Next Permutation of Array in DSA Python - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the goal of the Next Permutation algorithm?

To rearrange numbers into the lexicographically next greater permutation of numbers. If such arrangement is not possible, it rearranges it as the lowest possible order (sorted in ascending order).

intermediate

In the Next Permutation algorithm, what is the significance of finding the first decreasing element from the right?

It identifies the pivot point where we can make a change to get the next bigger permutation. This element is the first one that can be swapped to increase the sequence.

intermediate

Why do we reverse the suffix after swapping in the Next Permutation algorithm?

Because the suffix after the pivot is in descending order, reversing it makes it the smallest ascending order, ensuring the next permutation is just one step bigger.

beginner

What happens if the entire array is in descending order in the Next Permutation algorithm?

It means the current permutation is the highest possible. The algorithm then reverses the entire array to get the lowest possible order (ascending order).

intermediate

Show the step-by-step transformation of the array [1, 3, 2] using the Next Permutation algorithm.

1. Find pivot: from right, 1 < 3 (pivot at index 0) 2. Find element just larger than pivot: 2 at index 2 3. Swap pivot and this element: [2, 3, 1] 4. Reverse suffix after pivot: suffix is [3, 1], reversed to [1, 3] 5. Result: [2, 1, 3]

What is the first step in finding the next permutation of an array?

A Find the first element from the right that is smaller than the element next to it

B Reverse the entire array

C Sort the array in ascending order

D Swap the first and last elements

The first step is to find the pivot, which is the first element from the right that is smaller than its next element.

If no pivot is found (array is in descending order), what should the next permutation be?

A The array sorted in descending order

B The array reversed (ascending order)

C The same array

D Swap first two elements

If the array is in descending order, it is the highest permutation, so the next permutation is the lowest, which is the reversed array (ascending order).

After swapping the pivot with the next larger element, what is the next step?

A Swap the first and last elements

B Sort the entire array

C Reverse the suffix after the pivot

D Do nothing

Reversing the suffix after the pivot ensures the suffix is in ascending order, giving the next smallest permutation.

What is the time complexity of the Next Permutation algorithm?

A O(n)

B O(n^2)

C O(log n)

D O(1)

The algorithm scans the array a few times linearly, so the time complexity is O(n).

Given the array [1, 2, 3], what is the next permutation?

A [1, 2, 3]

B [3, 2, 1]

C [2, 1, 3]

D [1, 3, 2]

The next permutation after [1, 2, 3] is [1, 3, 2].

Explain the steps of the Next Permutation algorithm in your own words.

Think about how to get the next bigger number by changing the array minimally.

Why do we reverse the suffix after swapping in the Next Permutation algorithm?

Consider the order of numbers after the pivot.

Practice

1. What does the next permutation of an array represent?

easy

A. The next lexicographically greater arrangement of the array elements

B. The array sorted in ascending order

C. The array reversed

D. The smallest element moved to the front

Solution

1. Step 1: Understand the concept of permutation order

   Permutations can be ordered lexicographically like words in a dictionary.

2. Step 2: Define next permutation meaning

   The next permutation is the next arrangement that is just greater than the current one in this order.

3. Final Answer:

   The next lexicographically greater arrangement of the array elements -> Option A

4. Quick Check:

   Next permutation = next bigger arrangement [OK]

Hint: Next permutation means next bigger order arrangement [OK]

Common Mistakes:

• Thinking next permutation is just sorting ascending
• Confusing next permutation with reversing the array
• Assuming next permutation always moves smallest element front

2. Which of the following Python code snippets correctly swaps two elements arr[i] and arr[j] in an array?

easy

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

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

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

D. arr[i] = arr[j]

Solution

1. Step 1: Recall Python tuple unpacking for swapping

   Python allows swapping two variables in one line using tuple unpacking.

2. Step 2: Check each option

   arr[i], arr[j] = arr[j], arr[i] correctly swaps using arr[i], arr[j] = arr[j], arr[i]. arr[i] = arr[j]; arr[j] = arr[i] overwrites arr[i] before assigning arr[j]. swap(arr[i], arr[j]) is invalid syntax without a defined function. arr[i] = arr[j] only assigns one element.

3. Final Answer:

   arr[i], arr[j] = arr[j], arr[i] -> Option A

4. Quick Check:

   Python swap uses tuple unpacking [OK]

Hint: Use tuple unpacking to swap in Python [OK]

Common Mistakes:

• Overwriting one element before swapping
• Using undefined swap function
• Assigning only one element instead of both

3. What is the output of the following code snippet?

arr = [1, 3, 2]
# Apply next permutation logic
# Step 1: Find first decreasing element from right (index 0)
# Step 2: Find element just larger than arr[0] from right (index 2)
# Step 3: Swap arr[0] and arr[2]
# Step 4: Reverse elements after index 0

arr[0], arr[2] = arr[2], arr[0]
arr[1:] = arr[1:][::-1]
print(arr)

medium

A. [3, 1, 2]

B. [2, 1, 3]

C. [1, 2, 3]

D. [2, 3, 1]

Solution

1. Step 1: Identify the first decreasing element from right

   From right, arr[1]=3 > arr[2]=2 (not decreasing), arr[0]=1 < arr[1]=3 (decreasing at index 0)

2. Step 2: Find element just larger than arr[0] from right

   From right, arr[2]=2 > arr[0]=1, so index 2 chosen

3. Step 3: Swap arr[0] and arr[2]

   Swap 1 and 2 -> arr becomes [2, 3, 1]

4. Step 4: Reverse elements after index 0

   Reverse arr[1:] -> arr[1:] = [3,1] reversed to [1,3]

5. Final Answer:

   [2, 1, 3] -> Option B

6. Quick Check:

   Next permutation of [1,3,2] = [2,1,3] [OK]

Hint: Swap pivot with next bigger, then reverse suffix [OK]

Common Mistakes:

• Not reversing suffix after swap
• Swapping wrong elements
• Confusing increasing and decreasing order

4. The following code attempts to find the next permutation of arr = [3, 2, 1]. What is the error?

def next_permutation(arr):
    i = len(arr) - 2
    while i >= 0 and arr[i] >= arr[i + 1]:
        i -= 1
    if i >= 0:
        j = len(arr) - 1
        while arr[j] <= arr[i]:
            j -= 1
        arr[i], arr[j] = arr[j], arr[i]
    arr[i + 1:] = reversed(arr[i + 1:])

arr = [3, 2, 1]
next_permutation(arr)
print(arr)

medium

A. The while loop condition is incorrect

B. Swapping elements is done in wrong order

C. Using reversed() returns an iterator, not a list, causing wrong assignment

D. The function does not handle empty arrays

Solution

1. Step 1: Analyze the reversal line

   The line arr[i + 1:] = reversed(arr[i + 1:]) assigns an iterator to a list slice, which expects a list.

2. Step 2: Identify correct usage

   To assign reversed elements properly, convert iterator to list: arr[i + 1:] = list(reversed(arr[i + 1:])).

3. Final Answer:

   Using reversed() returns an iterator, not a list, causing wrong assignment -> Option C

4. Quick Check:

   reversed() needs list() for slice assignment [OK]

Hint: Use list(reversed(...)) when assigning slices [OK]

Common Mistakes:

• Assigning reversed() iterator directly to list slice
• Ignoring empty array edge case
• Swapping elements incorrectly

5. Given the array arr = [1, 5, 1], what is the next permutation after applying the standard algorithm? Consider duplicates carefully.

hard

A. [5, 1, 5]

B. [1, 1, 5]

C. [1, 5, 1]

D. [5, 1, 1]

Solution

1. Step 1: Find first decreasing element from right

   From right: arr[1]=5 > arr[2]=1 (not decreasing), arr[0]=1 < arr[1]=5 (decreasing at index 0)

2. Step 2: Find element just larger than arr[0] from right

   From right, arr[2]=1 not > 1, arr[1]=5 > 1, so index 1 chosen

3. Step 3: Swap arr[0] and arr[1]

   Swap 1 and 5 -> arr becomes [5, 1, 1]

4. Step 4: Reverse elements after index 0

   Reverse arr[1:] -> arr[1:] = [1,1] reversed is same [1,1]

5. Final Answer:

   [5, 1, 1] -> Option D

6. Quick Check:

   Duplicates handled, next permutation = [5,1,1] [OK]

Hint: Swap pivot with next bigger, reverse suffix even with duplicates [OK]

Common Mistakes:

• Ignoring duplicates when finding next bigger element
• Not reversing suffix after swap
• Assuming next permutation is sorted ascending always