## 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} 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 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^{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.

Original language | English |
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Title of host publication | Theory of Computing and Systems - ISTCS 1992, Israel Symposium, Proceedings |

Editors | Danny Dolev, Zvi Galil, Zvi Galil, Michael Rodeh |

Publisher | Springer Verlag |

Pages | 147-153 |

Number of pages | 7 |

ISBN (Print) | 9783540555537 |

DOIs | |

State | Published - 1992 |

Event | Israel Symposium on the Theory of Computing and Systems, ISTCS 1992 - Haifa, Israel Duration: 27 May 1992 → 28 May 1992 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 601 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | Israel Symposium on the Theory of Computing and Systems, ISTCS 1992 |
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Country/Territory | Israel |

City | Haifa |

Period | 27/05/92 → 28/05/92 |

## Keywords

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