Big O analysis for problem where number of items searched is unknown

Question

Consider this problem: you are searching an array of elements and are comparing the square of the current element to some number K. Essentially, you are looking to see if the square root of K is in the array. With this algorithm, chances are, you will not search the entire array because you will either find the square root or you will find that the square root is not in the array.

As such, you are searching only a fraction of the array, which lets say has M elements. Does this mean that the big O is still O(M), even though you are not searching all of the elements?

Answer 1

Big-O gives the upper bound function of any analysis. For an algorithm you could represent all (Worst, Average, Best) cases in Big-O notation.

In your case,

• Worst case: The required element is not present in the array (or is the last element of the array) : O(M)
• Best case: The element is found in the first position itself: O(1)
• Average case: Average of all possible case, i.e., the average time complexity of the cases where the element could be found at each possible position:

$$ \begin{align} = \frac{(1 + 2 + 3 + ... + M)}{M} \\ = \frac{M(M + 1)/2}{M} \\ = (M + 1)/2\\ = O(M) \end{align} $$

The average case is what you're looking for, which again O(M) only.

Answer 2

First, $O(M)$ means not more than $M$ modulo a constant factor when $M$ is large enough. So even if you find the element every time on the first try magically, $O(M)$ is still correct.

Second, suppose you meant it could be $o(M)$, which means strictly less than $M$ modul any given positive constant factor when $M$ is large enough. You are right in the following sense. The time-complexity of the best cases of this sequential search is $O(1)$.

Third, let us be interested in the average case. Since the better cases and the worse cases complement each other, the average time-complexity is $O(M/2)=O(M)$.

Lastly, the time-complexity of the worst cases is $O(M)$.