Product range spaces, sensitive sampling, and derandomization

We introduce the concept of a sensitive /spl epsi/-approximation, and use it to derive a more efficient algorithm for computing /spl epsi/-nets. We define and investigate product range spaces, for which we establish sampling theorems analogous to the standard finite VC-dimensional case. This generalizes and simplifies results from previous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(nlog/sup 3/ n) deterministic algorithm for computing the diameter of an n-point set in 3-dimensional space.<<ETX>>

[1]  Martin E. Dyer,et al.  A class of convex programs with applications to computational geometry , 1992, SCG '92.

[2]  Leonidas J. Guibas,et al.  Diameter, width, closest line pair, and parametric searching , 1993, Discret. Comput. Geom..

[3]  Bernard Chazelle,et al.  Cutting hyperplanes for divide-and-conquer , 1993, Discret. Comput. Geom..

[4]  Leonidas J. Guibas,et al.  Improved bounds on weak ε-nets for convex sets , 1993, STOC.

[5]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1990, Comb..

[6]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[7]  A. Heppes.  Beweis einer Vermutung von A. Vázsonyi , 1956 .

[8]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[9]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[10]  Jirí Matousek.  Cutting hyperplane arrangements , 1991, Discret. Comput. Geom..

[11]  Richard M. Dudley,et al.  Some special vapnik-chervonenkis classes , 1981, Discret. Math..

[12]  Nimrod Megiddo,et al.  Applying parallel computation algorithms in the design of serial algorithms , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[13]  Prabhakar Raghavan,et al.  Probabilistic construction of deterministic algorithms: Approximating packing integer programs , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

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

[15]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[16]  Jirí Matousek,et al.  Efficient partition trees , 1991, SCG '91.

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

[18]  Kenneth L. Clarkson,et al.  RANDOMIZED GEOMETRIC ALGORITHMS , 1992 .