Probability and algorithms enjoy an almost boisterous interaction that has led to an active, extensive literature that touches fields as diverse as number theory and the design of computer hardware. This article offer...
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Probability and algorithms enjoy an almost boisterous interaction that has led to an active, extensive literature that touches fields as diverse as number theory and the design of computer hardware. This article offers a gentle introduction to the simplest, most basic ideas that underlie this development.
Abstract: We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e., in the degree of th...
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Abstract: We present a probabilistic algorithm that finds the irreducible factors of a bivariate polynomial with coefficients from a finite field in time polynomial in the input size, i.e., in the degree of the polynomial and log (cardinality of field). The algorithm generalizes to multivariate polynomials and has polynomial running time for densely encoded inputs. A deterministic version of the algorithm is also discussed, whose running time is polynomial in the degree of the input polynomial and the size of the field.
It follows from the famous result of Cook about the NP-completeness of the Boolean satisfiability problem that there is no polynomial algorithm for this problem if P not equal NP. In this paper, we prove that the Bool...
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It follows from the famous result of Cook about the NP-completeness of the Boolean satisfiability problem that there is no polynomial algorithm for this problem if P not equal NP. In this paper, we prove that the Boolean satisfiability problem remains computationally hard on polynomial strongly generic subsets of formulas provided P not equal NP and P = BPP. Boolean formulas are represented in the natural way by labeled binary trees.
Skip lists, introduced by Pugh, provide an alternative to search trees, although a precise analysis of their behaviour had been elusive. The exact value of the expected cost for the search of the mth element in a skip...
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Skip lists, introduced by Pugh, provide an alternative to search trees, although a precise analysis of their behaviour had been elusive. The exact value of the expected cost for the search of the mth element in a skip list of n elements is derived first in terms of previously studied functions, and secondly as an asymptotic expression. The latter suggests that Pugh's upper bound of the expected search cost is fairly tight for the interesting cases. Assuming a uniform query distribution, the exact and an asymptotic value of the average (over all m) expected search cost in a skip list of n elements is also derived. Finally, all insert and delete costs are obtained.
Let AlAlAl be a polynomial time nondeterministic algorithm accepting a set A, and let a∈Aa∈Aa \in
Let AlAl be a polynomial time nondeterministic algorithm accepting a set A, and let a∈Aa∈A. As a consequence, an algorithm for testing primality is proposed (under the extended Riemann hypothesis) which is more efficient than ones proposed by Miller [Pros. 7th ACM Symp. Theory of Computing, 1975, pp. 234–239] and Vélu [SIGACT News, 10 (1978), pp. 58–59]. A probabilistic version is compared with the algorithm of Solovay and Strassen [SIAM J. Comput., 6 (1977), pp. 84–85; erratum, 7 (1978), p. 118].
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