Permutations

Clarify the problem:

• The problem requires us to generate all possible permutations of a given list of numbers.
• We need to implement a function that takes in a list of distinct integers and returns a list of all possible permutations.
• The order of the permutations does not matter.
• The input list can contain duplicate numbers, and the output should not contain duplicate permutations.

Analyze the problem:

• Input: List of distinct integers.
• Output: List of all possible permutations.
• Constraints:
  • The length of the input list is between 1 and 6.
  • The integers in the input list are distinct.

Design an algorithm:

• We can solve this problem using backtracking, which is a depth-first search (DFS) algorithm.
• We start with an empty permutation list and an empty visited set.
• We iterate through each number in the input list.
• For each number, if it is not visited, we add it to the permutation list and mark it as visited.
• Then, we recursively generate the permutations for the remaining unvisited numbers.
• After generating the permutations, we backtrack by removing the current number from the permutation list and marking it as unvisited.
• We repeat this process for all numbers in the input list.
• The base case of the recursion is when the permutation list has the same length as the input list, indicating that we have generated a valid permutation.
• We store the valid permutations in a result list and return it as the output.

Explain your approach:

• We will implement a backtracking algorithm to generate all possible permutations.
• First, we define a recursive helper function called generate_permutations that takes in the input list, a permutation list, a visited set, and a result list.
• Inside the helper function, we check the base case: if the permutation list has the same length as the input list, we have a valid permutation, so we add it to the result list.
• Otherwise, we iterate through each number in the input list.
• If the number is not visited, we add it to the permutation list and mark it as visited.
• Then, we recursively call the generate_permutations function with the updated permutation list, visited set, and result list.
• After the recursive call, we backtrack by removing the current number from the permutation list and marking it as unvisited.
• Finally, we define the main function permute that initializes the permutation list, visited set, and result list, and calls the generate_permutations function.
• The permute function returns the result list of permutations.

Write clean and readable code:

python

class Solution:
    def permute(self, nums):
        def generate_permutations(nums, permutation, visited, result):
            if len(permutation) == len(nums):
                result.append(permutation[:])
                return

            for num in nums:
                if num not in visited:
                    visited.add(num)
                    permutation.append(num)
                    generate_permutations(nums, permutation, visited, result)
                    permutation.pop()
                    visited.remove(num)

        result = []
        generate_permutations(nums, [], set(), result)
        return result

Test your code:

python

solution = Solution()

nums = [1, 2, 3]
permutations = solution.permute(nums)
print(permutations)  # Output: [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]

nums = [0, 1]
permutations = solution.permute(nums)
print(permutations)  # Output: [[0, 1], [1, 0]]

nums = [1]
permutations = solution.permute(nums)
print(permutations)  # Output: [[1]]

Optimize if necessary:

• The provided solution uses backtracking, which generates all possible permutations.
• The time complexity is O(N!), where N is the length of the input list, as there are N! possible permutations.
• The space complexity is O(N) for the recursion stack and the result list.

Handle error cases:

• The code assumes that the input for the permute function is a valid list of distinct integers.
• If the input list is empty, the function will return an empty list.

Discuss complexity analysis:

• The time complexity of the solution is O(N!), where N is the length of the input list. This is because there are N! possible permutations.
• The space complexity is O(N) for the recursion stack and the result list.
• The solution is efficient for the given problem, considering the nature of generating all possible permutations.