Quick Sort Algorithm in DSA Javascript - 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 with 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.

function quickSort(arr) {
  if (arr.length <= 1) return arr;
  const pivot = arr[arr.length - 1];
  const left = [];
  const right = [];
  for (let i = 0; i < arr.length - 1; i++) {
    if (arr[i] < pivot) left.push(arr[i]);
    else right.push(arr[i]);
  }
  return [...quickSort(left), pivot, ...quickSort(right)];
}
    

This code sorts an array by picking a pivot, splitting the array into smaller parts, and sorting those parts recursively.

Identify Repeating Operations

• Primary operation: The for-loop that compares each element to the pivot to split the array.
• How many times: This loop runs once for each element in the current array segment, repeated recursively on smaller parts.
• Recursion: The function calls itself twice on smaller arrays (left and right), repeating the splitting and sorting.

How Execution Grows With Input

When the array size doubles, the number of comparisons grows roughly by a bit more than double because of the recursive splitting.

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

Pattern observation: The operations grow faster than the input size but slower than the square of the input size.

Final Time Complexity

Time Complexity: O(n log n)

This means that as the list gets bigger, the time to sort grows a little faster than the size but much slower than checking every pair.

Common Mistake

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

[OK] Correct: The time depends on how balanced the splits are; if the pivot is always the smallest or largest, it takes longer.

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 the pivot? How would the time complexity change?"

Practice

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

easy

A. Sort the array by repeatedly selecting the smallest element

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

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

D. Merge two sorted arrays into one sorted array

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 by calling itself recursively until the whole array is sorted.

3. Final Answer:

   Divide the array 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 array, then sort parts recursively [OK]

Common Mistakes:

• Thinking Quick Sort compares all pairs directly
• Confusing Quick Sort with Selection Sort
• Believing Quick Sort merges arrays like Merge Sort

2. Which of the following is the correct syntax to swap two elements arr[i] and arr[j] in JavaScript during Quick Sort?

easy

A. arr[i], arr[j] = arr[j], arr[i];

B. temp = arr[i]; arr[i] = arr[j]; arr[j] = temp;

C. swap(arr[i], arr[j]);

D. arr[i] = arr[j]; arr[j] = arr[i];

Solution

1. Step 1: Identify correct swap method in JavaScript

   JavaScript does not support tuple unpacking like Python, so swapping requires a temporary variable.

2. Step 2: Check each option

   temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; uses a temporary variable to swap correctly. arr[i], arr[j] = arr[j], arr[i]; is Python syntax, swap(arr[i], arr[j]); is undefined, arr[i] = arr[j]; arr[j] = arr[i]; overwrites values incorrectly.

3. Final Answer:

   temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; -> Option B

4. Quick Check:

   Swap with temp variable = temp = arr[i]; arr[i] = arr[j]; arr[j] = temp; [OK]

Hint: Use a temp variable to swap in JavaScript [OK]

Common Mistakes:

• Using Python-style tuple swap in JavaScript
• Overwriting one element before saving it
• Calling undefined swap function

3. What will be the output of the following Quick Sort partition step on array [4, 2, 7, 1] with pivot as the last element?

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

let arr = [4, 2, 7, 1];
let pi = partition(arr, 0, arr.length - 1);
console.log(arr);

medium

A. [1, 2, 7, 4]

B. [1, 2, 4, 7]

C. [4, 2, 7, 1]

D. [2, 1, 4, 7]

Solution

1. Step 1: Trace the partition loop

   Pivot = 1. Loop checks elements less than 1. No elements are less, so i stays at -1.

2. Step 2: Swap pivot with element at i+1

   Swap arr[0] and arr[3]: 4 and 1 swap places. Array becomes [1, 2, 7, 4].

3. Final Answer:

   [1, 2, 7, 4] -> Option A

4. Quick Check:

   Pivot 1 placed at start, others unchanged [OK]

Hint: Pivot swaps with first bigger element after loop [OK]

Common Mistakes:

• Assuming pivot stays at end
• Swapping wrong elements after loop
• Not updating i correctly

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

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

medium

A. Loop should start from low, not low + 1

B. Pivot should be arr[high], not arr[low]

C. No error, function is correct

D. Swapping pivot with arr[i] should be with arr[i - 1]

Solution

1. Step 1: Understand pivot placement

   Pivot is arr[low]. After loop, pivot must be placed at correct position.

2. Step 2: Check swap of pivot with arr[i]

   Variable i points to next greater element, so pivot should swap with arr[i - 1], not arr[i].

3. Final Answer:

   Swapping pivot with arr[i] should be with arr[i - 1] -> Option D

4. Quick Check:

   Pivot swap index off by one [OK]

Hint: Pivot swap index is i-1 after partition loop [OK]

Common Mistakes:

• Swapping pivot with wrong index
• Confusing pivot position in array
• Changing loop start incorrectly

5. Which of the following pivot selection strategies in Quick Sort will result in the most balanced partitions and thus better performance?

hard

A. Choosing the median of first, middle, and last elements as pivot

B. Choosing the first element as pivot

C. Choosing the middle element as pivot

D. Choosing the last element as pivot

Solution

1. Step 1: Understand pivot selection strategies

   Choosing first or last can lead to worst-case O(n^2) on sorted arrays. Middle element alone may not be representative.

2. Step 2: Best strategy for balance

   Median-of-three selects a pivot likely closer to the array's true median, promoting balanced partitions and average O(n log n) performance.

3. Final Answer:

   Choosing the median of first, middle, and last elements as pivot -> Option A

4. Quick Check:

   Median-of-three pivot = balanced partitions [OK]

Hint: Median-of-three pivot improves balance and speed [OK]

Common Mistakes:

• Always picking first or last element as pivot
• Ignoring pivot choice impact on performance
• Assuming middle element is always best pivot