Sorting Algorithms - Quicksort, Mergesort, Heapsort

Introduction

Sorting algorithms are fundamental in computer science, used to arrange elements in a specific order. Understanding different sorting algorithms helps in choosing the right one for different scenarios based on time and space complexity.

Quick Sort

Quick sort is a divide-and-conquer algorithm that selects a pivot and partitions the array around it.

def quick_sort(arr):
    if len(arr) <= 1:
        return arr
    
    pivot = arr[len(arr) // 2]
    left = [x for x in arr if x < pivot]
    middle = [x for x in arr if x == pivot]
    right = [x for x in arr if x > pivot]
    
    return quick_sort(left) + middle + quick_sort(right)

# In-place quick sort
def quick_sort_inplace(arr, low, high):
    if low < high:
        pi = partition(arr, low, high)
        quick_sort_inplace(arr, low, pi - 1)
        quick_sort_inplace(arr, pi + 1, high)

def partition(arr, low, high):
    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

# Test
numbers = [64, 34, 25, 12, 22, 11, 90]
print("Quick Sort:", quick_sort(numbers))

arr = [64, 34, 25, 12, 22, 11, 90]
quick_sort_inplace(arr, 0, len(arr) - 1)
print("In-place Quick Sort:", arr)

Merge Sort

Merge sort divides the array into halves, sorts them, and merges them back together.

def merge_sort(arr):
    if len(arr) <= 1:
        return arr
    
    mid = len(arr) // 2
    left = merge_sort(arr[:mid])
    right = merge_sort(arr[mid:])
    
    return merge(left, right)

def merge(left, right):
    result = []
    i = j = 0
    
    while i < len(left) and j < len(right):
        if left[i] <= right[j]:
            result.append(left[i])
            i += 1
        else:
            result.append(right[j])
            j += 1
    
    result.extend(left[i:])
    result.extend(right[j:])
    
    return result

# Test
numbers = [64, 34, 25, 12, 22, 11, 90]
print("Merge Sort:", merge_sort(numbers))

Heap Sort

Heap sort uses a binary heap data structure to sort elements.

def heap_sort(arr):
    def heapify(arr, n, i):
        largest = i
        left = 2 * i + 1
        right = 2 * i + 2
        
        if left < n and arr[left] > arr[largest]:
            largest = left
        if right < n and arr[right] > arr[largest]:
            largest = right
        if largest != i:
            arr[i], arr[largest] = arr[largest], arr[i]
            heapify(arr, n, largest)
    
    n = len(arr)
    
    for i in range(n // 2 - 1, -1, -1):
        heapify(arr, n, i)
    
    for i in range(n - 1, 0, -1):
        arr[0], arr[i] = arr[i], arr[0]
        heapify(arr, i, 0)

# Test
numbers = [64, 34, 25, 12, 22, 11, 90]
heap_sort(numbers)
print("Heap Sort:", numbers)

Comparison

import time
import random

def measure_sort_time(sort_func, arr):
    arr_copy = arr.copy()
    start = time.time()
    sort_func(arr_copy)
    return time.time() - start

sizes = [100, 1000, 5000]
for size in sizes:
    arr = [random.randint(0, 10000) for _ in range(size)]
    print(f"\nArray size: {size}")
    print(f"  Quick Sort: {measure_sort_time(quick_sort, arr):.4f}s")
    print(f"  Merge Sort: {measure_sort_time(merge_sort, arr):.4f}s")
    arr_copy = arr.copy()
    print(f"  Heap Sort: {measure_sort_time(heap_sort, arr_copy):.4f}s")

Practice Problems

1. Implement quick sort with random pivot selection.
2. Write an in-place merge sort implementation.
3. Optimize heap sort to handle duplicates efficiently.
4. Compare the performance of all three sorting algorithms.
5. Implement a hybrid sorting algorithm (TimSort-style).