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title: "题解 - [Luogu P3524] [POI2011] IMP-Party" categories:
plain Byteasar intends to throw up a party
Naturally, he would like it to be a success. Furthermore, Byteasar is quite certain that to make it so it suffices if all invited guests know each other. He is currently trying to come up with a list of his friends he would like to invite
Byteasar has $n$ friends, where $n$ is divisible by 3
plain Fortunately, most of Byteasar's friends know one another
Furthermore, Byteasar recalls that he once attended a party where there were $\frac{2}{3}n$ of his friends, and where everyone knew everyone else
Unfortunately, Byteasar does not quite remember anything else from that party
In particular, he has no idea which of his friends attended it
Byteasar does not feel obliged to throw a huge party, but he would like to invite at least \frac{n}{3} of his friends
He has no idea how to choose them, so he asks you for help
In the first line of the standard input two integers, $n$ and $m$ ($3\le n\le 3\ 000$, $\frac{\frac{2}{3}n(\frac{2}{3}n-1)}{2}\leq m\leq \frac{n(n-1)}{2}$), are given,separated by a single space. These denote the number of Byteasar's friends and the number of pairs of his friends who know each other, respectively
Byteasar's friends are numbered from 1 to $n$
Each of the following $m$ lines holds two integers separated by a single space
The numbers in line no. $i+1$ (for $i=1,2,...,m$) are $a_i$ and $b_i$($1\le a_i<b_i\le n$), separated by a single space, which denote that the persons $a_i$ and $b_i$ know each other. Every pair of numbers appears at most once on the input
In the first and only line of the standard output your program should print $\frac{n}{3}$ numbers, separated by single spaces, in increasing order. These number should specify the numbers of Byteasar's friends whom he should invite to the party. As there are multiple solutions, pick one arbitrarily
6 10
2 5
1 4
1 5
2 4
1 3
4 5
4 6
3 5
3 4
3 6
2 4
给定一张 n 个点 m 条边的图 $n\equiv 0(mod\ 3)$, 保证存在一个大小为 $\frac{2}{3}n$ 的团, 要求输出一个大小为 $\frac{n}{3}$ 的团
讲一个效率和正确率都很高的非完美算法
我们发现如果团内每个点的度往往都大于不在团内每个点的度
所以我们自然想到把所有点按度降序排个序然后取前 $n\over3$ 个输出
但是这样是有 反例 的
{% note %}
我们可以构造这样的图
9 24
1 3
1 5
1 6
2 5
2 6
3 4
3 5
5 6
2 1
2 3
6 3
4 1
2 4
4 5
4 6
7 9
8 9
9 3
9 2
9 5
9 4
9 6
1 7
1 8
画出来是这样的:
我们发现 $1$ 和 $9$ 的度是 $7$, $7$ 和 $8$ 的度是 $2$, 其余点的度是 $6$ 按这样的做法就会输出
1 x 9
其中 $x\in{2,3,4,5,6}$
而 $1$ 和 $9$ 没有边, 所以这是错误答案
{% endnote %}
我们观察这个反例可以发现如果我们把取点策略换成: 在前 $2n\over 3$ 个点里随机取 $n\over3$ 的点, 就有很大概率正确
所以我们搞个 Las Vegas 随机就好了(本篇题解只是单纯的单次shuffle, 并没有加 Las Vegas 随机)
不考虑输入部分的话, 时间复杂度主要在排序上, 也就是 $O(n\log n)$, 这是比正解快的 (输入部分的时间复杂度是 $O(n^2)$)