Abstract
For a class of matrices defining exponents of variables in a system of monomials, a nontrivial lower bound of complexity is found (where the complexity is defined as the minimum number of multiplications required to compute the system starting from variables). An example of a sequence of matrices (systems of monomials, respectively) is also given so that the usage of inverse values of variables (in addition to the variables themselves) makes the complexity asymptotically two times less.
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References
N. Pippenger, “On Evaluation of Powers and Monomials,” SIAM J. Comput. 9(2), 230 (1980).
O. B. Lupanov, “On an Approach to Synthesis of Control Systems, the Principle of Local Coding,” Probl. Kibernetiki 14 31 (1965).
J. E. Savage, The Complexity of Computing, (Wiley-Interscience, 1976; Factorial, Moscow, 1998).
A. F. Sidorenko, “The Complexity of Additive Computation of Integer Linear Form Families,” Zapiski Nauch. Sem. LOMI 105, 53 (1981).
V. V. Kochergin, “On the Complexity of Computation of Monomials and Systems of Integer Linear Forms,” in Discrete Mathematics and its Applications: Collected Lectures of Youth Scientific Schools on Discrete Mathematics and its Applications, Vol. 3 (Izd-vo Keldysh IPM RAS, Moscow, 2007), pp. 3–63 [in Russian].
D. E. Knuth, The Art of Computer Programming, Vol. 2 (Addison-Wesley, 1969; Mir, Moscow, 1977).
D. E. Knuth and C. H. Papadimitriou, “Duality in Addition Chains,” Bull. Eur. Assoc. Theor. Comput. Sci. 13, 2 (1981).
J. Olivos, “On Vectorial Addition Chains,” J. Algorithms 2(1), 13 (1981).
S. B. Gashkov and V. V. Kochergin, “The Additive Vectorial Chains, Gate Circuits, and Complexity of the Powers Computation,” Metody Diskr. Analisa v Teorii Grafov i Slozhn. 52, 22 (1992).
J. Morgenstern, “Note on a Lower Bound of the Linear Complexity of the Fast Fourier Transform,” J. Assoc. Comput. Mach. 20, 305 (1973).
V. V. Kochergin, “The Complexity of Computation of a Pair of Monomials in Two Variables,” Diskr. Matem. Prilozh. 17(4), 116 (2005) [Discrete Math. and Appl. 15 (6), 547 (2005)].
V. V. Kochergin, “The Complexity of Computation of a Pair of Monomials in Two Variables,” in Proc. VII Int. Conf. “Discrete Models in Control Systems Theory,” Pokrovskoe, March 4–6, 2006 (MAKS Press, Moscow, 2006), pp. 185–190 [in Russian].
V. V. Kochergin, “The Complexity of Joint Computation of Three Monomials in Three Variables,” Matem. Voprosy Kibern. 15, 79 (2006).
V. V. Kochergin, “Asymptotics of the Complexity of Systems of Integer Linear Forms for Additive Computations,” Diskretn. Anal. Issled. Oper., Series 1, 13(2), 38, (2006).
V. V. Kochergin, “Maximal Complexity of a System of Elements of a Free Abelian Group Computation,” Vestn. Mosk. Univ., Matem. Mekhan. No. 3, 15 (2007).
V. V. Kochergin, “The Complexity of Joint Computation of Three Elements of a Free Abelian Group with Two Generators,” Diskretn. Anal. Issled. Oper., Series 1, 15(2), 23 (2008).
A. Brauer, “On Addition Chains,” Bull. Amer. Math. Soc. 45, 736 (1939).
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Original Russian Text © V.V. Kochergin, 2009, published in Vestnik Moskovskogo Universiteta, Matematika. Mekhanika, 2009, Vol. 64, No. 4, pp. 8–13.
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Kochergin, V.V. Relation between two measures of the computation complexity for systems of monomials. Moscow Univ. Math. Bull. 64, 144–149 (2009). https://doi.org/10.3103/S0027132209040020
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DOI: https://doi.org/10.3103/S0027132209040020