Question

The recursive sorting technique called quicksort uses the following basic algorithm for a one-dimensional array of values: a) Partitioning Step: Take the first element of the unsorted array and determine its final lo- cation in the sorted array (i.e., all values to the left of the element in the array are less than the element, and all values to the right of the element in the array are greater than the element-we show how to do this below). We now have one element in its proper location and two unsorted subarrays. b) Recursive Step: Perform Step \(I\) on each unsorted subarray. Each time Step 1 is performed on a subarray, another element is placed in its final location of the sorted array, and two unsorted subarrays are created. When a subarray consists of one element, that element is in its final location (because a one- element array is already sorted).

Short answer

Use quicksort: partition the array around a pivot, place the pivot, then recurse on subarrays.

Step-by-step solution

Choose a Pivot

In quicksort, the algorithm typically selects the first element of the array as the pivot. This pivot will be used to partition the array into elements less than the pivot and elements greater than the pivot.

Partition the Array

Rearrange the array so that all elements less than the pivot are on its left, and all elements greater than or equal to it are on the right. This can be done by iterating over the array and swapping elements to create two partitions: less-than-pivot and greater-than-pivot.

Place Pivot in Final Position

After partitioning, place the pivot in its correct position in the array such that elements to the left are less and elements to the right are greater, positioning the pivot correctly in terms of order.

Recursive Sorting of Subarrays

Perform the quicksort algorithm recursively on the left and right subarrays (elements less than pivot and elements greater than pivot, respectively). This involves choosing new pivots for each subarray, partitioning them, and placing the new pivots in their correct locations.

Base Case

This step completes the quicksort process. If a subarray has only one element, it is already sorted, and recursion halts for that subarray.

Key concepts

Recursive Sorting

Quicksort is a remarkable example of recursive sorting. Recursive sorting involves breaking down a problem into smaller, more manageable sub-problems of the same type. In the case of quicksort, we sort an array by dividing it into subarrays, sorting those subarrays, and then combining them to form a sorted array.
The process kicks off by selecting a 'pivot' element from the array. This element acts as a reference point for partitioning the rest of the array into smaller parts. These smaller parts, or subarrays, are then further sorted with the same method - selecting a new pivot, partitioning, and so on. This recursion continues until each subarray reaches a base case, which is when the subarray contains only one element.

With recursive sorting, the key is the repetitive application of the same sorting principles - divide, conquer, and combine. This recursive nature is what makes quicksort both powerful and efficient, often outperforming other algorithms such as bubble sort or insertion sort in practical scenarios.

Partitioning Technique

The partitioning technique is central to the quicksort algorithm. It refers to the method of rearranging the elements of an array around a pivot. Partitioning is done such that all elements less than the pivot are moved to its left, while elements greater than the pivot move to its right.

A common way to achieve this involves iterating over the array and using two pointers. One pointer scans from the left, and the other from the right of the array. When the left pointer finds an element larger than the pivot, and the right pointer finds an element smaller than the pivot, the two elements are swapped. This process continues until the pointers meet.

After partitioning, the pivot is placed at the junction of these two partitioned lists. This not only places the pivot in its final position within the array but also ensures that it effectively divides the array into two parts, both of which can be independently sorted through recursion. Mastering this partitioning technique is essential to implementing a swift quicksort operation.

Base Case in Recursion

In any recursive algorithm, including quicksort, the base case is what halts further recursion. It defines the simplest instance of the problem, which can be solved directly without further recursive calls. For quicksort, the base case occurs when a subarray has either zero or one element.

The logic is straightforward: a single-element array is already sorted by definition. Therefore, once quicksort divides an array into subarrays of a single element, no further sorting is needed, and the recursion stops for those subarrays.

• This base case is crucial to ensure quicksort doesn't fall into an infinite loop.
• It effectively puts a limit on the depth of recursive calls.
• By reaching the base case, quicksort completes the sorting process efficiently.

Understanding when and how recursion ends is imperative, as it ensures the algorithm runs smoothly and concludes correctly.