Diameter, width, closest line pair, and parametric searching

We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improved solutions for them. We obtain, for any fixed ε>0, anO(n1+ε) algorithm for computing the diameter of a point set in 3-space, anO(8/5+ε) algorithm for computing the width of such a set, and onO(n8/5+ε) algorithm for computing the closest pair in a set ofn lines in space. All these algorithms are deterministic.

[1]  Jirí Matousek,et al.  Linear optimization queries , 1992, SCG '92.

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

[3]  Sivan Toledo,et al.  Extremal polygon containment problems , 1991, SCG '91.

[4]  Kenneth L. Clarkson.  Further applications of random sampling to computational geometry , 1986, STOC '86.

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

[6]  Richard Cole,et al.  Slowing down sorting networks to obtain faster sorting algorithms , 2015, JACM.

[7]  Sivan Toledo,et al.  Computing a Segment Center for a Planar Point Set , 1993, J. Algorithms.

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

[9]  Micha Sharir,et al.  Planar geometric location problems and maintaining the width of a planar set , 1991, SODA '91.

[10]  Andrew Stein,et al.  Finding the repeated median regression line , 1992, SODA '92.

[11]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[12]  Godfried T. Toussaint,et al.  Computating the width of a set , 1985, SCG '85.

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

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

[15]  Leonidas J. Guibas,et al.  A Singly Exponential Stratification Scheme for Real Semi-Algebraic Varieties and its Applications , 1991, Theor. Comput. Sci..

[16]  Micha Sharir,et al.  On the zone of a surface in a hyperplane arrangement , 1993, Discret. Comput. Geom..

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

[18]  Bernard Chazelle,et al.  An optimal convex hull algorithm and new results on cuttings , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[19]  Leslie G. Valiant,et al.  Parallelism in Comparison Problems , 1975, SIAM J. Comput..

[20]  Jirí Matousek,et al.  Randomized Optimal Algorithm for Slope Selection , 1991, Inf. Process. Lett..

[21]  Jirí Matousek,et al.  Ray shooting and parametric search , 1992, STOC '92.

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

[23]  Jirí Matousek.  Linear Optimization Queries , 1993, J. Algorithms.

[24]  Otfried Cheong,et al.  Euclidean minimum spanning trees and bichromatic closest pairs , 1991, Discret. Comput. Geom..

[25]  Jirí Matousek.  Computing the Center of Planar Point Sets , 1990, Discrete and Computational Geometry.

[26]  Leonidas J. Guibas,et al.  Lines in space-combinators, algorithms and applications , 1989, STOC '89.

[27]  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).