Size-Depth Tradeoffs for Algebraic Formulae

We prove some tradeoffs between the size and depth of algebraic formulae. In particular, we show that, for any fixed ffl ? 0, any algebraic formula of size S can be converted into an equivalent formula of depth O(log S) and size O(S 1+ffl ). This result is an improvement over previously-known results where, to obtain the same depth bound, the formula-size is \Omega\Gamma S ff ), with ff 2. 1 Introduction A classical result, due to Brent (1974), implies that for any algebraic formula there is an algebraic circuit of "small" depth and "similar" size that computes the same function. More precisely, if the formula has size S then the circuit has depth O(log S) and size O(S). This result holds for formulae over any field. We believe that a natural question to consider is whether for any algebraic formula there is an equivalent formula of small depth and similar size. Since any circuit of depth O(log S) can be transformed into a formula of the same depth and with size polynomial in S,..