Approximating center points with iterative Radon points

We give a practical and provably good Monte Carlo algorithm for approximating center points. Let P be a set of n points in Rd. A point c∈Rd is a β-center point of P if every closed halfspace containing c contains at least βn points of P. Every point set has a 1/(d+1)-center point; our algorithm finds an Ω(1/d2)-center point with high probability. Our algorithm has a small constant factor and is the first approximate center point algorithm whose complexity is subexponential in d. Moreover, it can be optimally parallelized to require O(log2d log log n) time. Our algorithm has been used in mesh partitioning methods and can be used in the construction of high breakdown estimators for multivariate datasets in statistics. It has the potential to improve results in practice for constructing weak ∊-nets. We derive a variant of our algorithm whose time bound is fully polynomial in d and linear in n, and show how to combine our approach with previous techniques to compute high quality center points more quickly.

[1]  Kenneth L. Clarkson,et al.  A Las Vegas algorithm for linear programming when the dimension is small , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[2]  F. Frances Yao,et al.  A 3-space partition and its applications , 1983, STOC.

[3]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[4]  Gary L. Miller,et al.  Separators for sphere-packings and nearest neighbor graphs , 1997, JACM.

[5]  Philip E. Gill,et al.  Numerical Linear Algebra and Optimization , 1991 .

[6]  Gary L. Miller,et al.  Automatic Mesh Partitioning , 1992 .

[7]  S. Teng.  Points, spheres, and separators: a unified geometric approach to graph partitioning , 1992 .

[8]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[9]  Asish Mukhopadhyay,et al.  Computing a centerpoint of a finite planar set of points in linear time , 1993, SCG '93.

[10]  V. Klee,et al.  Helly's theorem and its relatives , 1963 .

[11]  Bernard Chazelle,et al.  On linear-time deterministic algorithms for optimization problems in fixed dimension , 1996, SODA '93.

[12]  Richard Cole,et al.  On k-Hulls and Related Problems , 1987, SIAM J. Comput..

[13]  Leslie G. Valiant,et al.  Short Monotone Formulae for the Majority Function , 1984, J. Algorithms.

[14]  B. Weide.  Statistical methods in algorithm design and analysis. , 1978 .

[15]  Gary L. Miller,et al.  Geometric Mesh Partitioning: Implementation and Experiments , 1998, SIAM J. Sci. Comput..

[16]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[17]  Noga Alon,et al.  Point Selections and Weak ε-Nets for Convex Hulls , 1992, Combinatorics, Probability and Computing.

[18]  D. Donoho,et al.  Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness , 1992 .

[19]  David Haussler,et al.  Epsilon-nets and simplex range queries , 1986, SCG '86.

[20]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[21]  Jirí Matousek,et al.  Approximations and optimal geometric divide-and-conquer , 1991, STOC '91.

[22]  J. Spencer.  Ten lectures on the probabilistic method , 1987 .

[23]  Nimrod Megiddo,et al.  Linear Programming in Linear Time When the Dimension Is Fixed , 1984, JACM.