USACO 2月 2021-2022 February Contest Silver银组 题解

要准备usaco的铁铁们可以参考这个文章哦！

想刷好USACO——看这篇文章就够了_GeekAlice的博客-优快云博客usaco 必刷网站https://blog.youkuaiyun.com/GeekAlice/article/details/122291933

Problem 1. Redistributing Gifts

Farmer John has N gifts labeled 1…N for his N cows, also labeled 1…N(1≤N≤500). Each cow has a wishlist, which is a permutation of all N gifts such that the cow prefers gifts that appear earlier in the list over gifts that appear later in the list.

FJ was lazy and just assigned gift i to cow i for all i. Now, the cows have gathered amongst themselves and decided to reassign the gifts such that after reassignment, every cow ends up with the same gift as she did originally, or a gift that she prefers over the one she was originally assigned.

For each i from 1 to N, compute the most preferred gift cow i could hope to receive after reassignment.

INPUT FORMAT (input arrives from the terminal / stdin):

The first line contains N. The next N lines each contain the preference list of a cow. It is guaranteed that each line forms a permutation of 1…N.

OUTPUT FORMAT (print output to the terminal / stdout):

Please output NN lines, the ii-th of which contains the most preferred gift cow ii could hope to receive after reassignment.

SAMPLE INPUT:

4
1 2 3 4
1 3 2 4
1 2 3 4
1 2 3 4

SAMPLE OUTPUT:

1
3
2
4

In this example, there are two possible reassignments:

• The original assignment: cow 1 receives gift 11, cow 22 receives gift 22, cow 33 receives gift 33, and cow 44 receives gift 44.
• Cow 11 receives gift 11, cow 22 receives gift 33, cow 33 receives gift 22, and cow 44 receives gift 44.

Observe that both cows 11 and 44 cannot hope to receive better gifts than they were originally assigned. However, both cows 22 and 33 can.

SCORING:

• Test cases 2-3 satisfy N≤8N≤8.
• Test cases 4-11 satisfy no additional constraints.

        只有当奶牛 i 接收奶牛 j 的分布对应于将 G 的顶点划分为一些包含边 i→j 的简单循环。由此可见，其中一个简单循环包含边 i→j。如果存在一个包含i→j的简单圈C。然后我们可以将 C 上的每头奶牛分配给循环中下一头奶牛的礼物，C 上的每头奶牛最终都会过得更好。让所有不在 C 沿线的奶牛都收到它们原来的礼物。这对应于有效分布。

#incl