J. Matousek, Tight upper bounds for the discrepancy of halfspaces. KAM Series (Tech. Report), Charles University, Prague 1994  Home/Search   Document Not in Database   Summary   Related Articles   Check

....can ensure the entropy requirement and that 2 f(s) O(n ) This gives a substantial partial coloring of [n] with j(A)j = O(n ) for all A 2 F . The iteration of this method to get a full coloring (without losing a logarithmic factor ) uses interesting but noncombinatorial ideas. Matousek[15] applied entropy to discrepancy of halfplanes. Let P be a set of n points in the plane and F the family of H P , H a halfplane. Here the decomposition is more difficult, the end result again being a family G so that all A 2 F are expressible in terms of B 2 G of distinct cardinalities 2 . ....

J. Matousek, Tight upper bounds for the discrepancy of halfspaces. KAM Series (Tech. Report), Charles University, Prague 1994

.... of a somewhat smaller size can be constructed by other methods (based on a connection of approximations with discrepancy) MWW93] If the primal shatter function is bounded by Cm d for some fixed C; d (d 1) then there exist (1=r) approximations of size O(r 2 Gamma2= d 1) MWW93] Mat95b] Similarly if S (m) Cm d for all m, 1=r) approximations of size O(r 2 Gamma2= d 1) log r) 1 Gamma1= d 1) exist. Ball packing. Consider a set system (X; S) of VC dimension d on an n point set X. Define a metric on S: The distance of two sets S 1 ; S 2 2 S is the cardinality of ....

....been obtained via semi random combinatorial constructions. A large part of known upper bounds in this direction is subsumed by the following theorem bounding discrepancy in terms of shatter functions. 3.2 Theorem Let S be a set system on an n point set X, and let C and d 1 be constants. i) Mat95b] If the primal shatter function satisfies S (m) Cm d for all m = 1; 2; jXj, then disc(S) O i n 1 2 Gamma 1 2d j : ii) MWW93] If the dual shatter function satisfies S (m) Cm d for all m = 1; 2; jSj, then disc(S) O i n 1 2 Gamma 1 2d p log n j ....

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J. Matousek. Tight upper bounds for the discrepancy of halfspaces. Discr. & Comput. Geom., 13:593--601, 1995.

....of x2 (see comments in [5] are such that we have not been able to obtain an algorithm that would output this coloring in time polynomial in n. Our proof involves variants of the probabilistic method, we give [1] as a general reference. The technique of our proof combines methods of [2] 5] [6]. Throughout the paper, we ll use the symbols c; c 0 etc. generically for denoting absolute constants, and in order to limit the number of symbols, we reuse them freely. 2 Entropy Let A 1 ; A v Omega Gamma A partial coloring is a map : Omega f Gamma1; 0; 1g. When (x) 0 we ....

J. Matousek. Tight upper bounds for the discrepancy of halfspaces. Submitted to Discrete & Computational Geometry, 1994.

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