Quick Sort Algorithm in DSA Java - Time & Space Complexity

Time Complexity: Quick Sort Algorithm

O(n log n)

Understanding Time Complexity

We want to understand how the time taken by Quick Sort changes as the list size grows.

How does the number of steps grow when sorting bigger lists?

Scenario Under Consideration

Analyze the time complexity of the following Quick Sort code snippet.

public 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);
  }
}

private 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

Look at what repeats as the input grows.

• Primary operation: The partition loop that compares and swaps elements.
• How many times: This loop runs once per recursive call, covering parts of the array each time.
• Recursion: The function calls itself twice on smaller parts after partitioning.

How Execution Grows With Input

As the list size grows, the number of comparisons depends on how well the pivot splits the list.

Input Size (n)	Approx. Operations
10	About 30 to 50 comparisons
100	About 700 to 1,000 comparisons
1000	About 10,000 to 15,000 comparisons

Pattern observation: If the pivot splits evenly, operations grow a bit more than n times log n; if splits are uneven, it can grow closer to n squared.

Final Time Complexity

Time Complexity: O(n log n)

This means sorting takes a bit more than n times the steps needed to split the list repeatedly in half.

Common Mistake

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

[OK] Correct: If the pivot is always the smallest or largest element, Quick Sort does more work and slows down to n squared time.

Interview Connect

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

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?

easy

A. Insert elements one by one into their correct position

B. Compare every element with every other element and swap if needed

C. Merge two sorted arrays into one sorted array

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

Solution

1. Step 1: Understand Quick Sort's approach

   Quick Sort picks a pivot element 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 them to get the sorted array.

3. Final Answer:

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

4. Quick Check:

   Quick Sort = Divide and conquer [OK]

Hint: Remember: Quick Sort uses a pivot to split and sort recursively [OK]

Common Mistakes:

• Confusing Quick Sort with Bubble Sort
• Thinking Quick Sort merges arrays like Merge Sort
• Assuming Quick Sort inserts elements one by one

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

easy

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

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

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

D. public static partition(int[] arr, int low, int high)

Solution

1. Step 1: Check Java method declaration syntax

   Java methods must specify access modifier, return type, method name, and parameter types with names.

2. Step 2: Validate each option

   public int partition(int[] arr, int low, int high) correctly declares a public method returning int with parameters typed and named properly.

3. Final Answer:

   public int partition(int[] arr, int low, int high) -> Option B

4. Quick Check:

   Java method syntax = public int partition(int[] arr, int low, int high) [OK]

Hint: Remember Java method syntax: access + return type + name + parameters [OK]

Common Mistakes:

• Omitting return type
• Missing parameter types
• Incorrect order of parameters

3. What will be the output array after the first partition call on arr = [8, 3, 7, 6, 2] with pivot as last element?

medium

A. [2, 3, 6, 7, 8]

B. [3, 2, 6, 7, 8]

C. [2, 3, 7, 6, 8]

D. [8, 3, 7, 6, 2]

Solution

1. Step 1: Choose pivot and partition

   Pivot is last element 2. Move elements smaller than 2 to left.

2. Step 2: Rearrange array

   No elements smaller than 2, so swap 2 with first element 8. Result: [2, 3, 6, 7, 8]

3. Step 3: Check order after partition

   Elements less than pivot (none) are left, others right. No further swaps in first partition.

4. Final Answer:

   [2, 3, 6, 7, 8] -> Option A

5. Quick Check:

   Pivot 2 placed correctly, smaller left, larger right [OK]

Hint: Pivot is last element; smaller elements go left [OK]

Common Mistakes:

• Not swapping pivot correctly
• Assuming all smaller elements are sorted immediately
• Confusing pivot position after partition

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) {
      swap(arr, i, j);
      i++;
    }
  }
  swap(arr, i, high);
  return i;
}

medium

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

B. Loop condition should be j <= high

C. Variable i should start at low - 1

D. Swap should happen only if arr[j] <= pivot

Solution

1. Step 1: Analyze initialization of i

   In Lomuto partition, i starts at low - 1 to track smaller elements index before loop.

2. Step 2: Check loop and swaps

   Starting i at low causes incorrect swaps and pivot placement.

3. Final Answer:

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

4. Quick Check:

   i = low - 1 for correct partition [OK]

Hint: i starts one before low to track smaller elements [OK]

Common Mistakes:

• Starting i at low causes wrong partition
• Changing pivot position incorrectly
• Using wrong loop bounds

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

hard

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

B. Always pick the first element as pivot

C. Use Bubble Sort instead for duplicates

D. Skip sorting duplicates by checking each element

Solution

1. Step 1: Understand problem with duplicates

   Standard Quick Sort performs poorly with many duplicates due to repeated swaps.

2. Step 2: Apply three-way partitioning

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

3. Step 3: Evaluate other options

   Picking first element or skipping sorting duplicates doesn't improve performance reliably; Bubble Sort is inefficient.

4. Final Answer:

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

5. Quick Check:

   Three-way partitioning = better with duplicates [OK]

Hint: Use three-way partitioning for arrays with many duplicates [OK]

Common Mistakes:

• Picking fixed pivot without randomization
• Ignoring duplicates in partitioning
• Switching to inefficient sorts unnecessarily