Worst case runtime for binary search

Question

The run time of binary search is O(log(n)).
log(8) = 3
It takes 3 comparisons to decide if an array of 8 elements contains a given element. It takes 4 comparisons in the example below.

python2.7

def binary_search(a_list, item, comparisons_inner=0):
    low = 0
    high = len(a_list) - 1

    while low <= high:
        mid = (low + high) / 2
        guess = a_list[mid]
        comparisons_inner += 1
        if guess == item:
            return mid, comparisons_inner
        if guess > item:
            high = mid - 1
        else:
            low = mid + 1
    return None, comparisons_inner


my_list = [5, 8, 11, 15, 21, 23, 100, 223]
index, comparisons = binary_search(my_list, 223)
print(index, comparisons)

log(8) < 4
Why is the run time of binary search O(log(n)) despite the 4 comparisons it takes in the example above?

Answer 1

Your algorithm makes $\lfloor \log n \rfloor+ 1$ comparisons for an array of length $n$, which is at most $2 \log n$, which is $O(\log n)$. Hence, $\lfloor \log n \rfloor+ 1 = O(\log n)$.

In more detail, recall that the big-O notation subsumes constant factors. A function $f(n)$ is said to be $O(\log n)$ if there exist positive constants $c$ and $n_0$ such that $f(n) \le c \log n$ for all $n \ge n_0$. In our case, we want to prove $\lfloor \log n \rfloor +1 = O(\log n)$. To prove the existence of positive constants $c$ and $n_0$ satisfying the desired condition, observe that $\lfloor \log n \rfloor+ 1 \le \log n + \log n = 2 \log n$, for all $n \ge 2$. So we can take $c = 2$ and $n_0 = 2$.