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Sorting#

Sorting rearranges elements into non-decreasing (or non-increasing) order. Choice of algorithm depends on stability, memory, worst-case guarantees, and whether keys are integers in a small range.

Complexity summary#

Algorithm Best Average Worst Space Stable
Bubble sort O(n) O(n²) O(n²) O(1) Yes
Insertion sort O(n) O(n²) O(n²) O(1) Yes
Selection sort O(n²) O(n²) O(n²) O(1) No
Merge sort O(n log n) O(n log n) O(n log n) O(n) Yes
Quick sort O(n log n) O(n log n) O(n²) O(log n) stack No
Heap sort O(n log n) O(n log n) O(n log n) O(1) No
Counting sort O(n + k) O(n + k) O(n + k) O(k) Yes
Radix sort O(d(n + k)) O(d(n + k)) O(d(n + k)) O(n + k) Yes
Bucket sort O(n) O(n + k) O(n²) O(n) Yes
Python sorted / .sort() O(n log n) O(n log n) O(n log n) O(n) Yes

k = range of key values; d = number of digits/pass in radix sort.

Interview default: use Python's Timsort (list.sort(), sorted()) unless asked to implement or keys are special (counting/radix for integers in bounded range).

What FAANG actually asks#

  • Default: sorted(arr) or arr.sort() — O(n log n), stable. Mention this first.
  • Implement merge/quick: only when explicitly requested; merge sort guarantees O(n log n) worst case; quicksort is in-place but O(n²) worst without random pivot.
  • Merge sort on linked lists: LC 148 — split with fast/slow, merge sorted halves.
  • K-th smallest: quickselect O(n) average, or size-k max-heap O(n log k).
  • Custom order: sorted(items, key=lambda x: ...) — never implement comparator classes unless asked.

Merge sort splits until size 1, then merges sorted halves — guarantees O(n log n) worst case (see diagram).

Merge sort divide and conquer tree

Simple Comparison-based Algorithms#

1. Bubble Sort#

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def bubble_sort(numbers):
    length = len(numbers)
    for i in range(length-1):
        for j in range(length-i-1):
            if numbers[j]>numbers[j+1]:
                numbers[j],numbers[j+1] = numbers[j+1], numbers[j]
    return numbers

numbers = [60,40,70,20,50,30,90,45]
sorted_numbers = bubble_sort(numbers)
print(sorted_numbers)

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def bubble_sort(numbers):
    length = len(numbers)
    for i in range(length-1):
        swapped = False
        for j in range(length-i-1):
            if numbers[j]>numbers[j+1]:
                numbers[j],numbers[j+1] = numbers[j+1], numbers[j]
                swapped = True
        if not swapped:
            break
    return numbers

numbers = [60,40,70,20,50,30,90,45]
sorted_numbers = bubble_sort(numbers)
print(sorted_numbers)

2. Insertion Sort#

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def insertion_sort(numbers):
    for i in range(1, len(numbers)):
        key = numbers[i]
        j = i - 1
        while j >= 0 and numbers[j] > key:
            numbers[j + 1] = numbers[j]
            j -= 1
        numbers[j + 1] = key
    return numbers

numbers = [60,40,70,20,50,30,90,45]
sorted_numbers = insertion_sort(numbers)
print(sorted_numbers)

3. Selection Sort#

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def selection_sort(numbers):
    length = len(numbers)
    for i in range(length-1):
        min_index = i
        for j in range(i+1, length):
            if numbers[j]<numbers[min_index]:
                min_index = j
        numbers[min_index],numbers[i] = numbers[i], numbers[min_index]
    return numbers

numbers = [60,40,70,20,50,30,90,45]
sorted_numbers = selection_sort(numbers)
print(sorted_numbers)

Efficient Comparison-based Algorithms#

4. Merge Sort#

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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

def merge_sort(numbers):

    if len(numbers) <= 1: return numbers
    mid = len(numbers) // 2

    left_half = merge_sort(numbers[:mid])
    right_half = merge_sort(numbers[mid:])

    return merge(left_half, right_half)

numbers = [60,40,70,20,50,30,90,45]
sorted_numbers = merge_sort(numbers)
print(sorted_numbers)

5. Quick Sort#

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def naive_partition(numbers, pivot):
    length = len(numbers)
    numbers[pivot], numbers[length-1] = numbers[length-1], numbers[pivot]
    temp = []

    for x in numbers:
        if x<=numbers[length-1]:
            temp.append(x)
    for x in numbers:
        if x>numbers[length-1]:
            temp.append(x)
    for i in range(length):
        numbers[i] = temp[i]

def quick_sort(numbers):
    if len(numbers) <= 1:
        return numbers
    pivot_index = len(numbers) - 1
    naive_partition(numbers, pivot_index)
    left = [x for x in numbers if x <= numbers[pivot_index] and x != numbers[pivot_index]]
    right = [x for x in numbers if x > numbers[pivot_index]]
    return quick_sort(left) + [numbers[pivot_index]] + quick_sort(right)

numbers = [60, 40, 70, 20, 50, 30, 90, 45]
sorted_numbers = quick_sort(numbers)
print(sorted_numbers)

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def lomuto_partition(numbers, low, high):
    pivot = numbers[high]
    i = low - 1
    for j in range(low, high):
        if numbers[j] <= pivot:
            i += 1
            numbers[i], numbers[j] = numbers[j], numbers[i]
    numbers[i + 1], numbers[high] = numbers[high], numbers[i + 1]
    return i + 1

def lomuto_quick_sort(numbers, low, high):
    if low < high:
        pi = lomuto_partition(numbers, low, high)
        lomuto_quick_sort(numbers, low, pi - 1)
        lomuto_quick_sort(numbers, pi + 1, high)

numbers = [60, 40, 70, 20, 50, 30, 90, 45]
lomuto_quick_sort(numbers, 0, len(numbers) - 1)
print("Lomuto Quick Sort:", numbers)

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def hoare_partition(numbers, low, high):
    pivot = numbers[low]
    i = low - 1
    j = high + 1
    while True:
        i += 1
        while numbers[i] < pivot:
            i += 1
        j -= 1
        while numbers[j] > pivot:
            j -= 1
        if i >= j:
            return j
        numbers[i], numbers[j] = numbers[j], numbers[i]

def hoare_quick_sort(numbers, low, high):
    if low < high:
        pi = hoare_partition(numbers, low, high)
        hoare_quick_sort(numbers, low, pi)
        hoare_quick_sort(numbers, pi + 1, high)

numbers = [60, 40, 70, 20, 50, 30, 90, 45]
hoare_quick_sort(numbers, 0, len(numbers) - 1)
print("Hoare Quick Sort:", numbers)

6. Heap Sort#

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def heap_sort(numbers):
    def heapify(numbers, n, i):
        largest = i
        left = 2 * i + 1
        right = 2 * i + 2

        if left < n and numbers[left] > numbers[largest]:
            largest = left
        if right < n and numbers[right] > numbers[largest]:
            largest = right

        if largest != i:
            numbers[i], numbers[largest] = numbers[largest], numbers[i]
            heapify(numbers, n, largest)

    n = len(numbers)

    for i in range(n // 2 - 1, -1, -1):
        heapify(numbers, n, i)

    for i in range(n - 1, 0, -1):
        numbers[i], numbers[0] = numbers[0], numbers[i]
        heapify(numbers, i, 0)

    return numbers

numbers = [60, 40, 70, 20, 50, 30, 90, 45]
sorted_numbers = heap_sort(numbers)
print("Sorted numbers:", sorted_numbers)

Hybrid Comparison-based Algorithms#

7. tim sort#

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def insertion_sort(arr, left, right):
    for i in range(left + 1, right + 1):
        key = arr[i]
        j = i - 1
        while j >= left and arr[j] > key:
            arr[j + 1] = arr[j]
            j -= 1
        arr[j + 1] = key

def merge(arr, left, mid, right):
    left_sub = arr[left:mid + 1]
    right_sub = arr[mid + 1:right + 1]
    i, j, k = 0, 0, left

    while i < len(left_sub) and j < len(right_sub):
        if left_sub[i] <= right_sub[j]:
            arr[k] = left_sub[i]
            i += 1
        else:
            arr[k] = right_sub[j]
            j += 1
        k += 1

    arr[k:k + len(left_sub) - i] = left_sub[i:]
    arr[k:k + len(right_sub) - j] = right_sub[j:]

def timsort(arr):
    n = len(arr)
    RUN = 32

    for start in range(0, n, RUN):
        end = min(start + RUN - 1, n - 1)
        insertion_sort(arr, start, end)

    size = RUN
    while size < n:
        for start in range(0, n, 2 * size):
            mid = min(n - 1, start + size - 1)
            end = min(start + 2 * size - 1, n - 1)
            if mid < end:
                merge(arr, start, mid, end)
        size *= 2

    return arr

numbers = [60, 40, 70, 20, 50, 30, 90, 45]
sorted_numbers = timsort(numbers)
print("Sorted array:", sorted_numbers)

8. Intro sort#

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import math

def insertion_sort(arr, left, right):
    for i in range(left + 1, right + 1):
        key = arr[i]
        j = i - 1
        while j >= left and arr[j] > key:
            arr[j + 1] = arr[j]
            j -= 1
        arr[j + 1] = key

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)

def heapsort(arr):
    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[i], arr[0] = arr[0], arr[i]
        heapify(arr, i, 0)

def quicksort(arr, left, right, depth_limit):
    if left < right:
        if right - left < 16:
            insertion_sort(arr, left, right)
        elif depth_limit == 0:
            heapsort(arr[left:right+1])
        else:
            pivot = partition(arr, left, right)
            quicksort(arr, left, pivot - 1, depth_limit - 1)
            quicksort(arr, pivot + 1, right, depth_limit - 1)

def partition(arr, left, right):
    pivot = arr[right]
    i = left - 1
    for j in range(left, right):
        if arr[j] <= pivot:
            i += 1
            arr[i], arr[j] = arr[j], arr[i]
    arr[i + 1], arr[right] = arr[right], arr[i + 1]
    return i + 1

def introsort(arr):
    depth_limit = int(math.log2(len(arr)) * 2)
    quicksort(arr, 0, len(arr) - 1, depth_limit)
    return arr

import random
numbers = [random.randint(1, 1000) for _ in range(500)]
sorted_numbers = introsort(numbers)
print("Sorted array:", sorted_numbers)

Non-Comparison-based Algorithms#

9. Counting Sort#

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def counting_sort(numbers):
    max_value = max(numbers)
    count = [0] * (max_value + 1)

    for num in numbers:
        count[num] += 1

    sorted_numbers = []
    for value in range(len(count)):
        sorted_numbers.extend([value] * count[value])

    return sorted_numbers

numbers = [60, 40, 70, 20, 50, 30, 90, 45]
sorted_numbers = counting_sort(numbers)
print("Sorted numbers:", sorted_numbers)

10. Radix Sort#

def counting_sort_for_radix(numbers, place):
    size = len(numbers)
    output = [0] * size
    count = [0] * 10

    for num in numbers:
        index = num // place
        count[index % 10] += 1

    for i in range(1, 10):
        count[i] += count[i - 1]

    for num in reversed(numbers):
        index = num // place
        output[count[index % 10] - 1] = num
        count[index % 10] -= 1

    for i in range(size):
        numbers[i] = output[i]

def radix_sort(numbers):
    max_value = max(numbers)
    place = 1
    while max_value // place > 0:
        counting_sort_for_radix(numbers, place)
        place *= 10

numbers = [60, 40, 70, 20, 50, 30, 90, 45]
radix_sort(numbers)
print("Sorted numbers:", numbers)

11. Bucket Sort#

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def insertion_sort(numbers):
    for i in range(1, len(numbers)):
        key = numbers[i]
        j = i - 1
        while j >= 0 and numbers[j] > key:
            numbers[j + 1] = numbers[j]
            j -= 1
        numbers[j + 1] = key

def bucket_sort(numbers):
    if not numbers:
        return numbers
    min_value = min(numbers)
    max_value = max(numbers)
    bucket_count = len(numbers)
    buckets = [[] for _ in range(bucket_count)]
    for num in numbers:
        index = (num - min_value) * (bucket_count - 1) // (max_value - min_value) if max_value != min_value else 0
        buckets[index].append(num)
    for bucket in buckets:
        insertion_sort(bucket)
    return [num for bucket in buckets for num in bucket]

import random
numbers = [random.randint(1, 1000) for _ in range(500)]
sorted_numbers = bucket_sort(numbers)
print("Sorted numbers:", sorted_numbers)