article published yes Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Leonidas Guibas author Micha Sharir author We show that the total number of edges ofm faces of an arrangement ofn lines in the plane isO(m 2/3– n 2/3+2 +n) for any&gt;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– n 2/3+2 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– n 2/3+2 +n (n) logm) for any&gt;0, where(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– n 2/3+2 log+n(n) log2 n logm). Springer1990 eng 0179-5376 1432-044410.1007/BF02187784 51161 - 196 yes H. Edelsbrunner, L. Guibas, M. Sharir, Discrete &#38; Computational Geometry 5 (1990) 161–196. Edelsbrunner H, Guibas L, Sharir M. The complexity and construction of many faces in arrangements of lines and of segments. <i>Discrete &#38; Computational Geometry</i>. 1990;5(1):161-196. doi:<a href="https://doi.org/10.1007/BF02187784">10.1007/BF02187784</a> H. Edelsbrunner, L. Guibas, and M. Sharir, “The complexity and construction of many faces in arrangements of lines and of segments,” <i>Discrete &#38; Computational Geometry</i>, vol. 5, no. 1. Springer, pp. 161–196, 1990. Edelsbrunner, Herbert, et al. “The Complexity and Construction of Many Faces in Arrangements of Lines and of Segments.” <i>Discrete &#38; Computational Geometry</i>, vol. 5, no. 1, Springer, 1990, pp. 161–96, doi:<a href="https://doi.org/10.1007/BF02187784">10.1007/BF02187784</a>. Edelsbrunner, Herbert, Leonidas Guibas, and Micha Sharir. “The Complexity and Construction of Many Faces in Arrangements of Lines and of Segments.” <i>Discrete &#38; Computational Geometry</i>. Springer, 1990. <a href="https://doi.org/10.1007/BF02187784">https://doi.org/10.1007/BF02187784</a>. Edelsbrunner H, Guibas L, Sharir M. 1990. The complexity and construction of many faces in arrangements of lines and of segments. Discrete &#38; Computational Geometry. 5(1), 161–196. Edelsbrunner, H., Guibas, L., &#38; Sharir, M. (1990). The complexity and construction of many faces in arrangements of lines and of segments. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href="https://doi.org/10.1007/BF02187784">https://doi.org/10.1007/BF02187784</a> 40722018-12-11T12:06:46Z2022-02-22T09:27:30Z