Vertical decompositions for triangles in 3-space

We prove that, for any constant ɛ>0, the complexity of the vertical decomposition of a set ofn triangles in three-dimensional space isO(n2+ɛ+K), whereK is the complexity of the arrangement of the triangles. For a single cell the complexity of the vertical decomposition is shown to beO(n2+ɛ). These bounds are almost tight in the worst case.We also give a deterministic output-sensitive algorithm for computing the vertical decomposition that runs inO(n2 logn+V logn) time, whereV is the complexity of the decomposition. The algorithm is reasonably simple (in particular, it tries to perform as much of the computation in two-dimensional spaces as possible) and thus is a good candidate for efficient implementations.The algorithm is extended to compute the vertical decomposition of arrangements ofn algebraic surface patches of constant maximum degree in three-dimensional space in timeO(nλq(n) logn +V logn), whereV is the combinatorial complexity of the vertical decomposition, λq(n) is a near-linear function related to Davenport-Schinzel sequences, andq is a constant that depends on the degree of the surface patches and their boundaries. We also present an algorithm with improved running time for the case of triangles which is, however, more complicated than the first algorithm.

[1]  Micha Sharir,et al.  Triangles in space or building (and analyzing) castles in the air , 1990, Comb..

[2]  Jirí Matousek,et al.  Range searching with efficient hierarchical cuttings , 1992, SCG '92.

[3]  Jirí Matousek,et al.  Ray Shooting and Parametric Search , 1993, SIAM J. Comput..

[4]  Leonidas J. Guibas,et al.  Lines in space - combinatorics, algorithms and applications , 1989, Symposium on the Theory of Computing.

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

[6]  Micha Sharir,et al.  Nonlinearity of davenport—Schinzel sequences and of generalized path compression schemes , 1986, FOCS.

[7]  Micha Sharir.  Almost tight upper bounds for lower envelopes in higher dimensions , 1994, Discret. Comput. Geom..

[8]  Micha Sharir,et al.  Output-sensitive hidden surface removal , 1989, 30th Annual Symposium on Foundations of Computer Science.

[9]  Mark de Berg,et al.  On lazy randomized incremental construction , 1994, STOC '94.

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

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

[12]  Ketan Mulmuley,et al.  Hidden surface removal with respect to a moving view point , 1991, STOC '91.

[13]  Leonidas J. Guibas,et al.  Combinatorial complexity bounds for arrangements of curves and spheres , 1990, Discret. Comput. Geom..

[14]  Michael T. Goodrich,et al.  Dynamic trees and dynamic point location , 1991, STOC '91.

[15]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1990, Comb..

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

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

[18]  Thomas Ottmann,et al.  Algorithms for Reporting and Counting Geometric Intersections , 1979, IEEE Transactions on Computers.

[19]  John Hershberger,et al.  Finding the Upper Envelope of n Line Segments in O(n log n) Time , 1989, Inf. Process. Lett..

[20]  Leonidas J. Guibas,et al.  Lines in space-combinators, algorithms and applications , 1989, STOC '89.

[21]  Tomás Recio,et al.  Algorithms in Real Algebraic Geometry and Applications to Computational Geometry , 1990, Discrete and Computational Geometry.

[22]  Bernard Chazelle,et al.  Convex Partitions of Polyhedra: A Lower Bound and Worst-Case Optimal Algorithm , 1984, SIAM J. Comput..

[23]  Ketan Mulmuley.  Randomized multidimensional search trees: further results in dynamic sampling , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[24]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[25]  Leonidas J. Guibas,et al.  A dichromatic framework for balanced trees , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[26]  Leonidas J. Guibas,et al.  A Singly Exponential Stratification Scheme for Real Semi-Algebraic Varieties and its Applications , 1991, Theor. Comput. Sci..

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

[28]  Roberto Tamassia,et al.  Efficient Point Location in a Convex Spatial Cell-Complex , 1989, SIAM J. Comput..

[29]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

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

[31]  Micha Sharir,et al.  Planar realizations of nonlinear davenport-schinzel sequences by segments , 1988, Discret. Comput. Geom..

[32]  Micha Sharir,et al.  Castles in the air revisited , 1994, Discret. Comput. Geom..

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

[34]  Micha Sharir,et al.  Sharp upper and lower bounds on the length of general Davenport-Schinzel sequences , 1989, J. Comb. Theory A.

[35]  Micha Sharir,et al.  Near-quadratic bounds for the motion planning problem for a polygon in a polygonal environment , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

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