Testing non-uniform k-wise independent distributions over product spaces (extended abstract)

A distribution D over Σ1× ⋯ ×Σ n is called (non-uniform) k-wise independent if for any set of k indices {i 1, ..., i k } and for any z1zki1ik, PrXD[Xi1Xik=z1zk]=PrXD[Xi1=z1]PrXD[Xik=zk]. We study the problem of testing (non-uniform) k-wise independent distributions over product spaces. For the uniform case we show an upper bound on the distance between a distribution D from the set of k-wise independent distributions in terms of the sum of Fourier coefficients of D at vectors of weight at most k. Such a bound was previously known only for the binary field. For the non-uniform case, we give a new characterization of distributions being k-wise independent and further show that such a characterization is robust. These greatly generalize the results of Alon et al. [1] on uniform k-wise independence over the binary field to non-uniform k-wise independence over product spaces. Our results yield natural testing algorithms for k-wise independence with time and sample complexity sublinear in terms of the support size when k is a constant. The main technical tools employed include discrete Fourier transforms and the theory of linear systems of congruences.National Science Foundation (U.S.) (NSF grant 0514771)National Science Foundation (U.S.) (grant 0728645)National Science Foundation (U.S.) (Grant 0732334)Marie Curie International Reintegration Grants (Grant PIRG03-GA-2008-231077)Israel Science Foundation (Grant 1147/09)Israel Science Foundation (Grant 1675/09)Massachusetts Institute of Technology (Akamai Presidential Fellowship

Sparse 0-1-matrices and forbidden hypergraphs

Available from TIB Hannover: RR 8071(97-11) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDeutsche Forschungsgemeinschaft (DFG), Bonn (Germany)DEGerman

An algorithm for Heilbronn's problem

Heilbronn conjectured that given arbitrary n points form R"2, located in the unit square (or circle), there must be three points which form a triangle of area at most O(1/n"2). This conjecture was proved false by a nonconstructive argument of Komlos, Pintz and Szemeredi [KPS] wo showed that there is a configuration of n points in the unit square where all triangles have area at least #OMEGA#(log n/n"2). In this paper, we provide a polynomial-time algorithm which for every n computes such as configuration of points. We then consider a generalization of this problem as introduced by Schmidt [Sc] to convex hulls of k #>=#4 points. We obtain the following result: for every k, there is a polynomial-time algorithm which on input n computes n points in the unit square such that the convex hull of any k points has area at least #OMEGA#(1/n"("k"-")"/"("k"-"2")). For k=4, the existence of such a configuration has been proved in [Sc]. (orig.)Available from TIB Hannover: RR 8071(97-1)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

The algorithmics aspects of uncrowded hypergraphs

We consider the problem of finding deterministically a large independent set of guaranteed size in a hypergraph on n vertices and with m edges. With respect to the Turan bound, the quality of our solutions is for hypergraphs with not too many small cycles by a logarithmic factor in the input size better. The algorithms are fast; they often have a running time of O(m)+o(n"3). Indeed, the denser the hypergraphs are the more close are the running times to the linear ones. This gives for the first time for some combinatorial problems algorithmic solutions with state-of-the-art quality, solutions of which so far only the existence was known. In some cases, the corresponding upper bounds match the lower bounds up to constant factors. The involved concepts are uncrowded hypergraphs. (orig.)Available from TIB Hannover: RR 8071(97-7)+a / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

Proper bounded edge-colorings

SIGLEAvailable from TIB Hannover: RR 8071(97-12) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman