Extending the Centerpoint Theorem to Multiple Points
The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set P. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median.
We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set Q of (few) points such that every halfspace that contains one point of Q contains a large fraction of the points of P and every halfspace that contains more of Q contains an even larger fraction of P. This setting is comparable to the well-studied concepts of weak epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n).
centerpoint
point sets
Tukey depth
Theory of computation~Design and analysis of algorithms
53:1-53:13
Regular Paper
https://arxiv.org/abs/1810.10231
Alexander
Pilz
Alexander Pilz
Institute of Software Technology, Graz University of Technology, Austria
https://orcid.org/0000-0002-6059-1821
Supported by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35.
Patrick
Schnider
Patrick Schnider
Department of Computer Science, ETH Zurich, Switzerland
10.4230/LIPIcs.ISAAC.2018.53
Greg Aloupis. Geometric measures of data depth. In Regina Y. Liu, Robert Serfling, and Diane L. Souvaine, editors, Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications, pages 147-158. DIMACS/AMS, 2003.
Greg Aloupis, Carmen Cortés, Francisco Gómez, Michael Soss, and Godfried Toussaint. Lower bounds for computing statistical depth. Comput. Statist. Data Anal., 40(2):223-229, 2002. URL: http://dx.doi.org/10.1016/S0167-9473(02)00032-4.
http://dx.doi.org/10.1016/S0167-9473(02)00032-4
Boris Aronov, Franz Aurenhammer, Ferran Hurtado, Stefan Langerman, David Rappaport, Carlos Seara, and Shakhar Smorodinsky. Small weak epsilon-nets. Comput. Geom., 42(5):455-462, 2009. URL: http://dx.doi.org/10.1016/j.comgeo.2008.02.005.
http://dx.doi.org/10.1016/j.comgeo.2008.02.005
Maryam Babazadeh and Hamid Zarrabi-Zadeh. Small Weak Epsilon-Nets in Three Dimensions. In Proc. 18th Canadian Conference on Computational Geometry (CCCG), 2006. URL: http://www.cs.queensu.ca/cccg/papers/cccg13.pdf.
http://www.cs.queensu.ca/cccg/papers/cccg13.pdf
Boris Bukh and Gabriel Nivasch. One-Sided Epsilon-Approximants. In Martin Loebl, Jaroslav Nešetřil, and Robin Thomas, editors, A Journey Through Discrete Mathematics: A Tribute to Jiří Matoušek, pages 343-356. Springer, 2017. URL: http://dx.doi.org/10.1007/978-3-319-44479-6_12.
http://dx.doi.org/10.1007/978-3-319-44479-6_12
Probal Chaudhuri. On a geometric notion of quantiles for multivariate data. J. American Statist. Assoc., 91(434):862-872, 1996.
David Haussler and Emo Welzl. ε-nets and simplex range queries. Discrete Comput. Geom., 2:127-151, 1987. URL: http://dx.doi.org/10.1007/BF02187876.
http://dx.doi.org/10.1007/BF02187876
Shreesh Jadhav and Asish Mukhopadhyay. Computing a Centerpoint of a Finite Planar Set of Points in Linear Time. Discrete Comput. Geom., 12:291-312, 1994. URL: http://dx.doi.org/10.1007/BF02574382.
http://dx.doi.org/10.1007/BF02574382
Stefan Langerman and William L. Steiger. Optimization in Arrangements. In 20th Symp. on Theoretical Aspects of Computer Science (STACS), volume 2607 of LNCS, pages 50-61, 2003. URL: http://dx.doi.org/10.1007/3-540-36494-3_6.
http://dx.doi.org/10.1007/3-540-36494-3_6
Jiří Matoušek. Computing the Center of Planar Point Sets. In Jacob E. Goodman, Richard Pollack, and William Steiger, editors, Discrete and Computational Geometry: Papers from the DIMACS Special Year, volume 6 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 221-230. DIMACS/AMS, 1990.
Jiří Matoušek. Construction of epsilon-Nets. Discrete Comput. Geom., 5:427-448, 1990. URL: http://dx.doi.org/10.1007/BF02187804.
http://dx.doi.org/10.1007/BF02187804
Jiří Matoušek. Tight upper bounds for the discrepancy of half-spaces. Discrete Comput. Geom., 13(3):593-601, 1995. URL: http://dx.doi.org/10.1007/BF02574066.
http://dx.doi.org/10.1007/BF02574066
Jiří Matoušek, Emo Welzl, and Lorenz Wernisch. Discrepancy and approximations for bounded VC-dimension. Combinatorica, 13(4):455-466, 1993. URL: http://dx.doi.org/10.1007/BF01303517.
http://dx.doi.org/10.1007/BF01303517
Nabil Mustafa and Kasturi Varadarajan. Epsilon-approximations and epsilon-nets. In Handbook of Discrete and Computational Geometry. HAL, 2017. URL: https://hal.archives-ouvertes.fr/hal-01468664.
https://hal.archives-ouvertes.fr/hal-01468664
Nabil H. Mustafa and Saurabh Ray. An optimal extension of the centerpoint theorem. Comput. Geom., 42(6):505-510, 2009. URL: http://dx.doi.org/10.1016/j.comgeo.2007.10.004.
http://dx.doi.org/10.1016/j.comgeo.2007.10.004
Sambuddha Roy and William Steiger. Some Combinatorial and Algorithmic Applications of the Borsuk-Ulam Theorem. Graphs Combin., 23:331-341, 2007. URL: http://dx.doi.org/10.1007/s00373-007-0716-1.
http://dx.doi.org/10.1007/s00373-007-0716-1
Mudassir Shabbir. Some results in computational and combinatorial geometry. PhD thesis, Rutgers The State University of New Jersey, 2014.
John W. Tukey. Mathematics and the picturing of data. In Proc. International Congress of Mathematicians, pages 523-531, 1975.
Alexander Pilz and Patrick Schnider
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