Bounds on parallel computation of multivariate polynomials

We consider the problem of fast parallel evaluation of multivariate polynomials over a field F. We define “maximal-degree” (max_deg) of a multivariate polynomial f as max_i deg_xi(f(x_i,⋯,x_n)) i=1,⋯, n. The first lower bound result states that if a circut G evaluates a multivariate polynomial f, where its nodes are capable of performing (+,*), then the depth(G) is not less than log₂[max_deg(f)]. This result is a generalization of Kung’s[K] results for a univariate polynomial which is log₂ [deg f]. In the second part, we consider the circuit G which evaluates an arbitrary polynomial f in n variables with max_deg(f)≜d_p>1. We present two algorithms that achieve better performance than the classical results of Hyafil[H] and Valiant et al. [VSBR] for most classes of multivariate polynomials. For the class of” dense” polynomials the results are closed to the theorethical bound log C, where C is the sequential complexity.The algorithms generalize Munro-Paterson[MP] method for the univariate case. It should be noticed that the bound obtained by Hyafil[H] and Valiant, Skyum, Berkowitz and Rackoff[VSBR] is not sufficiently tight for the worst-sequential case (dense multivariate polynomials) and their bound can be reduced by the factor of log d while the number of required processors is only O(C). The best improvement is achieved in a case of a”dense” multivariate polynomial. A polynomial is dense if the computation necessitates Ω(dⁿ_p) sequential steps. The simple algorithm requires only n log d_p+O(n√log d_p) parallel steps The second algorithm has parallel complexity, measured by the depth of the circuit, depth(G) ≤ n(log d_p+β)+log n+√log d_p where β ≤ √log d_p. If C=Ω(dⁿ_p) then it is less than log C+√n log C where C is the number of sequential steps, and it requires only O(C) processors. The second algorithm is slightly better than the” simple” one. The improvement is achieved when β is small. The improvement of both algorithms in the parallel complexity and the number of processors with respect to Valiant is significant for most classes of multivariate polynomials.