Quick Sort Algorithm in DSA Python - Time & Space Complexity

Time Complexity: Quick Sort Algorithm

O(n log n)

Understanding Time Complexity

We want to understand how the time to sort a list using Quick Sort changes as the list gets bigger.

How does the number of steps grow when the input size increases?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

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)

This code sorts a list by picking a pivot, splitting the list into parts, and sorting each part recursively.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Splitting the list into smaller lists around the pivot using list comprehensions.
• How many times: This splitting happens at each recursive call, which can be up to the number of elements in the list.

How Execution Grows With Input

Each time the list is split, the work done is proportional to the size of the current list. The total work depends on how balanced the splits are.

Input Size (n)	Approx. Operations
10	About 50 to 100 operations
100	About 700 to 1,000 operations
1000	About 10,000 to 15,000 operations

Pattern observation: When splits are balanced, operations grow roughly like n times log n; if splits are unbalanced, it can grow much faster.

Final Time Complexity

Time Complexity: O(n log n)

This means the time to sort grows a bit faster than the size of the list, but much slower than checking every pair.

Common Mistake

[X] Wrong: "Quick Sort always takes the same time regardless of input order."

[OK] Correct: If the pivot choices are poor, Quick Sort can take much longer, especially if the list is already sorted or nearly sorted.

Interview Connect

Understanding Quick Sort's time complexity helps you explain why it is fast on average and what cases to watch out for, showing your grasp of sorting efficiency.

Self-Check

"What if we always picked the first element as the pivot? How would the time complexity change?"

Practice

1. What is the main idea behind the Quick Sort algorithm?

easy

A. Sort the list by inserting elements into their correct position one by one

B. Sort the list by repeatedly swapping adjacent elements

C. Divide the list into smaller parts using a pivot and sort them recursively

D. Sort the list by merging two sorted halves

Solution

1. Step 1: Understand Quick Sort approach

   Quick Sort picks a pivot element and divides the list into parts smaller and larger than the pivot.

2. Step 2: Recursive sorting

   It then sorts these parts recursively, combining them to get the sorted list.

3. Final Answer:

   Divide the list into smaller parts using a pivot and sort them recursively -> Option C

4. Quick Check:

   Quick Sort = Divide and conquer [OK]

Hint: Remember: pivot divides list, then sort parts recursively [OK]

Common Mistakes:

• Confusing Quick Sort with Bubble Sort
• Thinking Quick Sort merges sorted halves
• Assuming Quick Sort inserts elements one by one

2. Which of the following is the correct way to define the partition function header in Python for Quick Sort?

easy

A. def partition(arr, low, high):

B. def partition(arr):

C. def partition(low, high):

D. def partition(arr, pivot):

Solution

1. Step 1: Identify required parameters

   The partition function needs the array and the current low and high indices to work on a sublist.

2. Step 2: Match function signature

   Only def partition(arr, low, high): correctly includes all three parameters: arr, low, and high.

3. Final Answer:

   def partition(arr, low, high): -> Option A

4. Quick Check:

   Partition needs array and index bounds [OK]

Hint: Partition needs array plus low and high indexes [OK]

Common Mistakes:

• Omitting low and high parameters
• Using pivot as parameter instead of indices
• Defining partition with only one parameter

3. What will be the output of the following code snippet?

def quick_sort(arr, low, high):
    if low < high:
        pi = partition(arr, low, high)
        quick_sort(arr, low, pi - 1)
        quick_sort(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

arr = [3, 6, 8, 10, 1, 2, 1]
quick_sort(arr, 0, len(arr) - 1)
print(arr)

medium

A. [1, 1, 2, 3, 6, 8, 10]

B. [10, 8, 6, 3, 2, 1, 1]

C. [3, 6, 8, 10, 1, 2, 1]

D. [1, 2, 1, 3, 6, 8, 10]

Solution

1. Step 1: Understand partition and recursive calls

   The partition function places the pivot in correct position and divides array into smaller and larger parts. Quick sort recursively sorts these parts.

2. Step 2: Trace sorting of given array

   Starting with [3,6,8,10,1,2,1], after full recursion, the array becomes sorted ascending: [1,1,2,3,6,8,10].

3. Final Answer:

   [1, 1, 2, 3, 6, 8, 10] -> Option A

4. Quick Check:

   Quick Sort output = sorted array ascending [OK]

Hint: Quick Sort outputs sorted array ascending by default [OK]

Common Mistakes:

• Thinking Quick Sort sorts descending by default
• Confusing original array with sorted output
• Ignoring pivot placement effect

4. Identify the error in this Quick Sort partition function:

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

medium

A. Swapping inside the loop is incorrect

B. Pivot should be arr[high] instead of arr[low]

C. Return statement should return 'i' instead of 'i-1'

D. The for loop should include 'high' index by using range(low + 1, high + 1)

Solution

1. Step 1: Check loop range for partitioning

   The for loop uses range(low + 1, high), which excludes the 'high' index element. It should include 'high' to consider the pivot's correct position.

2. Step 2: Correct loop range

   Changing to range(low + 1, high + 1) includes the last element, ensuring all elements are compared to pivot.

3. Final Answer:

   The for loop should include 'high' index by using range(low + 1, high + 1) -> Option D

4. Quick Check:

   Loop must cover full sublist including high index [OK]

Hint: Loop must include high index to compare all elements [OK]

Common Mistakes:

• Forgetting to include last element in loop
• Changing pivot position unnecessarily
• Returning wrong index after partition

5. You want to sort a list of strings by their length using Quick Sort. Which modification to the partition function is correct?

def partition(arr, low, high):
    pivot = arr[high]
    i = low - 1
    for j in range(low, high):
        if ???:
            i += 1
            arr[i], arr[j] = arr[j], arr[i]
    arr[i + 1], arr[high] = arr[high], arr[i + 1]
    return i + 1

What should replace ??? to sort by string length?

hard

A. arr[j].startswith(pivot)

B. len(arr[j]) <= len(pivot)

C. len(arr[j]) >= len(pivot)

D. arr[j] <= pivot

Solution

1. Step 1: Understand sorting by string length

   To sort strings by length, compare lengths of elements, not the strings themselves.

2. Step 2: Replace condition with length comparison

   Using len(arr[j]) <= len(pivot) ensures elements with length less or equal to pivot's length go left.

3. Final Answer:

   len(arr[j]) <= len(pivot) -> Option B

4. Quick Check:

   Compare lengths to sort strings by length [OK]

Hint: Compare lengths, not strings, to sort by length [OK]

Common Mistakes:

• Comparing strings lexicographically instead of length
• Using >= instead of <= causing reverse order
• Using unrelated string methods like startswith