  We present three algorithms to compute the complexity $\Vert n\Vert$ of all
natural numbers $ n\le N$. The first of them is a brute force algorithm,
computing all these complexities in time $O(N^2)$ and space $O(N\log^2 N)$. 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 $O(N^{1+\beta})$, where $1+\beta
=\log3/\log2\approx1.584963$. 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 $O(N^\alpha)$ and space $O(N\log\log N)$. (Here $\alpha =
1.230175$).
  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 $O(N^{1+\beta})$
to $O(N^\alpha)$.
  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 $O(N^\alpha)$ and space
$O(N^{(1+\beta)/2}\log\log N)$, where $\alpha=1.230175$ and
$(1+\beta)/2\approx0.792481$.
