[LeetCode] 765. Couples Holding Hands

N couples sit in 2N seats arranged in a row and want to hold hands. We want to know the minimum number of swaps so that every couple is sitting side by side. A swap consists of choosing any two people, then they stand up and switch seats.

The people and seats are represented by an integer from 0 to 2N-1, the couples are numbered in order, the first couple being (0, 1), the second couple being (2, 3), and so on with the last couple being (2N-2, 2N-1).

The couples' initial seating is given by row[i] being the value of the person who is initially sitting in the i-th seat.

Example 1:

Input: row = [0, 2, 1, 3]
Output: 1
Explanation: We only need to swap the second (row[1]) and third (row[2]) person.

Example 2:

Input: row = [3, 2, 0, 1]
Output: 0
Explanation: All couples are already seated side by side.

Note:

1. len(row) is even and in the range of [4, 60].
2. row is guaranteed to be a permutation of 0...len(row)-1.

情侣牵手。

题意是给偶数个人，以数组表示。他们的下标是从0到2N - 1。其中（2N - 2, 2N - 1）是情侣。现在请你做swap操作，请问至少需要swap多少次，可以让所有情侣牵手。

这个题有两种思路，一种是贪心，一种是并查集union find。我这里先给出贪心的思路。同时建议可以先做一下41题，思路类似。

这个题贪心的做法跟桶排序类似，还是试着将每个坐标上放应该放的数字。过一下第一个例子，[0, 2, 1, 3]。第一个数字是0，那么第二个数字的判断就是看他是否是第一个数字0的配偶。判断配偶的方式是看当前这个nums[i + 1]是不是等于nums[i] ^ 1。这一点不容易想到。因为配偶的座位排序要求不是非常严格，比如0和1，既可以01这样坐，也可以10这样坐。但是位运算的这个特点就可以无视0和1的顺序，来判断两个相邻的数字是否互为配偶。如果想不到这个位运算的办法，那么就只能通过先判断当前index上数字的奇偶性来判断index + 1那个位置上的数字是否是配偶。贪心的做法为什么对呢？我这里引用一个discussion给的例子，并且稍加解释。

The proof is easy. Consider a simple example: 7 1 4 6 2 3 0 5. At first step we have two choice to match the first couple: swap 7 with 0, or swap 1 with 6. Then we get 0 1 4 6 2 3 7 5 or 7 6 4 1 2 3 0 5. Pay attention that the first couple doesn't count any more. For the later part it is composed of 4 X 2 3 Y 5 (X=6 Y=7 or X=1 Y=0). Since different couples are unrelated, we don't care X Y is 6 7 pair or 0 1 pair. They are equivalent! Thus it means our choice doesn't count.
Therefore, just follow the distinction and you will get it right!

首先，input是[7 1 4 6 2 3 0 5]。然后第一次swap可以swap7和0，或者swap1和6。第一次swap完了之后，数组长这样，

4 X 2 3 Y 5 (X=6 Y=7 or X=1 Y=0)

后面的swap只跟XY有关，跟已经被swap好的第一组数字无关了。所以也就无所谓先swap哪些数字了。

时间O(n^2)

空间O(1)

Java实现

 1 class Solution {
 2     public int minSwapsCouples(int[] row) {
 3         int res = 0;
 4         for (int i = 0; i < row.length; i += 2) {
 5             if (row[i + 1] == (row[i] ^ 1)) {
 6                 continue;
 7             }
 8             res++;
 9             for (int j = i + 1; j < row.length; j++) {
10                 if (row[j] == (row[i] ^ 1)) {
11                     row[j] = row[i + 1];
12                     row[i + 1] = row[i] ^ 1;
13                     break;
14                 }
15             }
16         }
17         return res;
18     }
19 }

 

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