On the ultimate complexity of factorials

It has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (i.e. n ! ) by straight-line programs. Formally, we say that a sequenc...

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Vydáno v:	Theoretical computer science Ročník 326; číslo 1; s. 419 - 429
Hlavní autor:	Cheng, Qi
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 20.10.2004 Elsevier
Témata:	Algebra Algorithmics. Computability. Computer arithmetics Applied sciences Computational and structural complexity Computer science; control theory; systems Exact sciences and technology Mathematics Number theory Sciences and techniques of general use Theoretical computing Computational and structural complexity Integer Upper bound Computer theory Structural complexity Computational complexity Factorization Number theory
ISSN:	0304-3975, 1879-2294
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Abstract	It has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (i.e. n ! ) by straight-line programs. Formally, we say that a sequence of integers a n is ultimately f ( n ) -computable, if there exists a nonzero integer sequence m n such that for any n , a n m n can be computed by a straight-line program (using only additions, subtractions and multiplications) of length at most f ( n ) . Shub and Smale [12] showed that if n ! is ultimately hard to compute, then the algebraic version of NP ≠ P is true. Assuming a widely believed number theory conjecture concerning smooth numbers in a short interval, a subexponential upper bound ( exp ( c log n log log n ) ) for the ultimate complexity of n ! is proved in this paper, and a randomized subexponential algorithm constructing such a short straight-line program is presented as well.
AbstractList	It has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (i.e. n ! ) by straight-line programs. Formally, we say that a sequence of integers a n is ultimately f ( n ) -computable, if there exists a nonzero integer sequence m n such that for any n , a n m n can be computed by a straight-line program (using only additions, subtractions and multiplications) of length at most f ( n ) . Shub and Smale [12] showed that if n ! is ultimately hard to compute, then the algebraic version of NP ≠ P is true. Assuming a widely believed number theory conjecture concerning smooth numbers in a short interval, a subexponential upper bound ( exp ( c log n log log n ) ) for the ultimate complexity of n ! is proved in this paper, and a randomized subexponential algorithm constructing such a short straight-line program is presented as well.
Author	Cheng, Qi
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Keywords	Computational and structural complexity Integer Upper bound Computer theory Structural complexity Computational complexity Factorization Number theory
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Snippet	It has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly. In this paper, we...
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SubjectTerms	Algebra Algorithmics. Computability. Computer arithmetics Applied sciences Computational and structural complexity Computer science; control theory; systems Exact sciences and technology Mathematics Number theory Sciences and techniques of general use Theoretical computing
Title	On the ultimate complexity of factorials
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