Generate All Permutations of Array in DSA C++ - Time & Space Complexity

Time Complexity: Generate All Permutations of Array

O(n!)

Understanding Time Complexity

When we generate all permutations of an array, we want to know how the time needed grows as the array gets bigger.

We ask: How many steps does it take to list every possible order of the array elements?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

void permute(vector<int>& nums, int start, vector<vector<int>>& result) {
    if (start == nums.size()) {
        result.push_back(nums);
        return;
    }
    for (int i = start; i < nums.size(); i++) {
        swap(nums[start], nums[i]);
        permute(nums, start + 1, result);
        swap(nums[start], nums[i]);
    }
}

This code generates all possible orders (permutations) of the array by swapping elements and exploring each choice recursively.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Recursive calls combined with a loop that swaps elements.
• How many times: For an array of size n, the function explores n choices at the first level, then n-1 at the next, and so on, leading to n! (factorial) total permutations.

How Execution Grows With Input

Each time we add one more element, the number of permutations multiplies by that new size.

Input Size (n)	Approx. Operations
3	6 (3 x 2 x 1)
5	120 (5 x 4 x 3 x 2 x 1)
7	5040 (7 x 6 x 5 x 4 x 3 x 2 x 1)

Pattern observation: The number of operations grows very fast, multiplying by the next number each time we add an element.

Final Time Complexity

Time Complexity: O(n!)

This means the time needed grows very fast as the array size increases, because we must list every possible order.

Common Mistake

[X] Wrong: "The time grows like n squared or n cubed because of the loops."

[OK] Correct: The loops are inside recursive calls that multiply the work, so the total grows much faster than simple loops.

Interview Connect

Understanding this helps you explain why some problems take much longer as input grows, and shows you can analyze complex recursive code clearly.

Self-Check

"What if we only wanted permutations of length k from n elements? How would the time complexity change?"

Practice

1. What does the process of generating all permutations of an array mean?

easy

A. Listing all possible orders of the array elements

B. Sorting the array in ascending order

C. Finding the maximum element in the array

D. Removing duplicate elements from the array

Solution

1. Step 1: Understand permutations

   Permutations are all possible ways to arrange elements in order.

2. Step 2: Apply to array elements

   Generating all permutations means listing every possible order of the array elements.

3. Final Answer:

   Listing all possible orders of the array elements -> Option A

4. Quick Check:

   Permutations = all orders [OK]

Hint: Permutations = all possible orders of elements [OK]

Common Mistakes:

• Confusing permutations with sorting
• Thinking permutations remove duplicates
• Assuming permutations find max or min

2. Which of the following C++ function signatures is correct for a recursive function generating all permutations of an integer array arr with size n</> and current index start?

easy

A. void permute(int arr[], int start, int n)

B. int permute(int arr[], int n, int start)

C. void permute(int arr[], int n)

D. int permute(int arr[], int start)

Solution

1. Step 1: Check parameter order and types

   The function needs the array, current index (start), and total size (n) to generate permutations recursively.

2. Step 2: Validate return type and parameters

   It should return void (no value) and have parameters (int arr[], int start, int n) in that order.

3. Final Answer:

   void permute(int arr[], int start, int n) -> Option A

4. Quick Check:

   Correct signature = void permute(arr, start, n) [OK]

Hint: Recursive permute needs array, start index, and size [OK]

Common Mistakes:

• Wrong parameter order
• Incorrect return type
• Missing current index parameter

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

void permute(int arr[], int start, int n) {
  if (start == n) {
    for (int i = 0; i < n; i++) std::cout << arr[i] << ' ';
    std::cout << '\n';
    return;
  }
  for (int i = start; i < n; i++) {
    std::swap(arr[start], arr[i]);
    permute(arr, start + 1, n);
    std::swap(arr[start], arr[i]);
  }
}

int main() {
  int arr[] = {1, 2};
  permute(arr, 0, 2);
  return 0;
}

medium

A. 2 1 \n 1 2 \n

B. 1 2 \n 2 1 \n

C. 1 2 \n 1 2 \n

D. No output

Solution

1. Step 1: Trace first call with arr = {1, 2}

   Start = 0, n = 2. Swap arr[0] with arr[0] (no change), call permute with start=1.

2. Step 2: Trace second call with start=1

   Swap arr[1] with arr[1] (no change), call permute with start=2 which prints "1 2".

3. Step 3: Backtrack and swap arr[0] with arr[1]

   Array becomes {2, 1}, call permute with start=1 again.

4. Step 4: At start=1 again, swap arr[1] with arr[1], call permute with start=2 prints "2 1".

5. Final Answer:

   1 2 \n 2 1 \n -> Option B

6. Quick Check:

   All permutations of {1,2} printed [OK]

Hint: Swap and backtrack to print all orders [OK]

Common Mistakes:

• Not swapping back after recursion
• Printing before base case
• Incorrect loop range

4. Identify the error in the following code snippet for generating permutations of array arr of size n:

void permute(int arr[], int start, int n) {
  if (start == n) {
    for (int i = 0; i < n; i++) std::cout << arr[i] << ' ';
    std::cout << '\n';
    return;
  }
  for (int i = 0; i < n; i++) {
    std::swap(arr[start], arr[i]);
    permute(arr, start + 1, n);
    std::swap(arr[start], arr[i]);
  }
}

medium

A. Function should return int, not void

B. Base case condition should be start == n - 1

C. Swap should be done only once outside the loop

D. Loop should start from i = start, not 0

Solution

1. Step 1: Analyze the for loop range

   The loop swaps arr[start] with arr[i]. If i starts at 0 always, it swaps elements already fixed in previous recursion steps.

2. Step 2: Correct loop start index

   Loop should start from i = start to avoid repeating swaps and ensure correct permutations.

3. Final Answer:

   Loop should start from i = start, not 0 -> Option D

4. Quick Check:

   Loop from start to n fixes correct elements [OK]

Hint: Start loop from current index to avoid duplicate swaps [OK]

Common Mistakes:

• Starting loop from 0 causes repeated permutations
• Changing base case incorrectly
• Misplacing swap calls

5. Given an array arr = {1, 2, 2}, how many unique permutations will a correct C++ program generate if it avoids duplicates?

hard

A. 2

B. 6

C. 3

D. 4

Solution

1. Step 1: Calculate total permutations without duplicates

   For 3 elements with two 2's, total permutations = 3! / 2! = 3.

2. Step 2: List unique permutations

   They are {1,2,2}, {2,1,2}, {2,2,1} -- total 3 unique permutations.

3. Final Answer:

   3 -> Option C

4. Quick Check:

   3! / 2! = 3 unique permutations [OK]

Hint: Divide factorial by duplicates factorial for unique count [OK]

Common Mistakes:

• Counting duplicates as unique permutations
• Using 3! = 6 without division
• Ignoring repeated elements