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Improvement of the estimates of the computational complexity for monomials and sets of powers in Bellman’s and Knuth’s problems

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Abstract

We study two generalizations of the classical problem of fast exponentiation, namely: Bellman’s problem on computational complexity (the minimal number of multiplications) based only on the variables of a normalized monomial of m variables and Knuth’s problem on the complexity of the simultaneous calculation of a system of m powers of one variable. Some results for these problems are surveyed in the paper. The asymptotic complexity bounds for Bellman’s and Knuth’s problems are improved for the cases when m and complexity behave similarly (have the same growth order). The upper and lower complexity bounds for almost all sets of exponents for Bellman’s and Knuth’s problems that are established provide the complexity growth asymptotics for a wide range of relations between parameters (the number of variables or the computed exponents, the maximal power, and the product of all powers). Moreover, they guarantee the ratio of the upper and lower bounds not exceeding 5/3 for all relations between the parameters.

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  1. Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia

    V. V. Kochergin

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  1. V. V. Kochergin
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Correspondence to V. V. Kochergin.

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Original Russian Text © V.V. Kochergin, 2014, published in Diskretnyi Analiz i Issledovanie Operatsii, 2014, Vol. 21, No. 6, pp. 51–72.

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Kochergin, V.V. Improvement of the estimates of the computational complexity for monomials and sets of powers in Bellman’s and Knuth’s problems. J. Appl. Ind. Math. 9, 68–82 (2015). https://doi.org/10.1134/S1990478915010081

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  • DOI: https://doi.org/10.1134/S1990478915010081

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