Towards Optimal Depth-Reductions for Algebraic Formulas
Résumé
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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https://hal.science/hal-04007228
Soumis le : lundi 27 février 2023-22:05:33
Dernière modification le : vendredi 26 avril 2024-13:55:09
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Hervé Fournier, Nutan Limaye, Guillaume Malod, Srikanth Srinivasan, Sébastien Tavenas. Towards Optimal Depth-Reductions for Algebraic Formulas. 2023. ⟨hal-04007228⟩
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