Towards Optimal Depth-Reductions for Algebraic Formulas

TL;DR

This paper presents a refined depth reduction technique for algebraic formulas that depends on the polynomial degree, achieving depth O(log d) with polynomial size blow-up, especially effective for low-degree polynomials.

Contribution

It introduces a degree-dependent depth reduction for algebraic formulas, extending classical results to low-degree and monotone formulas, with implications for algebraic complexity lower bounds.

Findings

Depth reduction from size s to depth O(log d) for degree d polynomials.

Extension of results to monotone and non-commutative formulas.

Optimality of these bounds in the monotone setting.

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<<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.