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) Rectorsive Step: Perform Step 1 on each unsorted subarray, Each time Step \(I\) 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). The basic algorithm seems simple enough, but how do we determine the final position of the first element of each subarray? As an example, consider the following set of values (the element in bold is the partitioning element- it will be placed in its final location in the sorted array): \(\begin{array}{llllllllll}37 & 2 & 6 & 4 & 89 & 8 & 10 & 12 & 68 & 45\end{array}\) Starting from the rightmost element of the array, compare each element with 37 until an element less than 37 is found; then swap 37 and that element. The first element less than 37 is \(12,\) so 37 and 12 are swapped. The new array is Starting from the right, but beginning with the element before \(89,\) compare each element with 37 until an element less than 37 is found - then swap 37 and that element. The first element less than 37 is \(10,\) so 37 and 10 are swapped. The new array is \\[ \begin{array}{llllllllll} 12 & 2 & 6 & 4 & 10 & 8 & 37 & 89 & 68 & 45 \end{array} \\] Starting from the left, but beginning with the element after \(10,\) compare each element with 37 until an element greater than 37 is found - then swap 37 and that element. There are no more elements greater than \(37,\) so when we compare 37 with itself, we know that 37 has been placed in its final location in the sorted array. Every value to the left of 37 is smaller than it, and every value to the right of 37 is larger than it. Once the partition has been applied on the previous array, there are two unsorted subarrays. The subarray with values less than 37 contains 12,2,6,4,10 and \(8 .\) The subarray with values greater than 37 contains 89,68 and \(45 .\) The sort continues recursively, with both subarrays being partitioned in the same manner as the original array. Based on the preceding discussion, write recursive method quicksortHelper to sort a one-dimensional integer array. The method should receive as arguments a starting index and an ending index on the original array being sorted.

Short answer

Implement the quicksortHelper method to recursively sort an array. Start with partitioning around a pivot (first element), recursively applying the same logic to sub-arrays while handling the base case of a single-element or empty sub-array.

Step-by-step solution

- Understanding Quicksort

Quicksort is a divide-and-conquer algorithm. It 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 sub-arrays are then sorted recursively.

- Implementing the Partitioning Step

Choose the first element as the pivot and partition the array using the following process: scan from the end of the array to find the first element less than the pivot and swap it; then scan from the start of the array to find the first element greater than the pivot and swap it. Continue this scanning process until the pointers meet.

- Defining the Recursive Step

Apply the quicksort algorithm recursively to the sub-arrays formed by the partitioning step. The base case of the recursion is when the sub-array has zero or one element, which doesn't need to be sorted.

- Writing the quciksortHelper Method

The quicksortHelper method should take an array and two indices (start and end) as arguments and apply the partitioning and recursive steps to sort the array. If the start index is less than the end index, perform the partitioning and then call quicksortHelper for the two sub-arrays.

- Partitioning Process Detail

To perform partitioning, maintain two indexes. One starts at the beginning of the array and moves forward, while the other starts at the end of the array and moves backward. Swap elements such that elements less than pivot are on the left, and elements greater than pivot are on the right. Finally, the pivot should be in its correct sorted position.

- Implementing the Method

Here is a skeleton of the quicksortHelper method: \[\text{public void quicksortHelper(int[] array, int start, int end) \{\}}\] Inside the method, include base case checks and the logic for finding the pivot's final location. After partitioning, recursively call quicksortHelper for the left and right subarrays created.

- Completing the Quicksort

The method quicksortHelper will sort the array between the indices 'start' and 'end'. To sort the entire array initially, quicksortHelper is called with the start index as 0 and the end index as the length of the array minus one.

Key concepts

Recursive Sorting Technique

Recursive sorting techniques are a cornerstone of efficient sorting algorithms, and quicksort is a prime example. Recursion involves breaking down a problem into smaller, more manageable sub-problems, solving each one, and combining the results. In the context of sorting, this means dividing an array into subarrays and sorting each piece independently.

Quicksort takes advantage of recursion by applying the same sorting strategy to progressively smaller subarrays until each subarray is trivially sorted—either empty or containing a single element. The elegance of recursive sorting lies in its simplicity; once the base case is reached, the recursive process ensures that each subarray is sorted, leading to a fully sorted array. Embracing recursion is a powerful strategy, as it can turn complex sorting tasks into a series of simpler steps.

Partitioning Step in Quicksort

The partitioning step is the heart of the quicksort algorithm. It’s responsible for picking a pivot element and arranging the other elements around it, such that those less than the pivot end up on one side, while those greater end up on the other. The choice of pivot can be flexible—some implementations pick the first element, some the last, and others might choose a random or median value.

Once the pivot is chosen, the partitioning process involves scanning from both ends of the array to swap elements that are on the wrong side of the pivot. This pivotal action quite literally 'partitions' the array into two halves, one which will contain elements less than the pivot, and the other consists of elements greater, with the pivot then landing in its correct position within the sorted array. It's a neat trick that lays the groundwork for sorting each half independently.

Quicksort Implementation

Implementing quicksort involves integrating the partitioning step with recursive calls to perform sorting on subarrays. A typical quicksort function takes an array and two indexes as its parameters: the starting index and the ending index of a range within the array to be sorted.

The implementation detail is in executing the partition correctly and invoking the quicksort function recursively on the resulting subarrays. This means that the quicksort algorithm is in constant communication with itself, refining each subarray until the entire array is sorted. Implementation wisdom advises keeping a sharp eye on the base cases to avoid infinite recursion and undue complexity.

Divide and Conquer Algorithms

Quicksort is an exemplifying instance of a divide and conquer algorithm. This strategy involves three steps: divide the problem into smaller subproblems, conquer the subproblems by solving them (often recursively), and combine the solutions to solve the original problem.

By repeatedly dividing the array into halves around a pivot, quicksort makes the problem smaller each time. The conquering occurs as these smaller and smaller arrays reach the base case and are considered sorted. The combination phase is somewhat implicit in quicksort, as the action of sorting smaller arrays results in the larger array becoming sorted. Divide and conquer algorithms like quicksort shine in their ability to simplify complex problems and optimize the sorting process.

Algorithm Efficiency

When considering algorithm efficiency, quicksort stands out for its performance. The efficiency of an algorithm is generally measured in terms of its time complexity, which for quicksort is best and average case O(n log n), and in the worst case, O(n^2). The worst-case scenario typically occurs when the smallest or largest elements are chosen as pivots at each step. However, with a smart pivot selection strategy, such as using the median-of-three method, quicksort typically performs close to its best case.

Space complexity is another angle of efficiency, where quicksort does well again. As a recursive algorithm, it uses O(log n) stack space, which is modest compared to the array's size. The efficiency of quicksort, and sorting algorithms in general, hinges on balancing these factors and applying the right strategies for different datasets to achieve optimal performance.