# 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 log_{2}[*max*_{deg}(*f*)]. This result is a generalization of Kung's[K] results for a univariate polynomial which is log_{2} [*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* _{p} ^{n} ) 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* _{p} ^{n} ) 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.

## Key words

Complexity of parallel computation Multivariate polynomials Design and analysis of parallel algorithms Maximal-degree Dense polynomial## Preview

Unable to display preview. Download preview PDF.

## References

- [BM]Borodin, A., Munro, I.: The Computational Complexity of Algebraic and Numeric problems, American Elsevier Publishing Company, 1975.Google Scholar
- [K]Kung, H.: New Algorithm and Lower Bounds for the Parallel Evaluation of Certain Rational Expressions and Recurrences, J. ACM, vol.
**23**, no. 2 (1976) 252–261.Google Scholar - [H]Hyafil, K.: On the Parallel Evaluation of Multivariate Polynomials, Proc., 10th ACM Symposium on Theory of Computing, (1978) pp. 193–195.Google Scholar
- [VSBR]Valiant, L.G., Skyum, S., Berkowitz, S., Rackoff, C.: Fast Parallel Computation of Polynomials using Few Processors, SIAM J. Comput.,
**12**, No. 4, (1983) 641–644.Google Scholar - [MP]Munro, I., Paterson, M.: Optimal Algorithms for Parallel Polynomial Evaluation, J. of Computer and System Science
**7**(1973) 189–198.Google Scholar