Question

I have a homework problem for my algorithms class asking me to calculate the maximum size of a problem that can be solved in a given number of operations using an O(n log n) algorithm (ie: n log n = c). I was able to get an answer by approximating, but is there a clean way to get an exact answer?

Answer 1

There is no closed-form formula for this equation. Basically, you can transform the equation:

 n log n = c
log(n^n) = c
     n^n = exp(c)

Then, this equation has a solution of the form:

n = exp(W(c))

where W is Lambert W function (see especially “Example 2”). It was proved that W cannot be expressed using elementary operations.

However, f(n)=n*log(n) is a monotonic function. You can simply use bisection (here in python):

import math

def nlogn(c):
    lower = 0.0
    upper = 10e10
    while True:
        middle = (lower+upper)/2
        if lower == middle or middle == upper:
            return middle
        if middle*math.log(middle, 2) > c:
            upper = middle
        else:
            lower = middle