Abstract
Let S be a set of n points in d-space. A convex Steiner partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a convex Steiner partition with at most ⌈(n − 1)/d⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.
Establishing a tight lower bound for the maximum volume of a tile in a Steiner partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n) in any dimension d ≥ 2. Here we give a (1 − ε)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0,1]d.
Dumitrescu is supported in part by the NSF grant DMS-1001667; Har-Peled is supported in part by the NSF grant CCF-0915984; Tóth acknowledges support from the NSERC grant RGPIN 35586.
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References
Aichholzer, O., Krasser, H.: The point set order type data base: A collection of applications and results. In: Proc. 13th Canadian Conf. on Comput. Geom., Waterloo, pp. 17–20 (2001)
Alon, N., Bárány, I., Füredi, Z., Kleitman, D.: Point selections and weak ε-nets for convex hulls. Combinatorics, Probability & Computing 1, 189–200 (1992)
Bambah, R.P., Woods, A.C.: On a problem of Danzer. Pacific J. Math. 37(2), 295–301 (1971)
Beck, J., Chen, W.: Irregularities of Distributions. Cambridge Tracts in Mathematics, vol. 89. Cambridge University Press, Cambridge (1987)
de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer (2008)
Dasgupta, S., Papadimitriou, C., Vazirani, U.: Algorithms. McGraw-Hill, New York (2008)
Dumitrescu, A., Har-Peled, S., Tóth, C.D.: Minimum convex partitions and maximum empty polytopes, arXiv:1112.1124
Dumitrescu, A., Tóth, C.D.: Minimum weight convex Steiner partitions. Algorithmica 60(3), 627–652 (2011)
Eppstein, D., Overmars, M., Rote, G., Woeginger, G.: Finding minimum area k-gons. Discrete Comput. Geom. 7(1), 45–58 (1992)
Fevens, T., Meijer, H., Rappaport, D.: Minimum convex partition of a constrained point set. Discrete Appl. Math. 109(1-2), 95–107 (2001)
Füredi, Z., Pach, J.: Traces of finite sets: extremal problems and geometric applications. In: Frankl, P., Füredi, Z., Katona, G., Miklós, D. (eds.) Extremal Problems for Finite Sets, Budapest. Bolyai Soc. Math. Studies, vol. 3, pp. 251–282 (1994)
García López, J., Nicolás, C.: Planar point sets with large minimum convex partitions. In: Proc. 22nd European Workshop on Comput. Geom., pp. 51–54 (2006)
Grantson, M., Levcopoulos, C.: A Fixed Parameter Algorithm for the Minimum Number Convex Partition Problem. In: Akiyama, J., Kano, M., Tan, X. (eds.) JCDCG 2004. LNCS, vol. 3742, pp. 83–94. Springer, Heidelberg (2005)
Har-Peled, S.: An output sensitive algorithm for discrete convex hulls. Comput. Geom. Theory Appl. 10, 125–138 (1998)
Hershberger, J., Suri, S.: Applications of a semi-dynamic convex hull algorithm. BIT 32(2), 249–267 (1992)
Horton, J.: Sets with no empty convex 7-gons. Canadian Math. Bulletin 26, 482–484 (1983)
Knauer, C., Spillner, A.: Approximation Algorithms for the Minimum Convex Partition Problem. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 232–241. Springer, Heidelberg (2006)
Matoušek, J.: Geometric Discrepancy: An Illustrated Guide. Springer (1999)
Matoušek, J.: Lectures on Discrete Geometry. Springer (2002)
Neumann-Lara, V., Rivera-Campo, E., Urrutia, J.: A note on convex partitions of a set of points in the plane. Graphs and Combinatorics 20(2), 223–231 (2004)
Sakai, T., Urrutia, J.: Convex partitions of point sets in the plane. In: Proc. 7th Japan Conf. on Comput. Geom. and Graphs, Kanazawa. JAIST (2009)
Spillner, A.: A fixed parameter algorithm for optimal convex partitions. J. Discrete Algorithms 6(4), 561–569 (2008)
Valtr, P.: Sets in ℝd with no large empty convex subsets. Discrete Mathematics 108(1-3), 115–124 (1992)
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Dumitrescu, A., Har-Peled, S., Tóth, C.D. (2012). Minimum Convex Partitions and Maximum Empty Polytopes. In: Fomin, F.V., Kaski, P. (eds) Algorithm Theory – SWAT 2012. SWAT 2012. Lecture Notes in Computer Science, vol 7357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31155-0_19
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