Question

Give a specific answer that make quickhull run in quadratic time. Please explain explicitly and in as much detail as you can.

Answer 1

In worst case, the quickhull algorithm run in quadratic time. It's time complexity is quadratic( O(n^2) ).

In average case and in best case it's time complexity is O(n log(n)).

Explanation :

Let us consider the quickhull algorithm below

We can conclude that the partition is determined by two extreme points, the leftmost highest point l and the rightmost lowest point r. So, the partitioning requires O(n) time.

For the recursive relation, it takes n steps in determining the extreme point z, but the cost of those recursive calls is dependent on the size of A and size of B.

If both the partition is balanced, then it takes average running time which is O(n log(n)).

But, the worst case occurs if the partitions are not balanced. The recurrance relation in this case is

T(n)=T(n-1)+O(n) 
    =T(n-1)+cn

Repeated expansion of the above recurrance relation gives O(n^2). Therefore, the QuickHull algorithm run in quadratic time.