Towards Optimal Depth-Reductions for Algebraic Formulas

Electron. Colloquium Comput. Complex. Pub Date : 2023-02-14 DOI:10.48550/arXiv.2302.06984

Hervé Fournier, N. Limaye, Guillaume Malod, S. Srinivasan, Sébastien Tavenas

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引用次数: 1

Abstract

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we consider a fine-grained version of this result depending on the degree of the polynomial computed by the algebraic formula. Given a homogeneous algebraic formula of size s computing a polynomial P of degree d, we show that P can also be computed by an (unbounded fan-in) algebraic formula of depth O(log d) and size poly(s). Our proof shows that this result also holds in the highly restricted setting of monotone, non-commutative algebraic formulas. This improves on previous results in the regime when d is small (i.e., d<

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代数公式的最优深度约简

Brent, Kuck和Maruyama的经典结果(IEEE译)。计算机1973)和Brent (JACM 1974)表明，任何大小为s的代数公式都可以转换为深度为O(log s)的代数公式，而只需要在大小上增加一个多项式。在本文中，我们考虑这个结果的一个细粒度版本，这取决于由代数公式计算的多项式的程度。给定一个大小为s的齐次代数公式来计算阶数为d的多项式P，我们证明P也可以由深度为O(log d)和大小为poly(s)的(无界扇入)代数公式来计算。我们的证明表明，这一结果在单调非交换代数公式的高度限制情况下也成立。当d很小(即d<

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Electron. Colloquium Comput. Complex.

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