Quick Sort with Pivot Selection

Quick Sort is a divide-and-conquer sorting algorithm that picks a pivot element and partitions the array so that elements smaller than the pivot come before it and elements larger come after. The choice of pivot heavily influences the algorithm's real-world performance.

Common Pivot Selection Strategies

• First element: simple, but leads to O(n²) on already sorted arrays.
• Last element: the classic Lomuto partition scheme.
• Random element: reduces the chance of worst-case behavior on adversarial input.
• Median-of-three: picks the median of the first, middle, and last elements for a more balanced split.

import random

def quick_sort(arr, low=0, high=None):
    if high is None:
        high = len(arr) - 1
    if low < high:
        pivot_index = partition(arr, low, high)
        quick_sort(arr, low, pivot_index - 1)
        quick_sort(arr, pivot_index + 1, high)
    return arr

def partition(arr, low, high):
    rand = random.randint(low, high)
    arr[rand], arr[high] = arr[high], arr[rand]   # random pivot
    pivot = arr[high]
    i = low - 1
    for j in range(low, high):
        if arr[j] <= pivot:
            i += 1
            arr[i], arr[j] = arr[j], arr[i]
    arr[i + 1], arr[high] = arr[high], arr[i + 1]
    return i + 1

Complexity

Why Pivot Choice Matters

A poor pivot (e.g. always picking the first element on a sorted array) creates highly unbalanced partitions, degrading performance to O(n²). Random or median-of-three pivot selection keeps partitions balanced on average, preserving the O(n log n) behavior in practice.