Abstract
We show that the maximum number of edges boundingm faces in an arrangement ofn line segments in the plane isO(m 2/3 n 2/3+nα(n)+nlogm). This improves a previous upper bound of Edelsbrunner et al. [5] and almost matches the best known lower bound which is Ω(m 2/3 n 2/3+nα(n)). In addition, we show that the number of edges bounding anym faces in an arrangement ofn line segments with a total oft intersecting pairs isO(m 2/3 t 1/3+nα(t/n)+nmin{logm,logt/n}), almost matching the lower bound of Ω(m 2/3 t 1/3+nα(t/n)) demonstrated in this paper.
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References
B. Aronov, andM. Sharir: Triangles in space, or building (and analyzing) castles in the air,Combinatorica 10 (1990), 27–70.
R. J. Canham: A theorem on arrangements of lines in the plane,Israel J. Math. 7 (1969), 393–397.
K. L. Clarkson, H. Edelsbrunner, L. J. Guibas, M. Sharir, andE. Welzl: Combinatorial complexity bounds for arrangements of curves and spheres,Discrete Comput. Geom. 5 (1990), 99–160.
K. L. Clarkson, andP. W. Shor: Applications of random sampling in computational geometry II,Discrete Comput. Geom 4 (1989), 387–421.
H. Edelsbrunner, L. J. Guibas, andM. Sharir: The complexity and construction of many faces in arrangements of lines and of segments,Discrete Comput. Geom. 5 (1990), 161–217.
H. Edelsbrunner, J. O'Rourke, andR. Seidel: Constructing arrangements of lines and hyperplanes with applications,SIAM J. Computing 15 (1986), 341–363.
H. Edelsbrunner, R. Seidel, andM. Sharir: On the zone theorem for hyperplane arrangements, to appear inSIAM J. Computing.
H. Edelsbrunner, andE. Welzl: On the maximal number of edges of many faces in arrangements,J. Comb. Theory, Ser. A 41 (1986), 159–166.
L. J. Guibas, M. Sharir, andS. Sifrony: On the general motion planning problem with two degrees of freedom,Discrete Comput. Geom. 4 (1989), 491–521.
J. Matušek: Construction of ε-nets,Discrete Comput. Geom 5 (1990), 427–448.
R. Pollack, M. Sharir, andS. Sifrony: Separating two simple polygons by a sequence of translations,Discrete Comput. Geom. 3 (1988), 123–136.
A. Wiernik, andM. Sharir: Planar realization of non-linear Davenport-Schinzel sequences by segments,Discrete Comput. Geom. 3 (1988), 15–47.
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Work on this paper by the first and fourth authors has been partially supported by Office of Naval Research Grant N00014-87-K-0129, by National Science Foundation Grants DCR-83-20085 and CCR-89-01484. Work by the first author has also been supported by an AT&T Bell Laboratories Ph.D. scholarship at New York University and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center (NSF-STC88-09648). Work by the second author has been supported by NSF under Grants CCR-87-14565 and CCR-89-21421. Work by the fourth author has additionally been supported by grants from the U.S.-Israeli Binational Science Foundation, the NCRD (the Israeli National Council for Research and Development) and the Fund for Basic Research in Electronics, Computers and Communication, administered by the Israeli National Academy of Sciences.
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Aronov, B., Edelsbrunner, H., Guibas, L.J. et al. The number of edges of many faces in a line segment arrangement. Combinatorica 12, 261–274 (1992). https://doi.org/10.1007/BF01285815
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DOI: https://doi.org/10.1007/BF01285815