Esther Ezra
ABSTRACT
We consider the problem of bounding the combinatorial complexity of a single cell in an arrangement of k convex polyhedra in 3-space having n facets in total. We use a variant of the technique of Halperin and Sharir [17], and show that this complexity is O(nk1+ε), for any ε > 0, thus almost settling a conjecture of Aronov et. el. [5]. We then extend our analysis and show that the overall complexity of the zone of a low-degree algebraic surface, or of the boundary of an arbitrary convex set, in an arrangement of k convex polyhedra in 3-space with n facets in total, is also O(nk1+ε), for any ε > 0. Finally, we present a deterministic algorithm that constructs a single cell in an arrangement of this kind, in time O(nk1+ε log2n), for any ε 0.
- B. Aronov and M. Sharir. Triangles in space or building (and analyzing) castles in the air. Combinatorica, 10(2):137--173, 1990.Google Scholar
Cross Ref
- B. Aronov and M. Sharir. Castles in the air revisited. Discrete Comput.G om., 12:119--150, 1994.Google Scholar
- B. Aronov and M. Sharir. On translational motion planning of a convex polyhedron in 3-space. SIAM J.Comput., 26(6):1785--1803, 1997. Google ScholarDigital Library
- B. Aronov and M. Sharir. The common exterior of convex polygons in the plane. Comput. Geom. Theory Appl., 6(3):139--149, 1997. Google ScholarDigital Library
- B. Aronov, M. Sharir, and B. Tagansky. The union of convex polyhedra in three dimensions. SIAM J.Comput., 26(6):1670--1688, 1997. Google ScholarDigital Library
- S. Basu. On the combinatorial and topological complexity of a single cell. Discrete Comput. Geom., 29(1):41--59, 2003.Google ScholarCross Ref
- J. D. Boissonnat, M. Sharir, B. Tagansky and M. Yvinec. Voronoi diagrams in higher dimensions under certain polyhedral distance functions. Discrete Comput. Geom., 14:485--519, 1998.Google ScholarCross Ref
- B. Chazelle, H. Edelsbrunner, L. J. Guibas, and M. Sharir. Algorithms for bichromatic line segment problems and polyhedral terrains. Algorithmica, 11:116--132, 1994.Google ScholarDigital Library
- B. Chazelle, H. Edelsbrunner, L. J. Guibas, M. Sharir, and J. Snoeyink. Computing a face in an arrangement of line segments and related problems. SIAM J.Comput., 22(6):1286--1302, 1993. Google ScholarDigital Library
- B. Chazelle, H. Edelsbrunner. Fractional cascading: I. A data structuring technique. Algorithmica, 1(2):133--162, 1986.Google ScholarCross Ref
- L. P. Chew, K. Kedem, M. Sharir, B. Tagansky, and E. Welzl. Voronoi diagrams of lines in 3-space under polyhedral convex distance functions. J.Algorithms, 29(2):238--255, 1998. Google ScholarDigital Library
- K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discr te Comput. Geom., 4(5):387--421, 1989.Google ScholarDigital Library
- H. Edelsbrunner, L. J. Guibas, J. Pach, R. Pollack, R. Seidel, and M. Sharir. Arrangements of curves in the plane: Topology, combinatorics, and algorithms. Theoretical Comput.Sci., 92:319--336, 1992. Google ScholarDigital Library
- L. J. Guibas, M. Sharir, and S. Sifrony. On the general motion planning problem with two degrees of freedom. Discrete Comput. Geom., 4:491--521, 1989.Google ScholarDigital Library
- D. Halperin. Algorithmic Motion Planning via Arrangements of Curves and of Surfaces. Ph.D. thesis, Computer Science Department, Tel-Aviv University, Tel Aviv, July 1992.Google Scholar
- D. Halperin and M. Sharir. New bounds for lower envelopes in three dimensions with applications to visibility of terrains. Discrete Comput. Geom., 12:313--326, 1994.Google ScholarDigital Library
- D. Halperin and M. Sharir. Almost tight upper bounds for the single cell and zone problems in three dimensions. Discrete Comput. Geom., 14(4):385--410, 1995.Google ScholarDigital Library
- A. Hatcher. Algebraic Topology. Cambridge University Press, first edition, 2002.Google Scholar
- M. W. Hirsch. Differential Topology, Volume33 of Graduat Texts in Math maticsematics. Springer-Verlag, Berlin, Heidelberg, New York, 1976.Google Scholar
- D. P. Huttenlocher, K. Kedem, and M. Sharir. The upper envelope of Voronoi surfaces and its applications. Discrete Computat. Geom., 9:267--291, 1993.Google ScholarDigital Library
- D. G. Kirkpatrick and D. P. Dobkin. Fast detection of polyhedral intersection. Theoretical Computer Science, 27:241--253, 1983.Google ScholarCross Ref
- J. Pach, I. Safruti, and M. Sharir. The union of congruent cubes in three dimensions. Discrete Comput. Geom., 30:133--160, 2003.Google ScholarCross Ref
- L. Palazzi and J. Snoeyink. Counting and reporting red/blue segment intersections. CVGIP:Graph.Models Imag Process., 56(4):304--310, 1994. Google ScholarDigital Library
- M. Sharir and P. Agarwal. Davenport-Schinz l Sequences and Their Geometric Applications. Cambridge University Press, New York, 1995. Google ScholarDigital Library
- J. T. Schwartz, and M. Sharir. On the "Piano Movers" problem: II. General techniques for computing topological properties of real algebraic manifolds. Adv.Appl.Math.,4: 298--351, 1983.Google ScholarDigital Library
- O. Schwarzkopf, and M. Sharir. Vertical decomposition of a single cell in a 3-dimensional arrangement of surfaces. Discrete Comput. Geom., 18:269--288, 1997.Google ScholarCross Ref
- M. Sharir. Almost tight upper bounds for lower envelopes in higher dimensions. Discrete Comput. Geom., 12:327--345, 1994.Google ScholarDigital Library
- B. Tagansky. A new technique for analyzing substructures in arrangements of piecewise linear surfaces. Discrete Comput. Geom., 16(4):455--479, 1996.Google ScholarDigital Library
Index Terms
- Almost tight bound for a single cell in an arrangement of convex polyhedra in R3
Recommendations
A Single Cell in an Arrangement of Convex Polyhedra in ${\Bbb R}^3$
We show that the combinatorial complexity of a single cell in an arrangement of k convex polyhedra in 3-space having n facets in total is $O(nk^{1+\varepsilon})$, for any $\varepsilon > 0$, thus settling a conjecture of Aronov et al. We then extend our ...
Injective Convex Polyhedra
We characterize the convex polyhedra P in $${\mathbb {R}}^n$$Rn for which any family of n-dimensional axis-parallel hypercubes centered in P and intersected with P has the binary intersection property. The characterization is effective, concrete and ...
Convex hulls of spheres and convex hulls of disjoint convex polytopes
Given a set @S of spheres in E^d, with d>=3 and d odd, having a constant number of m distinct radii @r"1,@r"2,...,@r"m, we show that the worst-case combinatorial complexity of the convex hull of @S is @Q(@__ __"1"=<"i"<>"j"=<"mn"in"j^@__ __^d^2^@__ __), ...
Comments