Explain how pivot location affects on time complexities of the quick sort algorithm

The pivot selection and partitioning steps are crucial to the performance of the QuickSort algorithm. The time complexity of QuickSort depends heavily on the pivot selection.

1. Best Case Scenario (O(n log n)): The best case scenario occurs when the pivot element is always the median of the array being sorted. This ensures that the array is always divided in half, leading to a time complexity of O(n log n). This is because the algorithm makes n comparisons per level (to partition the array), and there are log n levels (since the array is halved at each level).

2. Worst Case Scenario (O(n^2)): The worst case scenario occurs when the pivot element is either the smallest or the largest element in the array. This leads to an unbalanced partition where one partition contains n-1 elements and the other contains 0 elements. This results in a time complexity of O(n^2) because the algorithm makes n comparisons at each level, and there are n levels (since the array is not halved at each level).

3. Average Case Scenario (O(n log n)): On average, the pivot will be an element that gives a reasonably balanced partition. This leads to a time complexity of O(n log n), which is the same as the best case scenario.

In conclusion, the pivot selection in QuickSort can greatly affect its time complexity. A good pivot selection strategy can help ensure that the algorithm performs closer to its best case scenario rather than its worst case scenario.