526 Beautiful Arrangement

526. Beautiful Arrangement

1. Question

Suppose you have N integers from 1 to N. We define a beautiful arrangement as an array that is constructed by these N numbers successfully if one of the following is true for the ith position (1 <= i <= N) in this array:

1. The number at the ith position is divisible by i.

2. i is divisible by the number at the ith position.

Now given N, how many beautiful arrangements can you construct?

Example 1:

Input: 2

Output: 2

Explanation:

The first beautiful arrangement is [1, 2]:

Number at the 1st position (i=1) is 1, and 1 is divisible by i (i=1).

Number at the 2nd position (i=2) is 2, and 2 is divisible by i (i=2).


The second beautiful arrangement is [2, 1]:

Number at the 1st position (i=1) is 2, and 2 is divisible by i (i=1).

Number at the 2nd position (i=2) is 1, and i (i=2) is divisible by 1.

Note:

1. N is a positive integer and will not exceed 15.

2. Implementation

(1) Backtracking

class Solution {
    public int countArrangement(int N) {
        boolean[] used = new boolean[N + 1];
        int[] count = new int[1];
        getArrangement(1, N, used, count);
        return count[0];
    }

    public void getArrangement(int curNum, int N, boolean[] used, int[] count) {
        if (curNum == N + 1) {
            ++count[0];
            return;
        }

        for (int i = 1; i <= N; i++) {
            if (!used[i] && (curNum % i == 0 || i % curNum == 0)) {
                used[i] = true;
                getArrangement(curNum + 1, N, used, count);
                used[i] = false;
            }
        }
    }
}

3. Time & Space Complexity

Backtracking: 时间复杂度O(N!), 空间复杂度O(N)