Multiple Product Modulo Arbitrary Numbers

AbstractLetnbinary numbers of lengthnbe given. The Boolean function “Multiple Product”MPnasks for (some binary representation of ) the value of their product. It has been shown (K.-Y. Siu and V. Roychowdhury, On optimal depth threshold circuits for multiplication and related problems,SIAM J. Discrete Math.7, 285–292 (1994)) that this function can be computed in polynomial-size threshold circuits of depth 4. For many other arithmetic functions, circuits of depth 3 are known. They are mostly based on the fact that the value of the considered function modulo some prime numbers p can be computed easily in threshold circuits of depth 2. In this paper, we investigate the complexity of computingMPnmodulomby depth-2 threshold circuits. It turns out that for all but a few integersm, exponential size is required. In particular, it is shown that form∈{2, 4, 8}, polynomial-size circuits exist, form∈{3, 6, 12, 24}, the question remains open and in all other cases, exponential-size circuits are required. The result still holds if we allowmto grow withn

The Average-Case Area of Heilbronn-Type Triangles

From among ${n \choose 3}$ triangles with vertices chosen from $n$ points in the unit square, let $T$ be the one with the smallest area, and let $A$ be the area of $T$. Heilbronn's triangle problem asks for the maximum value assumed by $A$ over all choices of $n$ points. We consider the average-case: If the $n$ points are chosen independently and at random (with a uniform distribution), then there exist positive constants $c$ and $C$ such that $c/n^3 < \mu_n < C/n^3$ for all large enough values of $n$, where $\mu_n$ is the expectation of $A$. Moreover, $c/n^3 < A < C/n^3$, with probability close to one. Our proof uses the incompressibility method based on Kolmogorov complexity; it actually determines the area of the smallest triangle for an arrangement in ``general position.''Comment: 13 pages, LaTeX, 1 figure,Popular treatment in D. Mackenzie, On a roll, {\em New Scientist}, November 6, 1999, 44--4

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

The Algorithmic 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 Tur&apos;an 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(n³). 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

MODp-tests, almost independence and small probability spaces. Random Structures Algorithms

In this paper, we consider approximations of probability distributions over Z n p. We present an approach to estimate the quality of approximations of probability distributions towards the construction of small probability spaces. These are used to derandomize algorithms. In contrast to results by Even, Goldreich, Luby, Nisan and Velickovic [EGLNV], our methods are simple, and for reasonably small p, we get smaller sample spaces. Our considerations are motivated by a problem which was mentioned in recent work of Azar, Motwani and Naor [AMN], namely, howto construct in time polynomial in n a good approximation to the joint probability distribution of the random variables X1;X2;:::;Xn where each Xi has values in f0; 1g and satis es Xi = 0 with probability q and Xi = 1 with probability 1,q where q is arbitrary. Our considerations improve on results by [EGLNV] and [AMN].

Sparse 0-1 matrices and forbidden hypergraphs

We consider the problem of determining the maximal number N(m; k; r) of columns of a 0-1-matrix with m rows and exactly r ones in each column such that every k columns are linearly independent over Z2. For xed integers k 4 and r 2 where k is even and gcd(k,1;r)= 1, we shall prove the probabilistic lower bound N(m; k; r) = (m kr 2(k,1) (ln m) 1 k,1). This improves on earlier results from [13] by the factor ((ln m) 1 k,1) and extends results from [11] where the case r =2was considered. Moreover, we give a polynomial time algorithm achieving this new lower bound.