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):

long n_to_the_power_of_m(int n, int m) {
    if(m == 0) return 1;
    long sum = 0;
    for(int i = 0; i < n; ++i)
        sum += n_to_the_power_of_m(n, m-1);
    return sum;
}

(needs more than 4 minutes to compute 10^10)

Or other way around: Are there any Problems, which cannot be solved better than in O(n^n)?

Answer 1

What you have coded in your example is very similar to a depth first search. So, that’s one answer.

A depth first search algorithm without any special characteristics ( like re-convergent paths that can be optimized out ), should be n^n.

This is actually not a contrived example. Chess programs operate on the same algorithm. Each move there are n moves to consider ( i.e. branches ), and you search d moves deep. So that becomes O(n^d)