Abstract
It has long been observed that certain factorization algorithms provide a way to write product of a lot of integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (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 algebraic version of NP ≠ P is true. Assuming a widely believed number theory conjecture concerning smooth numbers in 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 random subexponential algorithm constructing such a short straight-line program is presented as well.
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Cheng, Q. (2003). On the Ultimate Complexity of Factorials. In: Alt, H., Habib, M. (eds) STACS 2003. STACS 2003. Lecture Notes in Computer Science, vol 2607. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36494-3_15
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DOI: https://doi.org/10.1007/3-540-36494-3_15
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