The Exact Fitting Problem in Higher Dimensions

Abstract Let S be a family of n points in E d . The exact fitting problem is that of finding a hyperplane containing the maximum number of points of S . In this paper, we present an O(min {( n d m d−1 ) log ( n m ),n d }) time algorithm where m denotes the number of points in the hyperplane. This algorithm is based on upper bounds on the maximum number of incidences between families of points and families of hyperplanes in E d and on and algorithm to compute these incidences. We also show how the upper bound on the maximum number of incidences between families of points and families of hyperplanes can be used to derive new bounds on some well-known problems in discrete geometry.

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

[2]  Gilles Brassard,et al.  Algorithmics - theory and practice , 1988 .

[3]  Leo Moser,et al.  On The Different Distances Determined By n Points , 1952 .

[4]  Paul Erdös,et al.  Repeated distances in space , 1988, Graphs Comb..

[5]  Leonidas J. Guibas,et al.  The complexity of many cells in arrangements of planes and related problems , 1990, Discret. Comput. Geom..

[6]  Raimund Seidel,et al.  Erratum to Better Lower Bounds on Detecting Affine and Spherical Degeneracies , 1993, Discret. Comput. Geom..

[7]  Endre Szemerédi,et al.  The number of different distances determined by a set of points in the Euclidean plane , 1992, Discret. Comput. Geom..

[8]  Paul Erdös.  On some problems of elementary and combinatorial geometry , 1975 .

[9]  E. Szemerédi,et al.  Unit distances in the Euclidean plane , 1984 .

[10]  Jayadev Misra,et al.  Finding Repeated Elements , 1982, Sci. Comput. Program..

[11]  Fan Chung.  Sphere-and-point incidence relations in high dimensions with applications to unit distances and furthest-neighbor pairs , 1989 .

[12]  Leonidas J. Guibas,et al.  The exact fitting problem for points , 1991 .

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

[14]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[15]  Leonidas J. Guibas,et al.  Topologically sweeping an arrangement , 1986, STOC '86.

[16]  Godfried T. Toussaint,et al.  Computing the Width of a Set , 1988, IEEE Trans. Pattern Anal. Mach. Intell..

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

[18]  Robin Wilson,et al.  Graph theory and combinatorics , 1979 .

[19]  Hiroshi Imai,et al.  Orthogonal Weighted Linear L1 and L∞ Approximation and Applications , 1993, Discret. Appl. Math..

[20]  Jirí Matousek,et al.  Cutting hyperplane arrangements , 1990, SCG '90.

[21]  Jirí Matousek,et al.  Range searching with efficient hierarchical cuttings , 1992, SCG '92.

[22]  Gillea Brassard,et al.  Crusade for a better notation , 1985, SIGA.

[23]  H. Edelsbrunner,et al.  On the number of furthest neighbour pairs in a point set , 1989 .

[24]  Leonidas J. Guibas,et al.  Topological Sweeping in Three Dimensions , 1990, SIGAL International Symposium on Algorithms.

[25]  Leonidas J. Guibas,et al.  Combinatorial complexity bounds for arrangements of curves and spheres , 1990, Discret. Comput. Geom..

[26]  D. Avis.  The Number of Furthest Neighbour Pairs of a Finite Planar Set , 1984 .

[27]  P. Erdös.  On Sets of Distances of n Points , 1946 .

[28]  Pankaj K. Agarwal,et al.  Partitioning arrangements of lines II: Applications , 2011, Discret. Comput. Geom..

[29]  Michael E. Houle,et al.  Algorithms for Weak and Wide Separation of Sets , 1993, Discret. Appl. Math..

[30]  Kenneth L. Clarkson,et al.  A Randomized Algorithm for Closest-Point Queries , 1988, SIAM J. Comput..

[31]  Endre Szemerédi,et al.  Extremal problems in discrete geometry , 1983, Comb..