The inverse Ackermann function occurs often when analyzing algorithms. A great presentation of it is here: http://www.gabrielnivasch.org/fun/inverse-ackermann. $$\alpha_1(n) = [n/2]$$$$\alpha_2(n) = [\log_2 n]$$$$\alpha_3(n) = \log^* n$$$$...$$$$\alpha_k(n) = 1 + \alpha_k(\alpha_{k1}(n))$$ and$$\alpha(n) = \min\{k: \alpha_k(n)\leq 3\}$$[Notation: [x] means that we round up x to the nearest integer, while log is the iterated log function discussed here: http://en.wikipedia.org/wiki/Iterated_logarithm ]My question is: What is the function$$k(n) = \min \{k: \alpha_k(n) \leq k\}$$Clearly $1\ll k(n) \leq \alpha(n)$. What tighter bounds can one give on $k(n)$? Is $k(n) \leq \log\alpha(n)$?
Fun with inverse Ackermann
ds.algorithms;ds.data structures;bounds
Let $A_k$ be the inverse of $\alpha_k$. $A_1(x) = 2x, A_2(x) = 2^x, \dots$. I claim that $k^{-1}(x) = A_x(x)$.Since $x = \alpha_x(A_x(x))$, and since $\forall z, \alpha_y(z) > \alpha_x(z)$, $\alpha_y(A_x(x)) > \alpha_x(A_x(x)) = x$. As a result $k(A_x(x)) = x$.Now consider the value of $\alpha(k^{-1}(n)) = \alpha(A_n(n))$. By definition of $\alpha$, this is $\min_z \{\alpha_z(A_n(n)) \leq 3\}$. We know that $\alpha_n(A_n(n)) = n$, so $\alpha(A_n(n)) > n$. I claim that $\alpha(A_n(n)) \leq n+2$. $\alpha_{n+1}(A_n(n)) = 1+\alpha_{n+1}(n)$. Now $\alpha(n) = \min_z\{\alpha_z(n) \leq 3\}$, so $\alpha_{\alpha(n)}(n) \leq 3$. Since $n+1 > \alpha(n)$, $\alpha_{n+1}(n) \leq 3$, so $\alpha_{n+1}(A_n(n)) \leq 4$. Thus, $\alpha_{n+2}(A_n(n)) = 1 + \alpha_{n+2}(\alpha_{n+1}(n)) \leq 1 + \alpha_{n+2}(4) \leq 3$.So, we have $n < \alpha(k^{-1}(n)) \leq n+2$, so $k$ and $\alpha$ are essentially equal.