Quicksort is a widely used and efficient sorting algorithm, best known for its divide-and-conquer approach. The core idea behind Quicksort is to select a "pivot" element from the array and partition the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The process is repeated recursively to efficiently sort the list. A critical part of understanding Quicksort is grasping the pivot position after one partition.

Understanding the Pivot Position After One Partition

When implementing Quicksort, determining the pivot position after a single partition is crucial. This position effectively separates the array into two parts: elements smaller than the pivot and those larger.

Partitioning Process

The partitioning process works as follows:

1. Choose a pivot element from the array. This choice can significantly affect the algorithm's performance. Common strategies are picking the first element, the last element, a random element, or the median.
2. Rearrange the array elements so that all elements less than the pivot come before it and all greater elements come after it.
3. The pivot element is now in its final position, and this position splits the array into two partitions.

Example of the Partitioning

Consider the following array:

• Start with two pointers: `low` starts at the beginning, and `high` moves from the end toward the start.
• Move `low` rightward until an element greater or equal to the pivot is found.
• Move `high` leftward until an element smaller or equal to the pivot is found.
• If `low` is less than or equal to `high`, swap these elements.
• Repeat until `low` exceeds `high`.
• Swap the pivot with the element at the `high` pointer.
• Best Case: Occurs when the pivot consistently splits the array into two equal halves. The time complexity is $O(n \log n)$.
• Average Case: The time complexity remains $O(n \log n)$, attributed to random pivot selection.
• Worst Case: Happens when partitioning results in highly unbalanced partitions, such as when the smallest or largest element is consistently chosen as a pivot. This results in a time complexity of $O(n^2)$.
• First or Last Element: Simple but can often lead to poor performance.
• Randomized Pivot: Helps avoid worst-case scenarios by reducing chances of unbalanced partitions.
• Median of Three: Typically involves picking the median of the first, middle, and last elements, providing a good balance.