Quick Sort as Divide and Conquer in DSA C - Time & Space Complexity

Time Complexity: Quick Sort as Divide and Conquer

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.

Specifically, how many steps does Quick Sort take when sorting more items?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

void quickSort(int arr[], int low, int high) {
    if (low < high) {
        int pi = partition(arr, low, high);
        quickSort(arr, low, pi - 1);
        quickSort(arr, pi + 1, high);
    }
}

int partition(int arr[], int low, int high) {
    int pivot = arr[high];
    int i = low - 1;
    for (int j = low; j <= high - 1; j++) {
        if (arr[j] < pivot) {
            i++;
            int temp = arr[i]; arr[i] = arr[j]; arr[j] = temp;
        }
    }
    int temp = arr[i + 1]; arr[i + 1] = arr[high]; arr[high] = temp;
    return i + 1;
}
    

This code sorts an array by picking a pivot, dividing the array around it, and sorting the parts recursively.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: The partition function loops through the array segment once to rearrange elements.
• How many times: The quickSort function calls itself twice on smaller parts, repeating partition on smaller segments until sorted.

How Execution Grows With Input

When the array size doubles, the number of operations grows more than double but less than square.

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

Pattern observation: The steps grow roughly proportional to n times log n, which is faster than simple repeated scanning but slower than quadratic.

Final Time Complexity

Time Complexity: O(n log n)

This means the sorting time grows a bit more than linearly but much less 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, even up to n squared steps.

Interview Connect

Understanding Quick Sort's time complexity helps you explain why it is fast on average and what can slow it down, showing your grasp of efficient algorithms.

Self-Check

"What if we always picked the middle element as pivot instead of the last? How would the time complexity change?"

Practice

1. What is the main idea behind the Quick Sort algorithm in the divide and conquer approach?

easy

A. Compare each element with every other element to sort

B. Divide the array into smaller parts using a pivot and sort them recursively

C. Merge two sorted arrays into one sorted array

D. Insert elements one by one into the correct position

Solution

1. Step 1: Understand divide and conquer in Quick Sort

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

2. Step 2: Recursive sorting of parts

   It then sorts these parts recursively, combining results to get a sorted array.

3. Final Answer:

   Divide the array into smaller parts using a pivot and sort them recursively -> Option B

4. Quick Check:

   Quick Sort = Divide and conquer with pivot [OK]

Hint: Remember pivot splits array for recursive sorting [OK]

Common Mistakes:

• Confusing Quick Sort with Merge Sort
• Thinking Quick Sort compares all pairs
• Believing Quick Sort inserts elements one by one

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

easy

A. int partition(int arr[], int low, int high);

B. void partition(int arr[], int low, int high);

C. int partition(int arr, int low, int high);

D. int partition(int arr[], int low);

Solution

1. Step 1: Check parameter types for partition

   The partition function takes an array and two indices low and high to work on a subarray.

2. Step 2: Check return type

   It returns an integer index where the pivot ends after partitioning.

3. Final Answer:

   int partition(int arr[], int low, int high); -> Option A

4. Quick Check:

   Partition returns pivot index with array and bounds [OK]

Hint: Partition returns pivot index, needs array and bounds [OK]

Common Mistakes:

• Using void return type instead of int
• Passing array as int instead of int[]
• Missing one of the index parameters

3. What will be the output array after calling Quick Sort on arr = {3, 6, 8, 10, 1, 2, 1}?

medium

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

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

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

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

Solution

1. Step 1: Apply Quick Sort to the array

   Quick Sort rearranges elements so smaller come before pivot and larger after, recursively sorting subarrays.

2. Step 2: Sort the given array

   Sorting {3,6,8,10,1,2,1} results in {1,1,2,3,6,8,10} in ascending order.

3. Final Answer:

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

4. Quick Check:

   Sorted array = ascending order [OK]

Hint: Quick Sort outputs sorted array ascending [OK]

Common Mistakes:

• Assuming original array stays same
• Confusing ascending with descending order
• Misplacing duplicate elements

4. Identify the error in this Quick Sort partition code snippet:

int partition(int arr[], int low, int high) {
  int pivot = arr[high];
  int i = low;
  for (int j = low; j < high; j++) {
    if (arr[j] < pivot) {
      i++;
      swap(&arr[i], &arr[j]);
    }
  }
  swap(&arr[i + 1], &arr[high]);
  return i + 1;
}

medium

A. For loop should run until j <= high

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

C. Swap should be called outside the loop only

D. Variable i should start at low - 1

Solution

1. Step 1: Check initialization of i

   In partition, i should start at low - 1 to correctly track smaller elements before j.

2. Step 2: Analyze loop and swaps

   Starting i at low causes incorrect swaps and wrong pivot placement.

3. Final Answer:

   Variable i should start at low - 1 -> Option D

4. Quick Check:

   i = low - 1 fixes partition indexing [OK]

Hint: Initialize i to low - 1 before loop [OK]

Common Mistakes:

• Starting i at low causes off-by-one errors
• Changing pivot position incorrectly
• Loop running beyond high index

5. Given an array with many duplicate elements, which modification to Quick Sort helps improve performance?

hard

A. Avoid recursion and use iterative sorting only

B. Always pick the first element as pivot

C. Use a three-way partitioning to separate less, equal, and greater elements

D. Skip sorting duplicates to save time

Solution

1. Step 1: Understand problem with duplicates in Quick Sort

   Duplicates cause many equal elements to be repeatedly partitioned, slowing sorting.

2. Step 2: Apply three-way partitioning

   Three-way partitioning divides array into less than, equal to, and greater than pivot, reducing redundant work.

3. Final Answer:

   Use a three-way partitioning to separate less, equal, and greater elements -> Option C

4. Quick Check:

   Three-way partitioning optimizes duplicates [OK]

Hint: Three-way partition handles duplicates efficiently [OK]

Common Mistakes:

• Picking fixed pivot causes worst-case time
• Skipping duplicates breaks sorting correctness
• Avoiding recursion does not fix duplicate issue