Quicksort

Quicksort is one of the best sorting algorithms which is based on the divide and conquer technique. This is because it is one of the best sorting algorithms that are widely used due to their efficiency, simplicity and their ability to sort the data in place without the need of using more memory space. Quicksort is most favourable in big data set, while it is widely implemented in different software products and libraries.

Key Concepts of Quicksort 

1. Divide-and-Conquer Strategy: The quicksort partitions the given problem into sub problems, solving each sub problem and finally combining them to solute the given problem. This strategy assists in the easy understanding of the sorting process and makes it easier to implement it. 
2. Pivot Selection: One of the most important stages of the Quicksort algorithm is to select a so-called ”pivot” element from an array. The pivot is used to partition the array into two subarrays: It divides the given array into two sub-arrays; one sub-array contains the elements less than or equal to the pivot and the second sub-array contains the elements greater than the pivot. A pivot can greatly affect the performance of the algorithm and, therefore, its efficiency. 

3. Partitioning: 

- Once the pivot has been chosen then the array is divided into two sub-arrays based on the value of the pivot. All elements which are less than or equal to the pivot are shifted to the left of pivot and all elements which are greater than the pivot are shifted to the right of the pivot. It is then placed at the right position in the sorted array as shown in the next section.

4. Recursive Sorting: 

- Quicksort divides the array in two parts that are to be sorted separately using the same procedure as on the original array. This process of recursion goes on until base case is met which is when the sub-arrays consist of only one element or no element at all, that is the sub-arrays are already sorted. 

5. In-Place Sorting: 

- One more advantage of Quicksort is that this algorithm sorts the array in place, that means that it does not require other arrays or other data structures to store the elements during the sorting process. This property makes the Quicksort very space efficient and hence it can be considered as one of the best sorting algorithms.

Steps of the Quicksort Algorithm 

1. Choose a Pivot: 

- Choose some element at the array as the pivot. The following are some of the techniques used to determine the pivot element; they include; selecting the first element, the last element, middle element, or even the random element. 

2. Partition the Array: 

- The elements of the given array are rearranged in such a manner that all the elements which are less than the pivot are placed on the left side of the pivot and all the elements which are greater than the pivot are placed on the right side of it. The pivot is also in the right sorted position as it should be as shown below.

3. Recursively Apply Quicksort: 

- Repeat the same step to the left subarray which contains all the elements less than the pivot and the right subarray which contains all the elements greater than the pivot. This is continued until all sub-arrays of the said array are sorted.

4. Combine the Subarrays: 

- The sub arrays are sorted in place thus there are no further steps of combining to be done. This is the recursive step of the function and if a for loop is used here, the array that was passed to the function gets sorted.

Example of Quicksort 

Let’s take a look at the example: The array is [8, 3, 1, 7, 0, 10, 2].  

1. Now let us select a pivot, let it be ‘7’. 

2. Partition the array into two subarrays: It can be seen that the two sequences are 3, 1, 0, 2 and 8, 10 with 7 in between. 

3. Reapply the Quicksort algorithm to the subarrays [3, 1, 0, 2] and [8, 10]. 

4. After all the recursive calls the array is sorted to : [0, 1, 2, 3, 7, 8, 10]. 

Advantages of Quicksort 

1. Efficiency: 

- Quicksort algorithm appears to be very efficient since it has a best and average case of O(n log n) making it suitable for handling big data. In case of the worst scenario, the time complexity of this algorithm is O(n^2), though there are usually cases when such a situation can be prevented by proper selection of the pivot. 

2. In-Place Sorting: 

- Quicksort is a sorting algorithm which sorts the array in place i. e. it uses only a small, constant amount of extra memory. This is a great advantage over other sorting algorithms that needs extra space for storing. 

3. Versatility: 

- It should be noted that quick sort is suitable for sorting small and large data arrays. It is a generic comparison based sorting algorithm which has good efficiency for all kinds of data and data distributions. 

4. Tail Recursion Optimization: 

- It is possible to make quicksort tail recursive, which means that the overhead of recursion is also minimized enhancing the efficiency of the algorithm in real life. 

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Disadvantages and Considerations 

1. Worst-Case Performance: 

- The lower bound for the worst-case of Quicksort is O(n^2) it arises when the best choice of the pivot generates the most unbalanced partitions. This is possible when the first or the last element is used as the pivot for example. Nevertheless, this problem can be partly avoided through the use of randomized or median-of-three pivot selection schemes. 

2. Unstable Sort: 

- Quicksort is not stable, which means that when it sorts elements it may switch their positions and therefore the order of equal elements will be changed. This can be a disadvantage in the case where the arrangement of equal elements is valuable, for instance. 

3. Sensitive to Input: 

 - This is because the performance of Quicksort algorithm depends with the kind of pivot selected and the arrangement of the input data. Choosing of poor pivots results to skewed partitions and this slow down the efficiency of the algorithm. 

4. Recursive Depth: 

- The time complexity of using recursion is also high making the Quick sort unsuitable for large data set since there can be a large number of recursive calls. This may result into stack overflow complications if the recursion depth is bigger than the stack size of the operating system. Nevertheless, this problem can be avoided with the help of iterative versions or, in case of functional programming, tail recursion optimization. 

Conclusion 

Hence Quicksort is one of the fastest sorting technique which is employed under divide and conquer method to sort items in an array or a list. It functions by splitting the given array into sub arrays in an iterative manner, sorting them and THEN fusing them. It is also one of the most efficient sorting algorithms particularly for large amount of data and the average time complexity is given by \(O(n \log n)\). It is also memory optimistic since it uses a direct sort that sorts its data in place without availabilities to any other space. In spite of the fact that this algorithm is not free from some shortcomings, for example its upper limit time complexity and the fact that it is an unstable sorting algorithm, Quicksort has remained one of the most preferred and utilised sorting algorithms in the field of computer science.

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