewise linear functions: Tight bounds on the number of faces , 1989, Discret. Comput. Geom..

[23]  Mark de Berg,et al.  Efficient ray shooting and hidden surface removal , 1991, SCG '91.

[24]  Kenneth L. Clarkson,et al.  A Randomized Algorithm for Closest-Point Queries , 1988, SIAM J. Comput..

[25]  Vladlen Koltun.  Complexity Bounds for Vertical Decompositions of Linear Arrangements in Four Dimensions , 2001, WADS.

[26]  Vladlen Koltun.  Almost tight upper bounds for lower envelopes in higher dimensions , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[27]  Jirí Matousek,et al.  On range searching with semialgebraic sets , 1992, Discret. Comput. Geom..

[28]  Mark de Berg,et al.  Ray Shooting, Depth Orders and Hidden Surface Removal , 1993, Lecture Notes in Computer Science.