What is O(log* N)?

Question

What is O(log* N) and how is it different from O(log N)?

Answer 1

The log* N bit is an iterated algorithm which grows very slowly, much slower than just log N. You basically just keep iteratively 'logging' the answer until it gets below one (E.g: log(log(log(...log(N)))), and the number of times you had to log() is the answer.

Anyway, this is a five-year old question on Stackoverflow, but no code?(!) Let's fix that - here's implementations for both the recursive and iterative functions (they both give the same result):

public double iteratedLogRecursive(double n, double b)
{
    if (n > 1.0) {
        return 1.0 + iteratedLogRecursive( Math.Log(n, b),b );
    }
    else return 0;
}

public int iteratedLogIterative(double n, double b)
{
    int count=0;
    while (n >= 1) {
        n = Math.Log(n,b);
        count++;
    }
    return count;
}