An Olog n example

Introduction

Consider the following problem:

L is a sorted list containing n signed integers (n being big enough), for example [-5, -2, -1, 0, 1, 2, 4] (here, n has a value of 7). If L is known to contain the integer 0, how can you find the index of 0 ?

Naïve approach

The first thing that comes to mind is to just read every index until 0 is found. In the worst case, the number of operations is n, so the complexity is O(n).

This works fine for small values of n, but is there a more efficient way ?

Dichotomy

Consider the following algorithm (Python3):

a = 0
b = n-1
while True:
  h = (a+b)//2 ## // is the integer division, so h is an integer
  if L[h] == 0:
    return h
  elif L[h] > 0:
    b = h
  elif L[h] < 0:
    a = h

a and b are the indexes between which 0 is to be found. Each time we enter the loop, we use an index between a and b and use it to narrow the area to be searched.

In the worst case, we have to wait until a and b are equal. But how many operations does that take? Not n, because each time we enter the loop, we divide the distance between a and b by about two. Rather, the complexity is O(log n).

Explanation

Note: When we write “log”, we mean the binary logarithm, or log base 2 (which we will write “log_2”). As O(log_2 n) = O(log n) (you can do the math) we will use “log” instead of “log_2”.

Let’s call x the number of operations: we know that 1 = n / (2^x).

So 2^x = n, then x = log n

Conclusion

When faced with successive divisions (be it by two or by any number), remember that the complexity is logarithmic.