The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions

Authors Josep Diaz, Mordecai Golin

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Josep Diaz
  • Department of CS, UPC, Barcelona, Spain
Mordecai Golin
  • CSE Department, Hong Kong UST

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Josep Diaz and Mordecai Golin. The Expected Number of Maximal Points of the Convolution of Two 2-D Distributions. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 35:1-35:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The Maximal points in a set S are those that are not dominated by any other point in S. Such points arise in multiple application settings and are called by a variety of different names, e.g., maxima, Pareto optimums, skylines. Their ubiquity has inspired a large literature on the expected number of maxima in a set S of n points chosen IID from some distribution. Most such results assume that the underlying distribution is uniform over some spatial region and strongly use this uniformity in their analysis. This research was initially motivated by the question of how this expected number changes if the input distribution is perturbed by random noise. More specifically, let B_p denote the uniform distribution from the 2-dimensional unit ball in the metric L_p. Let delta B_q denote the 2-dimensional L_q-ball, of radius delta and B_p + delta B_q be the convolution of the two distributions, i.e., a point v in B_p is reported with an error chosen from delta B_q. The question is how the expected number of maxima change as a function of delta. Although the original motivation is for small delta, the problem is well defined for any delta and our analysis treats the general case. More specifically, we study, as a function of n,delta, the expected number of maximal points when the n points in S are chosen IID from distributions of the type B_p + delta B_q where p,q in {1,2,infty} for delta > 0 and also of the type B_infty + delta B_q where q in [1,infty) for delta > 0. For fixed p,q we show that this function changes "smoothly" as a function of delta but that this smooth behavior sometimes transitions unexpectedly between different growth behaviors.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • maximal points
  • probabilistic geometry
  • perturbations
  • Minkowski sum


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  1. Akash Agrawal, Yuan Li, Jie Xue, and Ravi Janardan. The most-likely skyline problem for stochastic points. Proc. 29th CCCG, pages 78-83, 2017. Google Scholar
  2. Zhi-Dong Bai, Luc Devroye, Hsien-Kuei Hwang, and Tsung-Hsi Tsai. Maxima in hypercubes. Random Struct. Algorithms, 27(3):290-309, 2005. Google Scholar
  3. I Bárány. The technique of M-regions and cap-coverings: a survey. Rendiconti di Palermo, 65:21-38, 2000. Google Scholar
  4. Yuri Baryshnikov. On expected number of maximal points in polytopes. In Discrete Mathematics and Theoretical Computer Science, pages 247-258, 2007. Google Scholar
  5. Stephan Börzsönyi, Donald Kossmann, and Konrad Stocker. The Skyline Operator. In Proceedings of the 17th International I.C.D.E.,, pages 421-430. IEEE Computer Society, 2001. Google Scholar
  6. Christian Buchta. On the average number of maxima in a set of vectors. Information Processing Letters, 33:63-65, 1989. Google Scholar
  7. Wei-Mei Chen, Hsien-Kuei Hwang, and Tsung-Hsi Tsai. Maxima-finding algorithms for multidimensional samples: A two-phase approach. Comput. Geometry: Theory and Applications, 45(1-2):33-53, 2012. Google Scholar
  8. Valentina Damerow. Average and smoothed complexity of geometric structures. PhD thesis, University of Paderborn, Germany, 2006. Google Scholar
  9. Valentina Damerow and Christian Sohler. Extreme Points Under Random Noise. In Algorithms – ESA 2004, pages 264-274, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg. Google Scholar
  10. Olivier Devillers, Marc Glisse, Xavier Goaoc, and Rémy Thomasse. Smoothed complexity of convex hulls by witnesses and collectors. Journal of Computational Geometry, 7(2):101-144, 2016. Google Scholar
  11. Luc Devroye. Lecture notes on bucket algorithms. Birkhauser Boston, 1986. Google Scholar
  12. Luc Devroye. Records, the maximal layer, and uniform distributions in monotone sets. Computers Math. Applic., 25(5):19-31, 1993. Google Scholar
  13. Josep Diaz and Mordecai Golin. Smoothed Analysis of the Expected Number of Maximal Points in Two Dimensions. arXiv preprint, 2018. URL:
  14. R A Dwyer. Kinder, gentler average-case analysis for convex hulls and maximal vectors. SIGACT News, 21(2):64-71, 1990. Google Scholar
  15. Marc Geilen, Twan Basten, Bart Theelen, and Ralph Otten. An algebra of Pareto points. Fundamenta Informaticae, 78(1):35-74, 2007. Google Scholar
  16. V. M. Ivanin. Asymptotic estimate for the mathematical expectation of the number of elements in the Pareto set. Cybernetics, 11(1):108-113, 1975. Google Scholar
  17. J.L.Bentley, H.T. Kung, M. Schkolnick, and C.D. Thompson. On the average number of maxima in a set of vectors and its applications. Jour. ACM, 25(4):536-543, 1978. Google Scholar
  18. H. T. Kung, Fabrizio Luccio, and Franco P. Preparata. On Finding the Maxima of a Set of Vectors. J. ACM, 22(4):469-476, 1975. Google Scholar
  19. Alfréd Rényi and Rolf Sulanke. Über die konvexe hülle von n zufällig gewählten punkten. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 2(1):75-84, 1963. Google Scholar
  20. Daniel A Spielman and Shang-Hua Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM (JACM), 51(3):385-463, 2004. Google Scholar
  21. Daniel A Spielman and Shang-Hua Teng. Smoothed analysis: an attempt to explain the behavior of algorithms in practice. Communications of the ACM, 52(10):76-84, 2009. Google Scholar
  22. Subhash Suri, Kevin Verbeek, and Hakan Yildiz. On the most likely convex hull of uncertain points. In European Symposium on Algorithms, pages 791-802. Springer, 2013. Google Scholar