Bounds on parallel computation of multivariate polynomials

Abstract

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 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(d ⁿ_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(d ⁿ_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.

This article was processed using the LAT_EX macro package with LLNCS style

This author was supported by Eshkol Fellowship No. 0375 administered by the Israeli Ministry of Science.