107 research outputs found
Algorithms for determining integer complexity
We present three algorithms to compute the complexity of all
natural numbers . The first of them is a brute force algorithm,
computing all these complexities in time and space . The
main problem of this algorithm is the time needed for the computation. In 2008
there appeared three independent solutions to this problem: V. V. Srinivas and
B. R. Shankar [11], M. N. Fuller [7], and J. Arias de Reyna and J. van de Lune
[3]. All three are very similar. Only [11] gives an estimation of the
performance of its algorithm, proving that the algorithm computes the
complexities in time , where . The other two algorithms, presented in [7] and
[3], were very similar but both superior to the one in [11]. In Section 2 we
present a version of these algorithms and in Section 4 it is shown that they
run in time and space . (Here ).
In Section 2 we present the algorithm of [7] and [3]. The main advantage of
this algorithm with respect to that in [11] is the definition of kMax in
Section 2.7. This explains the difference in performance from
to .
In Section 3 we present a detailed description a space-improved algorithm of
Fuller and in Section 5 we prove that it runs in time and space
, where and
.Comment: 21 pages. v2: We improved the computations to get a better bound for
$\alpha
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