On the Ultimate Complexity of Factorials

Cheng, Qi

doi:10.1007/3-540-36494-3_15

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…A nontrivial upper bound on the arithmetic complexity of computing certain multiples of n! has been published recently in Cheng (2003). This upper bound falls short of showing that n!…”

Section: Preliminariesmentioning

confidence: 91%

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Valiant?s model and the cost of computing integers

Koiran

2005

comput. complex.

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Abstract. Let τ (n) be the minimum number of arithmetic operations required to build the integer n ∈ N from the constants 1 and 2. A sequence x n is said to be "easy to compute" if there exists a polynomial p such that τ (x n ) ≤ p(log n) for all n ≥ 1. It is natural to conjecture that sequences such as 2 n ln 2 or n! are not easy to compute. In this paper we show that a proof of this conjecture for the first sequence would imply a superpolynomial lower bound for the arithmetic circuit size of the permanent polynomial. For the second sequence, a proof would imply a superpolynomial lower bound for the permanent or P = PSPACE.

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“…A nontrivial upper bound on the arithmetic complexity of computing certain multiples of n! has been published recently in Cheng (2003). This upper bound falls short of showing that n!…”

Section: Preliminariesmentioning

confidence: 91%

“…is not easy to compute: if it is then "factoring is easy" (see for instance Blum et al (1998), p. 126, andCheng (2003)). Note however that if division (computing remainder and quotient) is allowed, n!…”

Section: Preliminariesmentioning

confidence: 99%

Valiant?s model and the cost of computing integers

Koiran

2005

comput. complex.

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“…is easy or ultimately easy. The reader is referred to [25,22,13] for more references. A goal of this section is to relate these questions to the complexity of ACIT.…”

Section: Posslp Lies In Chmentioning

confidence: 99%

On the Complexity of Numerical Analysis

Allender

Kjeldgaard-Pedersen²,

Bürgisser

et al.

21st Annual IEEE Conference on Computational Complexity (CCC'06)

334

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Abstract. We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis:• The Blum-Shub-Smale model of computation over the reals.• A problem we call the "Generic Task of Numerical Computation," which captures an aspect of doing numerical computation in floating point, similar to the "long exponent model" that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer N , decide whether N > 0.• In the Blum-Shub-Smale model, polynomial time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture.• The Generic Task of Numerical Computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy -the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE.In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.

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“…over Z or modulo p takes about n 1=2 arithmetic operations in the corresponding ring, see [1], [2]. The best known algorithm to compute n!…”

Section: Introductionmentioning

confidence: 99%

Exponential sums and congruences with factorials

Garaev

Luca

Shparlinski³

2005

Journal für die reine und angewandte Mathematik (Crelles Journa

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We estimate the number of solutions of certain diagonal congruences involving factorials. We use these results to bound exponential sums with products of two factorials n!m! and also derive asymptotic formulas for the number of solutions of various congruences with factorials. For example, we prove that the products of two factorials n!m! with maxfn; mg < p 1=2þe are uniformly distributed modulo p, and that any residue class modulo p is representable in the form m!n! þ n 1 ! þ Á Á Á þ n 47 ! with maxfm; n; n 1 ; . . . ; n 47 g < p 1350=1351þe .

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On the Ultimate Complexity of Factorials

Cited by 4 publications

References 14 publications

Valiant?s model and the cost of computing integers

Valiant?s model and the cost of computing integers

On the Complexity of Numerical Analysis

Exponential sums and congruences with factorials

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