Quicksort is a popular sorting algorithm that has an average case time complexity of O(n log n). 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.

The time complexity of Quicksort is determined by the number of comparisons made between elements in the array. In the best case, the partitioning algorithm produces two sub-arrays of equal size, and the pivot is the median of the array. In this case, the time complexity is O(n log n). However, in the worst case, the partitioning algorithm produces one sub-array of size n-1 and one sub-array of size 1, resulting in a time complexity of O(n^2). The worst case occurs when the pivot is either the smallest or largest element in the array.

In practice, Quicksort is very efficient and is often faster than other popular sorting algorithms such as Merge Sort and Heap Sort. The reason for this is that Quicksort is an in-place algorithm and has good cache locality, which means that it is well-suited to modern computer architectures.

Quicksort can be implemented in a variety of programming languages, and there are many libraries and modules available that provide Quicksort functionality. In this notebook, we use the numpy library to implement Quicksort.

To analyze the performance of Quicksort, we can compare the execution time of the algorithm with arrays of varying sizes. We can measure the execution time using the time module in Python.

We generate arrays of sizes ranging from 100 to 100,000 and sort them using Quicksort. We then measure the execution time and plot the results using the matplotlib library.

The results show that the execution time of Quicksort increases with the size of the array. However, the increase in execution time is not linear and is much slower than O(n^2). This demonstrates the efficiency of Quicksort and its suitability for sorting large datasets.

Quicksort is a fast and efficient sorting algorithm with an average time complexity of O(n log n). It is well-suited to modern computer architectures and can handle large datasets with ease. By analyzing the performance of Quicksort using arrays of varying sizes, we can see that it is a highly scalable algorithm that is suitable for a wide range of applications.