quicksort in python explained with example and code

introduction
quicksort is a highly efficient and widely used sorting algorithm that employs a divide-and-conquer strategy to sort lists or arrays. developed by tony hoare in 1960, it is known for its simplicity and effectiveness in practice.
how quicksort works
choose a pivot: the core of the quicksort algorithm is selecting a 'pivot' element from the array or list. this element will help partition the array into two sub-arrays.
partitioning: rearrange the elements in such a way that all elements less than the pivot come before it, and all elements greater than the pivot come after it. after this step, the pivot is in its final position.
recursively apply: apply the same process to the sub-arrays formed by dividing the array around the pivot. this is done recursively for each sub-array until the base case is reached, which is when the sub-array has one or zero elements.
base case: the recursion ends when the sub-array has zero or one element, as such sub-arrays are already sorted.
key concepts
pivot selection: the choice of pivot can influence the performance of the algorithm. common strategies include selecting the first element, the last element, the middle element, or using a random element as the pivot.
partition scheme: different partitioning schemes can be used. the most common one is the lomuto partition scheme, which partitions the array around the pivot. another scheme is the hoare partition scheme, which is more efficient in some cases.
time complexity: on average, quicksort has a time complexity of o(n log n), where n is the number of elements. however, in the worst-case scenario, it can degrade to o(n^2), especially if the pivot selection is poor.
space complexity: quicksort requires o(log n) space on average due to the recursion stack. the in-place partitioning ensures minimal additional space usage.
advantages of quicksort
disadvantages of quicksort
example
suppose we have an array [3, 6, 8, 10, 1, 2, 1].
by repeating these steps recursively, the entir ...

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