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<<s). In particular, for the setting of d=O(log s), along with a result of Raz (STOC 2010, JACM 2013), our result implies the same depth reduction even for inhomogeneous formulas. This is particularly interesting in light of recent algebraic formula lower bounds, which work precisely in this ``low-degree" and ``low-depth" setting. We also show that these results cannot be improved in the monotone setting, even for commutative formulas.

翻译：Brent、Kuck和Maruyama（《IEEE Transactions on Computers》，1973年）以及Brent（《JACM》，1974年）的经典结果表明，任何规模为s的代数公式都可以转化为深度为O(log s)的公式，且规模仅多项式增长。本文考虑该结果的一个细粒度版本，其依赖于代数公式所计算多项式的次数。给定一个计算d次多项式P的齐次代数公式，其规模为s，我们证明P也可以用深度为O(log d)且规模为poly(s)的（无扇入限制）代数公式计算。我们的证明表明，该结果在单调、非交换代数公式的高度受限设定下依然成立。这改进了当d较小时（即d<<s）的先前结果。特别地，对于d=O(log s)的设定，结合Raz的结果（STOC 2010，JACM 2013），我们的结果意味着即使对于非齐次公式也能实现相同的深度归约。鉴于近期代数公式下界结果恰好适用于这一“低次数”和“低深度”设定，这一点尤为值得关注。我们还证明，在单调设定下，即使对于交换公式，这些结果也无法进一步改进。