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The amplification of probabilistic Boolean formulas refers to combining independent copies of such formulas to reduce the error probability. Les Valiant used the amplification method to produce monotone Boolean formulas of size O(n5.3) for the majority function of n variables. In this paper we show that the amount of amplification that Valiant obtained is optimal. In addition, using the amplification...
Let d = d(n) be the minimum d such that for every sequence of n subsets F1, F2, . . . , Fn of {1, 2, . . . , n} there exist n points P1, P2, . . . , Pn and n hyperplanes H1, H2 .... , Hn in Rd such that Pj lies in the positive side of Hi iff j ∈ Fi. Then n/32 ≤ d(n) ≤ (1/2 + 0(1)) ?? n. This implies that the probabilistic unbounded-error 2-way complexity of almost all the Boolean functions of 2p variables...
A distributed algorithm is presented that constructs the minimum-weight spanning tree of an undirected connected graph with distinct edge weights and distinct node identities. Initially each node knows only the weight of each of its adjacent edges. When the algorithm terminates, each node knows which of its adjacent edges are edges of the tree. For a graph with n nodes and e edges, the total number...
An algorithm is described that solves the all pairs shortest path problem for a nonnegatively weighted graph. The algorithm has an average requirement on quite general classes of random graphs of O(n2logn) time, where n is the number of vertices in the graph.
We present exponential lower bounds on the size of depth-k Boolean circuits for computing certain functions. These results imply that there exists an oracle set A such that, relative to A, all the levels in the polynomial-time hierarchy are distinct, i.e., ΣkP,A is properly contained in Σk+1P,A for all k.
We prove that the three extensions of first-order logic by means of positive inductions, monotone inductions, and so-called non-monotone (in our terminology, inflationary) inductions respectively, all have the same expressive power in the case of finite structures. As a by-product, the collapse of the corresponding fixed-point hierarchies can be deduced.
The verification problem for probabilistic concurrent finite-state program is to decide whether such a program satisfies its linear temporal logic specification. We describe an automata-theoretic approach, whereby probabilistic quantification over sets of computations is reduced to standard quantification over individual computations. Using new determinization construction for ω-automata, we manage...
A monotonic priority set is a new data structure which supports maximum-finding and deletions over a set of weighted points in the plane. Global updates to the weights can also be made, incrementing the weights of all points above a given threshold in one of the coordinates. The weights are assumed to be always monotonic in both coordinates. An efficient implementation of this structure is presented...
We introduce an efficient new algorithm for dynamic Huffman coding, called Algorithm V. It performs one-pass coding and transmission in real-time, and uses at most one more bit per letter than does the standard two-pass Huffman algorithm; this is optimum in the worst case among all one-pass schemes. We also analyze the dynamic Huffman algorithm due to Faller, Gallager, and Knuth. In each algorithm,...
We develop fast parallel solutions to a number of basic problems involving solvable and nilpotent permutation groups. Testing solvability is in NC, and RNC includes, for solvable groups, finding order, testing membership, finding the derived series and finding a composition series. Additionally, for nilpotent groups, one can, in RNC, find the center, a central composition series, and point-wise stabilizers...
We consider PRAM's with arbitrary computational power for individual processors, infinitely large shared memory and "priority" writeconflict resolution. The main result is that sorting n integers with n processors requires Ω(√log n) steps in this strong model. We also show that computing any symmetric polynomial (e.g. the sum or product) of n integers requires exactly log2n steps, for any...
We give an algorithm to construct a cell decomposition of Rd, including adjacency information, defined by any given set of rational polynomials in d variables. The algorithm runs in single exponential parallel time, and in NC for fixed d. The algorithm extends a recent algorithm of Ben-Or, Kozen, and Reif for deciding the theory of real closed fields.
We evaluate all the real and complex zeros λ1,...,λn of an n-th degree univariate polynomial with the relative precision 1/2nc for a given positive constant c. If for all g,h, log |λg/λh-1| ≥ 1/2O(n) unless λg = λh, then we need O(n3log2n) arithmetic operations or O(n2log n) steps, n log n processors. O(n2log n) operations or O(n log n) parallel steps, n processors suffice if either all the zeros...
This paper presents an algorithm for maximum matching on general graphs with integral edge weights, running in time O(n3/4m lg N), where n, m and N are the number of vertices, number of edges, and largest edge weight magnitude, respectively. The best previous bound is O(n(mlg lg lgd n + n lg n)) where d is the density of the graph. The algorithm finds augmenting paths in batches by scaling the weights...
We show that many Boolean functions (including, in a certain sense, "almost all" Boolean functions) have the property that the number of noisy gates needed to compute them differs from the number of noiseless gates by at most a constant factor. This may be contrasted with results of von Neumann, Dobrushin and Ortyukov to the effect that (1) for every Boolean function, the number of noisy...
The problem of scheduling a set of n jobs on m identical machines so as to minimize the makespan time is perhaps the most well-studied problem in the theory of approximation algorithms for NP-hard optimization problems. In this paper we present the strongest possible type of result for this problem, a polynomial approximation scheme. More precisely, for each ε, we give an algorithm that runs in time...
Several questions related to the complexity of communication over channels with noise are addressed. We compare some of our results to wellknown results in information theory. In particular we compare the following two problems. Assuming that the communication channel between two processors P1 and P2 makes an error with probability ε≫0, the identification problem is to determine whether P1 and P2...
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