Question

Suppose we modify the quicksort algorithm from Special Topic 12.3, selecting the middle element instead of the first one as pivot. What is the running time on a list that is already sorted?

Short answer

The running time is \(O(n \log n)\).

Step-by-step solution

Understand Quicksort Operation

Quicksort works by selecting a 'pivot' element from the array and partitioning the other elements into two sub-arrays, according to whether they are less than or greater than the pivot. The process is recursive and continues until the base case, an array of zero or one element, is reached.

Determine Pivot Selection

In traditional quicksort, usually the first element is chosen as the pivot. However, in the modified version, we select the middle element of the list. For a sorted list, the middle element will always be at the median position, effectively creating two nearly equal halves for partitioning.

Analyze Recursive Behavior on Sorted List

In a sorted list, selecting the middle as the pivot results in balanced partitions every time. Since the partition process is efficient, we partition the list into parts of nearly equal size repeatedly.

Calculate Recurrent Relation

For this setup, the recurrence relation follows: \[ T(n) = 2T\left(\frac{n}{2}\right) + O(n) \]where \(O(n)\) is the cost of partitioning and choosing the pivot.

Solve the Recurrence for Running Time

The recurrence \( T(n) = 2T\left(\frac{n}{2}\right) + O(n) \) is the classic form for the balanced quicksort, which solves to \(O(n \log n)\) using the Master Theorem.

Key concepts

Quicksort Algorithm

Quicksort is a popular sorting algorithm known for its efficiency, especially for large datasets. It's a comparison sort, meaning it compares elements to reorder them. The algorithm works in the following way:

• Select a 'pivot' element from the array.
• Partition the other elements into two sub-arrays: elements less than the pivot and elements greater than the pivot.
• Recursively apply the same operation to the sub-arrays.

This process continues until sub-arrays reach a size of zero or one, which are inherently sorted. The choice of pivot and partitioning strategy are critical in determining the algorithm's efficiency.

Pivot Selection

The method of selecting a pivot in the quicksort algorithm can significantly influence its performance. In traditional quicksort, the first or last element is often chosen as the pivot. However, this may not always give optimal results, particularly when dealing with already sorted lists.

In our modified quicksort method, we select the middle element as the pivot. This approach aims to achieve a balanced partition, significantly improving performance in scenarios where the list is sorted or nearly sorted.

• This strategy reduces the risk of the worst-case time complexity, which occurs for strictly increasing or decreasing lists when a poor pivot is chosen.
• With a good pivot, the algorithm splits the array into two evenly sized smaller problems, increasing its efficiency.

Running Time Analysis

Analyzing the running time of quicksort involves understanding how the algorithm's steps contribute to its overall efficiency. In the worst case, where the smallest or largest elements are consistently chosen as pivots, the algorithm degrades to an \(O(n^2)\) performance. However, in most practical scenarios, quicksort achieves its average-case time complexity of \(O(n \log n)\).

• By choosing the middle element as the pivot, as in our modified version, the list is consistently divided into two halves, achieving a more balanced and efficient partition.
• This reduces the depth of recursion needed, which optimizes the speed by minimizing the number of recursive calls needed.

Recurrence Relation

The recurrence relation is a powerful tool to express the performance of recursive algorithms like quicksort. It allows us to model the algorithm's complexity based on its subproblem patterns. In the case of our modified quicksort:

\[ T(n) = 2T\left(\frac{n}{2}\right) + O(n) \]

This relation describes two recursive calls on sub-arrays of half the original size, plus linear time \(O(n)\) spent on partitioning around the pivot. Solving this recurrence relation communicates the efficiency of the algorithm when handled with good pivot selection.

Master Theorem

The Master Theorem provides a straightforward way to determine the time complexity of recurrence relations common in divide-and-conquer algorithms. For the recurrence relation:

\[ T(n) = 2T\left(\frac{n}{2}\right) + O(n) \]

The theorem helps solve it to \(O(n \log n)\).

• This reveals that while quicksort involves many recursive calls, if the list is split evenly, the number of calls grows logarithmically in relation to the size of the list, greatly improving efficiency.
• Using the Master Theorem, we validate the effectiveness of choosing the middle element as the pivot in achieving the balance needed for good performance.