What is the O of this operation?

Question

Say for a given positive integer number n, you have to find a level k so that

1 + 2 + 3 + ... + k = S is below or equals n

but S + k + 1 is above n.

For example in python:

def find_level(n):
    level = 1
    while n > 0:
        n -= level
        level += 1
    return level - 1

What is the O magnitude of this function?

Answer 1

1. Simplify:

   def find_level(n):
       level = 0
       while n > 0:
           level += 1
           n -= level
       return level
   

2. Get the closed form for the highest n per level:

   n = sum(x = 0 to level, x) = level * (level + 1) / 2

3. Solve that for level using the quadratic formula or some other method:

   0.5 * level² + 0.5 * level - n = 0

   level = -.5 + sqrt(.25 + 2 * n)

Your algorithm is obviously O(sqrt(n)).