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In this article we state and prove a corrected version of Theorem 3.5 in [SIAM J. Comput., 18 (1989), pp. 859--881]. [PUBLICATION ABSTRACT]

Pseudorandom generators are fundamental to many theoretical and applied aspects of computing. We show how to construct a pseudorandom generator from any one-way function. Since it is easy to construct a one-way function from a pseudorandom generator, this result shows that there is a pseudorandom generator if and only if there is a one-way function.

By a sleeping bag for a baby snake in d dimensions we mean a subset of R^sup d^ which can cover, by rotation and translation, every curve of unit length. We construct sleeping bags which are smaller than any previously known in dimensions 3 and higher. In particular, we construct a three-dimensional sleeping bag of volume approximately 0.075803. For large d we construct d -dimensional sleeping bags with volume less than (c\\sqrt log d )^sup d^ / d^sup 3d/2^ for some constant c . To obtain the last result, we show that every curve of unit length in R^sup d^ lies between two parallel hyperplanes at distance at most c^sub 1^ d^sup -3/2^ \\sqrt log d , for some constant c^sub 1^ .[PUBLICATION ABSTRACT]

In this article we state and prove a corrected version of Theorem 3.5 in [SIAM J. Comput., 18 ( 1989), pp. 859-881].

We prove that if we hit a de Morgan formula of size L with a random restriction from Rp, then the expected remaining size is at most $O(p^2(\\log \\frac {1}{p})^{3/2}L+p\\sqrt L)$. As a corollary we obtain an $\\Omega(n^{3-o(1)})$-formula-size lower bound for an explicit function in P. This is the strongest known lower bound for any explicit function in NP.

In this paper, we analyze the stochastic behavior of backoff protocols for multiple access channels such as the Ethernet. In particular, we prove that binary exponential backoff is unstable if the arrival rate of new messages at each station is $\\tfrac{\\lambda }{N}$ for any $\\lambda > \\frac{1}{2}$ and the number of stations $N$ is sufficiently large. For small $N$, we prove that $\\lambda \\geqslant \\lambda _0 + \\frac{1}{{4N - 2}}$` implies instability, where $\\lambda _0 \\approx .567$. More importantly, we also prove that any superlinear polynomial backoff protocol (e.g., quadratic backoff) is stable for any set of arrival rates that sum to less than one and any number of stations. The results significantly extend the previous work in the area and provide the first examples of acknowledgment-based protocols known to be stable for a nonnegligible overall arrival rate distributed over an arbitrarily large number of stations. The results also disprove a popular assumption that exponential backoff is the best choice among acknowledgment-based protocols for systems with large overall arrival rates. Finally, we prove that any linear or sublinear backoff protocol is unstable if the arrival rate at each station is $\\frac{\\lambda }{N}$ for any fixed $\\lambda $ and sufficiently large $N$.

We introduce the notion of covering complexity of a veri er for probabilistically checkable proofs (PCPs). Such a veri er is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The veri er is also given a random string and decides whether to accept the proof or not, based on the given random string. We de ne the covering complexity of such a veri er, on a given input, to be the minimum number of proofs needed to satisfy the veri er on every random string; i.e., on every random string, at least one of the given proofs must be accepted by the veri er. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems and, in particular, (hyper) graph coloring problems. We present a PCP veri er for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a superconstant covering complexity for statements not in the language. Moreover, the acceptance predicate of this veri er is a simple not-all-equal check on the four bits it reads. This enables us to prove that, for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors and also yields a superconstant inapproximability result under a stronger hardness assumption.

This paper considers variants and generalizations of the following computational problem. Given a real input ${\\bf x} \\in \\mathbb{R}^n $, find a small integer relation ${\\bf m}$ for $x$ that is a nonzero vector $m \\in \\mathbb{Z}^n $ orthogonal to ${\\bf x}$, or prove that no integer relation ${\\bf m}$ exists with $\\|{\\bf m}\\| \\leqq 2^\\lambda $. An algorithm is presented that solves this problem in $O(n^3 (k + n))$ arithmetic operations over real numbers. The algorithm is a variation of the multidimensional Euclidean algorithm proposed by Ferguson and Forcade [Bull. Amer. Math. Soc., 1(1979), pp. 912-914] and Bergman [Notes on Ferguson and Forcade's Generalized Euclidean Algorithm, University of California, Berkeley, CA, 1980]. A connection between such multidimensional Euclidean algorithms and the Lattice Basis Reduction Algorithm of Lenstra, Lenstra Jr., and Lovász [Math. Ann., 21 (1982), pp. 515-534] is shown. Polynomial time solutions are also established for finding linearly independent sets of small integer relations and for finding small simultaneous integer relations for several real vectors, using real input vectors and counting arithmetic operations over real numbers at unit cost. For integer input vectors ${\\bf x}$ a different algorithm is given for finding integer relations (that always exist) that uses at most $O(n^3 \\log \\|x\\|)$ arithmetic operations on $O(n + \\log \\|{\\bf x}\\|)$ bit integers.

It is proved that if $n$ is a power of 2, then there is a threshold function on $n$ inputs that requires weights of size around $2^{( n \\log n )/2 - n} $. This almost matches the known upper bounds.