TY - GEN
T1 - Bounds on parallel computation of multivariate polynomials
AU - Sade, Ilan
AU - Averbuch, Amir
N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1992.
PY - 1992
Y1 - 1992
N2 - We consider the problem of fast parallel evaluation of multivariate polynomials over a field F. We define “maximal-degree” (maxdeg) of a multivariate polynomial f as maxi degxi(f(xi,⋯,xn)) 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[maxdeg(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 maxdeg(f)≜dp>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(dnp) sequential steps. The simple algorithm requires only n log dp+O(n√log dp) parallel steps The second algorithm has parallel complexity, measured by the depth of the circuit, depth(G) ≤ n(log dp+β)+log n+√log dp where β ≤ √log dp. If C=Ω(dnp) 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.
AB - We consider the problem of fast parallel evaluation of multivariate polynomials over a field F. We define “maximal-degree” (maxdeg) of a multivariate polynomial f as maxi degxi(f(xi,⋯,xn)) 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[maxdeg(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 maxdeg(f)≜dp>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(dnp) sequential steps. The simple algorithm requires only n log dp+O(n√log dp) parallel steps The second algorithm has parallel complexity, measured by the depth of the circuit, depth(G) ≤ n(log dp+β)+log n+√log dp where β ≤ √log dp. If C=Ω(dnp) 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.
KW - Complexity of parallel computation
KW - Dense polynomial
KW - Design and analysis of parallel algorithms
KW - Maximal-degree
KW - Multivariate polynomials
UR - http://www.scopus.com/inward/record.url?scp=85029764296&partnerID=8YFLogxK
U2 - 10.1007/bfb0035174
DO - 10.1007/bfb0035174
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AN - SCOPUS:85029764296
SN - 9783540555537
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 147
EP - 153
BT - Theory of Computing and Systems - ISTCS 1992, Israel Symposium, Proceedings
A2 - Dolev, Danny
A2 - Galil, Zvi
A2 - Galil, Zvi
A2 - Rodeh, Michael
PB - Springer Verlag
T2 - Israel Symposium on the Theory of Computing and Systems, ISTCS 1992
Y2 - 27 May 1992 through 28 May 1992
ER -