Abstract
Dujmović and Langerman (Discrete Comput Geom 49(1):74–88, 2013) proved a ham-sandwich cut theorem for an arrangement of lines in the plane. Recently, Xue and Soberón (Discrete Comput Geom 66:1150–1167, 2021) generalized it to balanced convex partitions of lines in the plane. In this paper, we study the computational problems of computing a ham-sandwich cut balanced convex partitions for an arrangement of lines in the plane. We show that both problems can be solved in polynomial time.
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References
Bereg, S.: Orthogonal equipartitions. Comput. Geom. Theory Appl. 42(4), 305–314 (2009). https://doi.org/10.1016/j.comgeo.2008.09.004
Bespamyatnikh, S., Kirkpatrick, D., Snoeyink, J.: Generalizing ham sandwich cuts to equitable subdivisions. Discrete Comput. Geom. 24(4), 605–622 (2000)
Brönnimann, H., Chazelle, B.: Optimal slope selection via cuttings. Comput. Geom. Theory Appl. 10(1), 23–29 (1998)
Cole, R., Salowe, J., Steiger, W., Szemerédi, E.: An optimal-time algorithm for slope selection. SIAM J. Comput. 18(4), 792–810 (1989)
Dillencourt, M.B., Mount, D.M., Netanyahu, N.S.: A randomized algorithm for slope selection. Intern. J. Comput. Geom. Appl. 2, 1–27 (1992)
Dujmovic, V., Langerman, S.: A center transversal theorem for hyperplanes and applications to graph drawing. Discrete Comput. Geom. 49(1), 74–88 (2013)
Edelsbrunner, H., Guibas, L.J.: Topologically sweeping an arrangement. J. Comput. Syst. Sci., 38, 165–194 (1989). Corrigendum in 42 (1991), 249–251
Edelsbrunner, H., Souvaine, D.L.: Computing median-of-squares regression lines and guided topological sweep. J. Am. Stat. Assoc. 85, 115–119 (1990)
Fredman, M.L.: On computing the length of longest increasing subsequences. Discrete Math. 11(1), 29–35 (1975)
Ito, H., Uehara, H., Yokoyama, M.: 2-dimension ham sandwich theorem for partitioning into three convex pieces. In: Proceedings of Japan Conference Discrete Computational Geometry’98 1763 of Lecture Notes Comput. Sci., pp. 129–157. Springer (2000)
Kano, M., Urrutia, J.: Discrete geometry on colored point sets in the plane—a survey. Graphs Comb. 37(1), 1–53 (2021)
Katz, M.J., Sharir, M.: Optimal slope selection via expanders. Inform. Process. Lett. 47, 115–122 (1993)
Knaster, B., Kuratowski, C., Mazurkiewicz, S.: Ein beweis des fixpunktsatzes für \(n\)-dimensionale simplexe. Fundam. Math. 14(1), 132–137 (1929)
Matoušek, J.: Randomized optimal algorithm for slope selection. Inform. Process. Lett. 39, 183–187 (1991)
Roldán-Pensado, E., Soberón, P.: A survey of mass partitions. Bull. Am. Math. Soc. 59, 227–267 (2021)
Sakai, T.: Balanced convex partitions of measures in \(R^2\). Graphs Comb. 18(1), 169–192 (2002)
Schensted, C.: Longest increasing and decreasing subsequences. Can. J. Math. 13, 179–191 (1961)
Xue, A., Soberón, P.: Balanced convex partitions of lines in the plane. Discrete Comput. Geom. 66, 1150–1167 (2021)
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Bereg, S. Computing Balanced Convex Partitions of Lines. Algorithmica 85, 2515–2528 (2023). https://doi.org/10.1007/s00453-022-01082-z
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DOI: https://doi.org/10.1007/s00453-022-01082-z