Quick Sort Algorithm in DSA Rust - 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 code snippet.

fn quick_sort(arr: &mut [i32]) {
    if arr.len() <= 1 {
        return;
    }
    let pivot_index = partition(arr);
    let (left, right) = arr.split_at_mut(pivot_index);
    quick_sort(left);
    quick_sort(&mut right[1..]);
}

fn partition(arr: &mut [i32]) -> usize {
    let pivot = arr[arr.len() - 1];
    let mut i = 0;
    for j in 0..arr.len() - 1 {
        if arr[j] <= pivot {
            arr.swap(i, j);
            i += 1;
        }
    }
    arr.swap(i, arr.len() - 1);
    i
}

This code sorts an array by choosing 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 once to rearrange elements.
• How many times: Quick Sort calls itself recursively on smaller parts until the whole array is sorted.

How Execution Grows With Input

When the pivot divides the array evenly, the work splits in half each time, leading to a balanced pattern.

Input Size (n)	Approx. Operations
10	About 30 to 40 steps
100	About 700 to 800 steps
1000	About 10,000 to 12,000 steps

Pattern observation: The steps grow a bit faster than n times but much slower than n squared, roughly n times log n.

Final Time Complexity

Time Complexity: O(n log n)

This means sorting takes longer as the list grows, but the increase is moderate and efficient for most cases.

Common Mistake

[X] Wrong: "Quick Sort always sorts in the same fast time no matter the input."

[OK] Correct: If the pivot is chosen poorly, like always the smallest or largest element, Quick Sort can slow down to much longer times.

Interview Connect

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

Self-Check

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

Practice

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

easy

A. Merge two sorted lists into one sorted list

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

C. Sort the list by repeatedly finding the minimum element

D. Divide the list 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 list 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 list.

3. Final Answer:

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

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 lists like Merge Sort
• Believing Quick Sort always picks the first element as pivot

2. Which of the following is the correct Rust function signature for a Quick Sort function that sorts a mutable slice of integers?

easy

A. fn quick_sort(arr: &mut [i32])

B. fn quick_sort(arr: Vec<i32>) -> Vec<i32>

C. fn quick_sort(arr: &[i32]) -> &[i32]

D. fn quick_sort(arr: &mut Vec<i32>) -> ()

Solution

1. Step 1: Identify correct parameter type for in-place sorting

   Quick Sort usually sorts in place, so it needs a mutable slice reference: &mut [i32].

2. Step 2: Check function signature correctness

   fn quick_sort(arr: &mut [i32]) matches this: it takes &mut [i32] and returns nothing (unit type implied).

3. Final Answer:

   fn quick_sort(arr: &mut [i32]) -> Option A

4. Quick Check:

   Mutable slice parameter = fn quick_sort(arr: &mut [i32]) [OK]

Hint: Quick Sort sorts in place, so use &mut slice parameter [OK]

Common Mistakes:

• Using immutable slice &[] which cannot be changed
• Returning a new vector instead of sorting in place
• Using Vec<i32> without mutability for in-place sort

3. What will be the output of the following Rust code snippet after calling quick_sort(&mut arr)?

fn quick_sort(arr: &mut [i32]) {
    if arr.len() <= 1 { return; }
    let pivot = arr[0];
    let mut left = 1;
    let mut right = arr.len() - 1;
    while left <= right {
        if arr[left] <= pivot {
            left += 1;
        } else {
            arr.swap(left, right);
            if right == 0 { break; }
            right -= 1;
        }
    }
    arr.swap(0, right);
    quick_sort(&mut arr[0..right]);
    quick_sort(&mut arr[right+1..]);
}

fn main() {
    let mut arr = [4, 2, 7, 1, 3];
    quick_sort(&mut arr);
    println!("{:?}", arr);
}

medium

A. [1, 2, 3, 7, 4]

B. [1, 2, 3, 4, 7]

C. [4, 2, 7, 1, 3]

D. [7, 4, 3, 2, 1]

Solution

1. Step 1: Understand partitioning around pivot

   The pivot is 4. Elements <=4 move left, >4 move right. After partition, array becomes [3, 2, 1, 4, 7] with pivot at index 3.

2. Step 2: Recursive calls sort left and right parts

   Left part [3, 2, 1] sorts to [1, 2, 3], right part [7] is already sorted. Combined final array is [1, 2, 3, 4, 7].

3. Final Answer:

   [1, 2, 3, 4, 7] -> Option B

4. Quick Check:

   Sorted array = [1, 2, 3, 4, 7] [OK]

Hint: Trace pivot swaps and recursive calls carefully [OK]

Common Mistakes:

• Ignoring pivot swaps and recursive sorting
• Assuming array remains unchanged
• Mixing up left and right partitions

4. Identify the bug in this Rust Quick Sort implementation:

fn quick_sort(arr: &mut [i32]) {
    if arr.len() <= 1 { return; }
    let pivot = arr[0];
    let mut left = 1;
    let mut right = arr.len() - 1;
    while left < right {
        if arr[left] <= pivot {
            left += 1;
        } else {
            arr.swap(left, right);
            right -= 1;
        }
    }
    arr.swap(0, right);
    quick_sort(&mut arr[0..right]);
    quick_sort(&mut arr[right+1..]);
}

medium

A. The while loop condition should be <= instead of <

B. Pivot should be chosen as the last element, not the first

C. The recursive calls should use inclusive ranges

D. The swap after the loop should be with left, not right

Solution

1. Step 1: Analyze the while loop condition

   The loop uses while left < right, but it must include the case when left == right to correctly partition.

2. Step 2: Effect of incorrect loop condition

   Stopping early misses some comparisons, causing incorrect partition and sorting errors.

3. Final Answer:

   The while loop condition should be <= instead of < -> Option A

4. Quick Check:

   Loop condition must cover all indices [OK]

Hint: Check loop boundaries carefully in partition step [OK]

Common Mistakes:

• Ignoring edge case when left equals right
• Changing pivot choice unnecessarily
• Misusing slice ranges in recursive calls

5. You want to sort a large list of integers using Quick Sort in Rust. Which strategy helps avoid worst-case performance when the input is already sorted or nearly sorted?

hard

A. Always pick the first element as pivot

B. Use Bubble Sort instead for sorted inputs

C. Pick a random element as pivot each time

D. Sort only the first half of the list

Solution

1. Step 1: Understand worst-case in Quick Sort

   Picking the first element as pivot on sorted input causes unbalanced partitions, leading to worst-case O(n²) time.

2. Step 2: Use random pivot to avoid worst-case

   Choosing a random pivot spreads partitions more evenly, improving average performance and avoiding worst-case.

3. Final Answer:

   Pick a random element as pivot each time -> Option C

4. Quick Check:

   Random pivot = better average case [OK]

Hint: Randomize pivot to prevent worst-case input issues [OK]

Common Mistakes:

• Always picking first element as pivot
• Switching to Bubble Sort unnecessarily
• Sorting only part of the list