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:布伦特、库克和马鲁亚马(IEEE Trans. computers 1973)和布伦特(JACM 1974)的经典结果显示,任何大小的代数公式都可以转换为深度O(logs)和体积的O(logs) 。在本文中,我们考虑这一结果的细微版取决于代数公式所计算的多元值程度。考虑到一个同一的代数公式,其大小计算为多数值P度d,我们显示,P也可以用深度O(logd)和体积聚体(JACM 1974)的(未受限制的扇形)代数公式转换为深度O(logs)和体积聚体积的O(logs)。我们的证据显示,这一结果还存在于单体系、非对等数的变数公式的高度限制的设置中。这与制度下值(i.e, d.e, d., d.)的代数计算为O(logs) 和Raz(STOC,2010, JACM, 和slimal-l) 公式的精确的公式的数值定值显示,我们最近的数值定定值也无法以相同的深度的数值的数值显示,这表示,这个数值的数值的数值的数值的数值的数值的数值的数值的底值的底值显示。