leetcode

README

526. Beautiful Arrangement

题目

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

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.

题目大意

假设有从 1 到 N 的 N 个整数，如果从这 N 个数字中成功构造出一个数组，使得数组的第 i 位 (1 <= i <= N) 满足如下两个条件中的一个，我们就称这个数组为一个优美的排列。条件：

• 第 i 位的数字能被 i 整除
• i 能被第 i 位上的数字整除

现在给定一个整数 N，请问可以构造多少个优美的排列？

解题思路

• 这一题是第 46 题的加强版。由于这一题给出的数组里面的数字都是不重复的，所以可以当做第 46 题来做。
• 这题比第 46 题多的一个条件是，要求数字可以被它对应的下标 + 1 整除，或者下标 + 1 可以整除下标对应的这个数字。在 DFS 回溯过程中加入这个剪枝条件就可以了。
• 当前做法时间复杂度不是最优的，大概只有 33.3%