Are there any real O(n^n) algorithms?

Question

Is there any real Algorithm with a time complexity O(n^n), that isn\'t just a gimmick?

I can create such an Algorithm, like computing n^n in O(n^n) / Θ(n^n):

<         


        

Answer 1

The program that takes a description of a (terminating) Turing machine, and returns the number of steps it takes to terminate. This is a relatively simple program to write -- it can simply emulate the Turing machine, and count the steps.

The complexity of this program has no computable upper bound (and in particular grows faster than any computable function), so certainly grows faster than O(n^n).

The worst-case run-time on an input of size n is BB(n), the Busy Beaver sequence, which starts 0, 1, 4, 6, 13, is unknown after this (although lower bounds exists -- for example the next two values are at least 47176870 and 7.412×10^36534 respectively) and uncomputable for n large enough.