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On the degree of symmetric functions on the Boolean cube
, 2010
"... In this paper we study the degree of nonconstant symmetric functions f: {0, 1} n → {0, 1,..., c}, where c ∈ N, when represented as polynomials over the real numbers. We show that as long as c < n it holds that deg(f) = Ω(n). As we can have deg(f) = 1 when c = n, our result shows a surprising t ..."
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threshold phenomenon. The question of lower bounding the degree of symmetric functions on the Boolean cube was previously studied by von zur Gathen and Roche [GR97] who showed the lower bound deg(f) ≥ n+1 c+1 and so our result greatly improves this bound. When c = 1, namely the function maps the Boolean
Generalized Shuffle Permutations on Boolean Cubes
 J. PARALLEL AND DISTRIBUTED COMPUTING
, 1991
"... In a generalized shuffle permutation an address (a q\Gamma1 a q\Gamma2 : : : a 0 ) receives its content from an address obtained through a cyclic shift on a subset of the q dimensions used for the encoding of the addresses. Bitcomplementation may be combined with the shift. We give an algorithm th ..."
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Cited by 10 (2 self)
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In a generalized shuffle permutation an address (a q\Gamma1 a q\Gamma2 : : : a 0 ) receives its content from an address obtained through a cyclic shift on a subset of the q dimensions used for the encoding of the addresses. Bitcomplementation may be combined with the shift. We give an algorithm that requires K 2 + 2 exchanges for K elements per processor, when storage dimensions are part of the permutation, and concurrent communication on all ports of every processor is possible. The number of element exchanges in sequence is independent of the number of processor dimensions oe r in the permutation. With no storage dimensions in the permutation our best algorithm requires (oe r + 1)d K 2oe r e element exchanges. We also give an algorithm for oe r = 2, or the real shuffle consists of a number of cycles of length two, that requires K 2 +1 element exchanges in sequence when there is no bit complement. The lower bound is K 2 for both real and mixed shuffles with no bit compl...
On the degree of symmetric functions on the Boolean cube
"... The research thesis was done under the supervision of Assoc. Prof. Amir ..."
Harmonicity and Invariance on Slices of the Boolean Cube
, 2015
"... In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for lowinfluence, lowdegree functions. Here we provide an alternative proof for general lowdegree functions, with no constraints on the influences. We show that any realvalued function on the slice, whose de ..."
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proofs for the facts that 1) one cannot distinguish between the slice and the cube based on functions of o(n) coordinates and 2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is o( n). ∗This material is based upon work
A brief introduction to Fourier analysis on the Boolean cube
 Theory of Computing Library– Graduate Surveys
, 2008
"... Abstract: We give a brief introduction to the basic notions of Fourier analysis on the ..."
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Cited by 34 (4 self)
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Abstract: We give a brief introduction to the basic notions of Fourier analysis on the
Friedgut–Kalai–Naor theorem for slices of the Boolean cube
, 2014
"... The Friedgut–Kalai–Naor theorem states that if a Boolean function f: {±1}n → {±1} is close (in L2distance) to an affine function `(x1,..., xn) = c0+ i cixi, then f is close to a Boolean affine function (which necessarily depends on at most one coordinate). We prove a similar theorem for functions ..."
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Cited by 2 (1 self)
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The Friedgut–Kalai–Naor theorem states that if a Boolean function f: {±1}n → {±1} is close (in L2distance) to an affine function `(x1,..., xn) = c0+ i cixi, then f is close to a Boolean affine function (which necessarily depends on at most one coordinate). We prove a similar theorem for functions
COMPUTING THE PARTITION FUNCTION OF A POLYNOMIAL ON THE BOOLEAN CUBE
, 2015
"... Abstract. For a polynomial f: {−1, 1}n − → C, we define the partition function as the average of eλf(x) over all points x ∈ {−1, 1}n, where λ ∈ C is a parameter. We present an algorithm, which, given such f, λ and > 0 approximates the partition function within a relative error of in NO(lnn−ln ) ..."
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Abstract. For a polynomial f: {−1, 1}n − → C, we define the partition function as the average of eλf(x) over all points x ∈ {−1, 1}n, where λ ∈ C is a parameter. We present an algorithm, which, given such f, λ and > 0 approximates the partition function within a relative error of in NO(lnn−ln ) time provided λ  ≤ (2L√d)−1, where d is the degree, L is (roughly) the Lipschitz constant of f and N is the number of monomials in f. We apply the algorithm to approximate the maximum of a polynomial f: {−1, 1}n − → R. 1. Introduction and
BiLipschitz Bijection between the Boolean Cube and the Hamming Ball
, 2013
"... We construct a biLipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n ∈ N there exists an explicit bijection ψ: {0, 1}n → {x ∈ {0, 1}n+1: x > n/2} such that for every x 6 = y ∈ {0, 1}n it holds that 1 5 ≤ distance(ψ(x), ψ(y) ..."
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Cited by 3 (2 self)
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We construct a biLipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n ∈ N there exists an explicit bijection ψ: {0, 1}n → {x ∈ {0, 1}n+1: x > n/2} such that for every x 6 = y ∈ {0, 1}n it holds that 1 5 ≤ distance(ψ(x), ψ
INCREASING THE EFFICIENCY OF PROLOG LEXICAL DATABASES WITH NGRAM BOOLEAN CUBES
"... PROLOG has been shown to be an effective tool for expressing the logic of many problems dealing with parsing, natural language processing, and spelling verification [1,7,8,9,12]. As a class, these problems deal with the manipulation of lexical databases as Horn clauses. Since PROLOG does not general ..."
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PROLOG has been shown to be an effective tool for expressing the logic of many problems dealing with parsing, natural language processing, and spelling verification [1,7,8,9,12]. As a class, these problems deal with the manipulation of lexical databases as Horn clauses. Since PROLOG does not generally differentiate between program clauses and data clauses, the internal representation and manipulation of data may not be optimal for a particular application. This paper discusses an alternative method of representing and manipulating lexical databases through the use of Ngram analysis, prefiltering, and integration with another high level language.
Construction of Hamiltonian Cycles with a Given Spectrum of Edge Directions in an nDimensional Boolean Cube
"... AbstractThe spectrum of a Hamiltonian cycle (of a Gray code) in an ndimensional Boolean cube is the series a = (a 1 , . . . , a n ), where a i is the number of edges of the ith direction in the cycle. The necessary conditions for the existence of a Gray code with the spectrum a are available: the ..."
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AbstractThe spectrum of a Hamiltonian cycle (of a Gray code) in an ndimensional Boolean cube is the series a = (a 1 , . . . , a n ), where a i is the number of edges of the ith direction in the cycle. The necessary conditions for the existence of a Gray code with the spectrum a are available
Results 1  10
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6,751