LeetCode 526. Beautiful Arrangement

Description

https://leetcode.com/problems/beautiful-arrangement/

Suppose you have n integers labeled 1 through n. A permutation of those n integers perm (1-indexed) is considered a beautiful arrangement if for every i (1 <= i <= n), either of the following is true:

  • perm[i] is divisible by i.
  • i is divisible by perm[i].

Given an integer n, return the number of the beautiful arrangements that you can construct.

Example 1:

Input: n = 2
Output: 2
Explanation: 
The first beautiful arrangement is [1,2]:
    - perm[1] = 1 is divisible by i = 1
    - perm[2] = 2 is divisible by i = 2
The second beautiful arrangement is [2,1]:
    - perm[1] = 2 is divisible by i = 1
    - i = 2 is divisible by perm[2] = 1

Example 2:

Input: n = 1
Output: 1

Constraints:

  • 1 <= n <= 15

Explanation

Use the backtracking approach to find all the combinations where each position meets nums[i] is divisible by i or i is divisible by nums[i] condition.

Python Solution

class Solution:
    def countArrangement(self, n: int) -> int:
        numbers = [i for i in range(1, n + 1)]


        results = []
        visited = set()
        self.helper(results, numbers, [], visited)

        return len(results)

    def helper(self, results, numbers, combination, visited):
        
        if len(combination) == len(numbers):
            results.append(list(combination))            
            return

        for num in numbers:
            if num in visited:
                continue

            if num % (len(combination) + 1) != 0 and (len(combination) + 1) % num != 0:
                continue
            
            combination.append(num)
            visited.add(num)

            self.helper(results, numbers, combination, visited)

            combination.pop()
            visited.remove(num)
  • Time Complexity: O(N).
  • Space Complexity: O(N).

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