The complexity and construction of many faces in arrangements of lines and of segments
 Herbert Edelsbrunner,
 Leonidas J. Guibas,
 Micha Sharir
 … show all 3 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m ^{2/3−δ } n ^{2/3+2δ }+n) for anyδ>0. The proof takes an algorithmic approach, that is, we describe an algorithm for the calculation of thesem faces and derive the upper bound from the analysis of the algorithm. The algorithm uses randomization and its expected time complexity isO(m ^{2/3−δ } n ^{2/3+2δ } logn+n logn logm). If instead of lines we have an arrangement ofn line segments, then the maximum number of edges ofm faces isO(m ^{2/3−δ } n ^{2/3+2δ }+nα (n) logm) for anyδ>0, whereα(n) is the functional inverse of Ackermann's function. We give a (randomized) algorithm that produces these faces and takes expected timeO(m ^{2/3−δ } n ^{2/3+2δ } log+nα(n) log^{2} n logm).
 Agarwal, P. K. An efficient algorithm for partitioning arrangements of lines and its applications. InProc. 5th ACM Symp. Comput. Geom., 1989, pp. 11–22.
 Aronov, B., Edelsbrunner, H., Guibas, L., and Sharir, M. Improved bounds on the number of edges of many faces in arrangements of line segments. Report UIUCDCSR891527, Department of Computer Science, University of Illinois, Urbana, Illinois, 1989.
 Aronov, B., and Sharir, M. Triangles in space, or: Building (and analyzing) castles in the air. InProc. 4th ACM Symp. Comput. Geom., 1988, pp. 381–391.
 Bentley, J. L., and Ottmann, T. A. Algorithms for reporting and counting geometric intersections.IEEE Trans. Comput. 28 (1979), 643–647.
 Canham, R. J. A theorem on arrangements of lines in the plane.Isreal J. Math. 7 (1969), 393–397.
 Chazelle, B., and Dobkin, D. P. Intersection of convex objects in two and three dimensions.J. Assoc. Comput. Mach. 34 (1987), 1–27.
 Clarkson, K. New applications of random sampling in computational geometry.Discrete Comput. Geom. 2 (1987), 195–222.
 Clarkson, K., Edelsbrunner, H., Guibas, L. J., Sharir, M., and Welzl, E. Combinatorial complexity bounds for arrangements of curves and spheres.Discrete Comput. Geom., this issue, 99–160.
 Cole, R., Sharir M., and Yap, C. K. Onkhulls and related problems.SIAM J. Comput. 16 (1987), 61–77.
 Edelsbrunner, H.Algorithms in Combinatorial Geometry. SpringerVerlag, Heidelberg, 1987.
 Edelsbrunner, H., Guibas, L. J., Hershberger, J., Seidel, R., Sharir, M., Snoeyink, J., and Welzl, E. Implicitly representing arrangements of lines or segments.Discrete Comput. Geom. 4 (1989), 433–466.
 Edelsbrunner, H., Guibas, L. J., and Sharir, M. The complexity of many cells in arrangements of planes and related problems.Discrete Comput. Geom., this issue, 197–216.
 Edelsbrunner, H., Guibas, L. J., and Stolfi, J. Optimal point location in a monotone subdivision.SIAM J. Comput. 15 (1986), 317–340.
 Edelsbrunner, H., O'Rourke, J., and Seidel, R. Constructing arrangements of lines and hyperplanes with applications.SIAM J. Comput. 15 (1986), 341–363.
 Edelsbrunner, H., and Sharir, M. The maximum number of ways to stabn convex nonintersecting sets in the plane is 2n − 2.Discrete Comput. Geom. 5 (1990), 35–42.
 Edelsbrunner, H., and Welzl, E. On the maximal number of edges of many faces in an arrangement.J. Combin. Theory Ser. A 41 (1986), 159–166.
 Edelsbrunner, H., and Welzl, E. Halfplanar range search in linear space andO(n ^{0.695}) query time.Inform. Process. Lett. 23 (1986), 289–293.
 Grünbaum, B.Convex Polytopes. Wiley, London, 1967.
 Guibas, L. J., Overmars, M. H., and Sharir, M. Counting and reporting intersections in arrangements of line segments. Tech. Report 434, Computer Science Department, NYU, 1989.
 Guibas, L. J., Sharir, M., and Sifrony, S. On the general motion planning problem with two degrees of freedom. InProc. 4th ACM Symp. Comput. Geom., 1988, pp. 289–298.
 Hart, S., and Sharir, M. Nonlinearity of DavenportSchinzel sequences and of generalized path compression schemes.Combinatorica 6 (1986), 151–177.
 Haussler, D., and Welzl, E. Epsilonnets and simplex range queries.Discrete Comput. Geom. 2 (1987), 127–151.
 Moise, E. E.Geometric Topology in Dimension 2 and 3. SpringerVerlag, New York, 1977.
 O'Rourke, J. The signature of a plane curve.SIAM J. Comput. 15 (1986), 34–51.
 Pollack, R., Sharir, M., and Sifrony, S. Separating two simple polygons by a sequence of translations.Discrete Comput. Geom. 3 (1988), 123–136.
 Preparata, F. P., and Shamos, M. I.Computational Geometry—An Introduction. SpringerVerlag, New York, 1985.
 Schmitt, A., Müller, H., and Leister, W. Ray tracing algorithms—theory and practice. InTheoretical Foundations of Computer Graphics and CAD (R. A. Earnshaw, Ed.), NATO ASI Series, Vol. F40, SpringerVerlag, Berlin, 1988, pp. 997–1030.
 Szemerédi, E., and Trotter, W. T. Extremal problems in discrete geometry.Combinatorica 3 (1983), 381–392.
 Wiernik, A., and Sharir, M. Planar realization of nonlinear DavenportSchinzel sequences by segments.Discrete Comput. Geom. 3 (1988), 15–47.
 Title
 The complexity and construction of many faces in arrangements of lines and of segments
 Journal

Discrete & Computational Geometry
Volume 5, Issue 1 , pp 161196
 Cover Date
 19901201
 DOI
 10.1007/BF02187784
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 Herbert Edelsbrunner ^{(1)}
 Leonidas J. Guibas ^{(2)} ^{(3)}
 Micha Sharir ^{(4)} ^{(5)}
 Author Affiliations

 1. Department of Computer Science, University of Illinois at UrbanaChampaign, 61801, Urbana, Ill, USA
 2. DEC Systems Research Center, 130 Lytton Avenue Palo Alto, 94301, Ca, USA
 3. Department of Computer Science, Stanford University, 94305, Ca, USA
 4. Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA
 5. School of Mathematical Science, Tel Aviv University, 69978, Tel Aviv, Israel