POJ 1466 Girls and Boys（最大独立集）

最新推荐文章于 2019-08-16 16:46:55 发布

原创最新推荐文章于 2019-08-16 16:46:55 发布 · 565 阅读

0 ·

CC 4.0 BY-SA版权

文章标签：

#匈牙利 #最大独立集 #算法 #poj

匹配问题专栏收录该内容

24 篇文章

订阅专栏

本文介绍了一种通过最大匹配算法解决最大独立集问题的方法。针对一组学生之间的浪漫关系，目标是找出最大的互不相关的子集。通过将问题转化为图的最大匹配问题，并利用匈牙利算法进行求解。

摘要生成于 C知道，由 DeepSeek-R1 满血版支持，前往体验 >

题目地址：点击打开链接

题意：

有n个学生，一些学生之间存在关系，现在要把学生弄到一个集合，这个集合中的学生两两不能有关系，求这个集合的

最大学生数是多少。

思路：

将每个点拆成两个点，一个当作是男的点，一个当作是女的点了，然后连边。由于关系是相互的，这样就造成了边的重

复。也就是边集是刚才的二倍，从而导致了最大匹配变成了

二倍。那么答案就是最大独立集=顶点数-最大匹配/2

代码：

#include<iostream>
#include<cstdio>
#include<cstring>
#include<vector>
using namespace std;
const int maxn = 505;
vector<int> g[maxn];
int n, m, match[maxn];
bool vis[maxn];

bool dfs(int x)
{
    for(int i = 0; i < g[x].size(); i++)
    {
        int v = g[x][i];
        if(!vis[v])
        {
            vis[v] = 1;
            if(match[v] == -1 || dfs(match[v]))
            {
                match[v] = x;
                return 1;
            }
        }
    }
    return 0;
}

int Hungary()
{
    int res = 0;
    for(int i = 1; i <= n; i++)
    {
        memset(vis, 0, sizeof(vis));
        res += dfs(i);
    }
    return res;
}

int main(void)
{
    while(cin >> n)
    {
        memset(match, -1, sizeof(match));
        for(int i = 0; i < maxn; i++)
            g[i].clear();
        for(int i = 1; i <= n; i++)
        {
            int t = 0;
            scanf("%d: (%d)", &t, &m);
            while(m--)
            {
                scanf("%d", &t);
                t++;
                g[i].push_back(t);
                g[t].push_back(i);
            }
        }
        printf("%d\n", n-Hungary()/2);
    }
    return 0;
}

Girls and Boys

Time Limit: 5000MS		Memory Limit: 10000K
Total Submissions: 12485		Accepted: 5578

Description

In the second year of the university somebody started a study on the romantic relations between the students. The relation "romantically involved" is defined between one girl and one boy. For the study reasons it is necessary to find out the maximum set satisfying the condition: there are no two students in the set who have been "romantically involved". The result of the program is the number of students in such a set.

Input

The input contains several data sets in text format. Each data set represents one set of subjects of the study, with the following description:

the number of students
the description of each student, in the following format
student_identifier:(number_of_romantic_relations) student_identifier1 student_identifier2 student_identifier3 ...
or
student_identifier:(0)

The student_identifier is an integer number between 0 and n-1 (n <=500 ), for n subjects.

Output

For each given data set, the program should write to standard output a line containing the result.

Sample Input

7
0: (3) 4 5 6
1: (2) 4 6
2: (0)
3: (0)
4: (2) 0 1
5: (1) 0
6: (2) 0 1
3
0: (2) 1 2
1: (1) 0
2: (1) 0

Sample Output

5
2

Source