*805*

**Quicksort** is an in-place sorting algorithm it is still a commonly used algorithm for sorting. When implemented well, it can be somewhat faster than merge sort and about two or three times faster than heapsort.

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

. For this reason, it is sometimes called **partition-exchange sort**.

The sub-arrays are then sorted recursively. This can be done in-place, requiring small additional amounts of memory to perform the sorting.

Quicksort is a comparison sort, meaning that it can sort items of any type for which a “less-than” relation (formally, a total order) is defined. Efficient implementations of Quicksort are not a stable sort, meaning that the relative order of equal sort items is not preserved.

Mathematical analysis of quicksort shows that, on average, the algorithm takes O(*n* log *n*) comparisons to sort *n* items.

Quicksort is a type of divide and conquer algorithm for sorting an array, based on a partitioning routine; the details of this partitioning can vary somewhat, so that quicksort is really a family of closely related algorithms.

Applied to a range of at least two elements, partitioning produces a division into two consecutive non empty sub-ranges, in such a way that no element of the first sub-range is greater than any element of the second sub-range.

After applying this partition, quicksort then recursively sorts the sub-ranges, possibly after excluding from them an element at the point of division that is at this point known to be already in its final location.

Due to its recursive nature, quicksort (like the partition routine) has to be formulated so as to be callable for a range within a larger array, even if the ultimate goal is to sort a complete array. The steps for in-place quicksort are:

- If the range has less than two elements, return immediately as there is nothing to do. Possibly for other very short lengths a special-purpose sorting method is applied and the remainder of these steps skipped.
- Otherwise pick a value, called a
*pivot*, that occurs in the range (the precise manner of choosing depends on the partition routine, and can involve randomness). *Partition*the range: reorder its elements, while determining a point of division, so that all elements with values less than the pivot come before the division, while all elements with values greater than the pivot come after it; elements that are equal to the pivot can go either way. Since at least one instance of the pivot is present, most partition routines ensure that the value that ends up at the point of division is equal to the pivot, and is now in its final position (but termination of quicksort does not depend on this, as long as sub-ranges strictly smaller than the original are produced).- Recursively apply the quicksort to the sub-range up to the point of division and to the sub-range after it, possibly excluding from both ranges the element equal to the pivot at the point of division. (If the partition produces a possibly larger sub-range near the boundary where all elements are known to be equal to the pivot, these can be excluded as well.)

### Code

def partition(nums, low, high): pivot = nums[(low + high) // 2] i = low - 1 j = high + 1 while True: i += 1 while nums[i] < pivot: i += 1 j -= 1 while nums[j] > pivot: j -= 1 if i >= j: return j # If an element at i (on the left of the pivot) is larger than the # element at j (on right right of the pivot), then swap them nums[i], nums[j] = nums[j], nums[i] def quick_sort(nums): # Create a helper function that will be called recursively def _quick_sort(items, low, high): if low < high: # This is the index after the pivot, where our lists are split split_index = partition(items, low, high) _quick_sort(items, low, split_index) _quick_sort(items, split_index + 1, high) _quick_sort(nums, 0, len(nums) - 1) # Verify it works arr = [95, 56, 25, 75, 43, 11, 80, 32, 64] quick_sort(arr) print(arr)