poj1236 强连通缩点

Posted 2020-07-30 sweatOtt

Network of Schools

Time Limit: 1000MS		Memory Limit: 10000K
Total Submissions: 15211		Accepted: 6040

Description

A number of schools are connected to a computer network. Agreements have been developed among those schools: each school maintains a list of schools to which it distributes software (the “receiving schools”). Note that if B is in the distribution list of school A, then A does not necessarily appear in the list of school B 
You are to write a program that computes the minimal number of schools that must receive a copy of the new software in order for the software to reach all schools in the network according to the agreement (Subtask A). As a further task, we want to ensure that by sending the copy of new software to an arbitrary school, this software will reach all schools in the network. To achieve this goal we may have to extend the lists of receivers by new members. Compute the minimal number of extensions that have to be made so that whatever school we send the new software to, it will reach all other schools (Subtask B). One extension means introducing one new member into the list of receivers of one school. 

Input

The first line contains an integer N: the number of schools in the network (2 <= N <= 100). The schools are identified by the first N positive integers. Each of the next N lines describes a list of receivers. The line i+1 contains the identifiers of the receivers of school i. Each list ends with a 0. An empty list contains a 0 alone in the line.

Output

Your program should write two lines to the standard output. The first line should contain one positive integer: the solution of subtask A. The second line should contain the solution of subtask B.

Sample Input

5
2 4 3 0
4 5 0
0
0
1 0

Sample Output

1
2

题意:

N(2<N<100)各学校之间有单向的网络，每个学校得到一套软件后，可以通过单向网络向周边的学校传输，问题1：初始至少需要向多少个学校发放软件，使得网络内所有的学校最终都能得到软件。2，至少需要添加几条传输线路(边)，使任意向一个学校发放软件后，经过若干次传送，网络内所有的学校最终都能得到软件。

给定一个有向图，求：

1) 至少要选几个顶点，才能做到从这些顶点出发，可以到达全部顶点

2) 至少要加多少条边，才能使得从任何一个顶点出发，都能到达全部顶点

思路:

问题1答案就是入度为0的点的个数。

在DAG上要加几条边，才能使得DAG变成强连通的，问题2的答案就是多少

加边的方法：

要为每个入度为0的点添加入边，为每个出度为0的点添加出边

假定有 n 个入度为0的点，m个出度为0的点，如何加边？

把所有入度为0的点编号 0,1,2,3,4 ....N -1

每次为一个编号为i的入度0点可达的出度0点，添加一条出边，连到编号为(i+1)%N 的那个出度0点,

这需要加n条边若 m <= n，则加了这n条边后，已经没有入度0点，则问题解决，一共加了n条边

若 m > n，则还有m-n个入度0点，则从这些点以外任取一点，和这些点都连上边，即可，这还需加m-n条边。所以，max(m,n)就是第二个问题的解。

/*
 * Author:  sweat123
 * Created Time:  2016/6/25 15:49:19
 * File Name: main.cpp
 */
#include<set>
#include<map>
#include<queue>
#include<stack>
#include<cmath>
#include<string>
#include<vector>
#include<cstdio>
#include<time.h>
#include<cstring>
#include<iostream>
#include<algorithm>
#define INF 1<<30
#define MOD 1000000007
#define ll long long
#define lson l,m,rt<<1
#define rson m+1,r,rt<<1|1
#define pi acos(-1.0)
using namespace std;
const int MAXN = 10010;
struct node{
    int from;
    int to;
    int next;  
}edge[MAXN*10];
int pre[MAXN],vis[MAXN],dfn[MAXN],low[MAXN],n,m,ind;
int f[MAXN],siz[MAXN],num,dep,out[MAXN],in[MAXN];
stack<int>s;
void add(int x,int y){
    edge[ind].from = x;
    edge[ind].to = y;
    edge[ind].next = pre[x];
    pre[x] = ind ++;   
}
void dfs(int rt){
    dfn[rt] = low[rt] = ++dep;
    vis[rt] = 1;
    s.push(rt);
    for(int i = pre[rt]; i != -1; i = edge[i].next){
        int t = edge[i].to;
        if(!dfn[t]){
            dfs(t);
            low[rt] = min(low[rt],low[t]);   
        } else if(vis[t]){
            low[rt] = min(low[rt],dfn[t]);   
        }
    }  
    if(low[rt] == dfn[rt]){
        ++num;
        while(!s.empty()){
            int  tp = s.top();
            s.pop();
            vis[tp] = 0;
            f[tp] = num;
            siz[num] ++;
            if(tp == rt)break;
        }   
    }
}
void setcc(){
    num = 0;
    dep = 0;
    memset(in,0,sizeof(in));
    memset(out,0,sizeof(out));
    memset(dfn,0,sizeof(dfn));
    memset(low,0,sizeof(low)); 
    for(int i = 1; i <= n; i++){
        if(!dfn[i]){
            dfs(i);   
        }
    }
    int ret = ind;
    for(int i = 0; i < ret; i++){
        int x = f[edge[i].from];
        int y = f[edge[i].to];
        if(x == y)continue;
        out[x] ++;
        in[y] ++;
    }
    int ans1,ans2;
    ans1 = ans2 = 0;
    for(int i = 1; i <= num; i++){
        if(in[i] == 0){
            ans1 ++;
        } 
        if(out[i] == 0){
            ans2 ++;   
        }
    }
    ans2 = max(ans2,ans1);
    if(num == 1)ans2 = 0;
    printf("%d\n%d\n",ans1,ans2);
}
int main(){
    while(~scanf("%d",&n)){
        ind = 0;
        while(!s.empty())s.pop();
        memset(pre,-1,sizeof(pre));
        memset(vis,0,sizeof(pre));
        memset(f,-1,sizeof(f));
        for(int i = 1; i <= n; i++){
            while(1){
                int x;
                scanf("%d",&x);
                if(x == 0)break;
                add(i,x);
            }
        }
        setcc();
    }
    return 0;   
}