Question

I have a random ordered array of 30 elements with only 3 distinct keys (TRUE, FALSE and NULL) that I want to sort using insertion sort. What will be the time complexity? Will it be O(n²) assuming worst case or O(n) assuming best case since there are only 3 different keys?

Answer 1

n refers to the size of the array, not the possible elements of the array. Thus, the complexity is the same:

• Worst-case: O(n²)
• Best-case: O(n)
• Average-case: O(n²)

Having 3 distinct elements will reduce the amount of elements you have to check during the “insertion” phase, but only by a constant factor. This will not change the asymptotic run-time.

For example, in the average-case, instead of insert checking n elements, it will check ⁿ/₃ elements. This is better, but not asymptotically.