An Eecient Multi-dimensional Searching Technique and Its Applications an Eecient Multi-dimensional Searching Technique and Its Applications

This paper describes an improved algorithm for the multi-dimensional searching problem introduced by Megiddo. As a result, we obtain a d O(d) n time deterministic algorithms for linear programming in R d with n constraints, for computing the Euclidean 1-center of a set of n points in R d , for computing the minimum enclosing ellipsoid of a set of n points in R d , etc. Our techniques also improve the running time of known algorithms for a number of parametric graph searching problems, including that of nding zero cycles in dynamic graphs 7].

[1]  Raimund Seidel,et al.  Small-dimensional linear programming and convex hulls made easy , 1991, Discret. Comput. Geom..

[2]  Edith Cohen,et al.  Strongly polynomial-time and NC algorithms for detecting cycles in dynamic graphs , 1989, STOC '89.

[3]  David Fernández-Baca,et al.  Weighted Multidimensional Search and Its Application to Convex Optimization , 1996, SIAM J. Comput..

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

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

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

[7]  L. G. H. Cijan.  A polynomial algorithm in linear programming , 1979 .

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

[9]  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.

[10]  Micha Sharir,et al.  A subexponential bound for linear programming , 1992, SCG '92.

[11]  David Fernández-Baca,et al.  Parametric Problems on Graphs of Bounded Tree-Width , 1992, SWAT.

[12]  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.

[13]  Jirí Matousek,et al.  Spanning trees with low crossing number , 1991, RAIRO Theor. Informatics Appl..

[14]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

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

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

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

[18]  Éva Tardos,et al.  Using separation algorithms in fixed dimension , 1990, SODA '90.

[19]  Martin E. Dyer,et al.  On a Multidimensional Search Technique and its Application to the Euclidean One-Centre Problem , 1986, SIAM J. Comput..

[20]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.