POJ2186 Popular Cows 【强连通分量】+【Kosaraju】+【Tarjan】+【Garbow】

Description

Input

Output

Sample Input

3 3
1 2
2 1
2 3

Sample Output

1

Hint

Source

题意：能够转换成“给定一些有向路，求有多少个点能够由其余的随意点到达。

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题解：第一道强连通分量的题，大致总结下Kosaraju算法：求强连通分量主要是为了简化图的构造，假设分量外的一个点能到达分量内的当中一个点，那么它必然能到达分量内的全部点，所以某种程度上。强连通分量能够简化成一个点。详细的求解过程是：1、随意选定一个点開始对原图进行深搜，记录每一个点离开时的时间（更确切的说是求每一个时间相应哪个点离开）。2、对原图的反图进行深搜，步骤一中最后离开的点最先開始深搜。每次将同一棵树中的点都哈希成同一个值。最后有多少棵树就有多少个强连通分量。

这题最后全部点都哈希完毕后实际上构成了一个DAG。假设新图中出度为0的点仅仅有一个那么有解，解为该出度为0的强连通分量中原来点的个数。若出度为0的点不止一个，那么无解，由于有两群牛互不崇拜，此时答案为0.在推断连通分量是否有出度时有个小技巧，就是在对反图DFS时若发现连接到的点已訪问且它的哈希值与当前訪问点的哈希值不同。那么这个被连接到的点相应的联通分量是有出度的。然后还需记录每一个连通分量的点数。

#include <stdio.h>
#include <string.h>
#define maxn 10002
#define maxm 50002

int head0[maxn], head1[maxn], id;
int count[maxn], num[maxn], hash[maxn];
struct Node{
    int t0, next0, t1, next1;
} E[maxm];
bool vis[maxn], out[maxn];

void addEdge(int u, int v)
{
    E[id].t0 = v; E[id].next0 = head0[u];
    head0[u] = id; E[id].t1 = u;
    E[id].next1 = head1[v]; head1[v] = id++;
}

void getMap(int n, int m)
{
    int i, u, v; id = 0;
    memset(head0, -1, sizeof(int) * (n + 1)); //save time
    memset(head1, -1, sizeof(int) * (n + 1));
    for(i = 0; i < m; ++i){
        scanf("%d%d", &u, &v);
        addEdge(u, v);
    }
}

void DFS0(int pos, int& sig)
{
    vis[pos] = 1; int i;
    for(i = head0[pos]; i != -1; i = E[i].next0){
        if(!vis[E[i].t0]) DFS0(E[i].t0, sig);
    }
    num[++sig] = pos;
}

void DFS1(int pos, int sig)
{
    vis[pos] = 1; hash[pos] = sig;
    int i; ++count[sig];
    for(i = head1[pos]; i != -1; i = E[i].next1){
        if(!vis[E[i].t1]) DFS1(E[i].t1, sig);
        else if(hash[E[i].t1] != hash[pos]) out[hash[E[i].t1]] = 1;
    }
}

void solve(int n) //Kosaraju
{
    int i, sig = 0, tmp = 0, ans;
    memset(vis, 0, sizeof(bool) * (n + 1));
    for(i = 1; i <= n; ++i)
        if(!vis[i]) DFS0(i, sig);
    memset(vis, 0, sizeof(bool) * (n + 1));
    memset(count, 0, sizeof(int) * (n + 1));
    memset(out, 0, sizeof(bool) * (n + 1));
    i = sig; sig = 0;
    for(; i; --i)
        if(!vis[num[i]]) DFS1(num[i], ++sig);
    for(i = 1; i <= sig; ++i)
        if(!out[i]) ++tmp, ans = count[i];
    //printf("sig%d\n", sig);
    if(tmp == 1) printf("%d\n", ans);
    else printf("0\n");
}

int main()
{
    int n, m;
    while(scanf("%d%d", &n, &m) == 2){
        getMap(n, m);
        solve(n);
    }
    return 0;
}

Tarjan解法：

#include <stdio.h>
#include <string.h>
#define maxn 10002
#define maxm 50002

int head[maxn], vis[maxn], id, id2, scc_num, sec;
int dfn[maxn], low[maxn], sta[maxn], count[maxn];
bool out[maxn];
struct Node{
    int to, next;
} E[maxm];

int min(int a, int b){
    return a < b ?
 a : b;
}

void addEdge(int u, int v)
{
    E[id].to = v; 
    E[id].next = head[u];
    head[u] = id++;
}

void getMap(int n, int m)
{
    int i, u, v; id = 0;
    memset(head, -1, sizeof(int) * (n + 1));
    memset(vis, 0, sizeof(int) * (n + 1));
    memset(out, 0, sizeof(bool) * (n + 1));
    memset(count, 0, sizeof(int) * (n + 1));
    for(i = 0; i < m; ++i){
        scanf("%d%d", &u, &v);
        addEdge(u, v);
    }
}

void DFS(int pos) //强连通分量必然是该树的子树
{
    dfn[pos] = low[pos] = ++sec;
    vis[pos] = 1; sta[id2++] = pos;
    int i, u, v;
    for(i = head[pos]; i != -1; i = E[i].next){
        v = E[i].to;
        if(!vis[v]) DFS(v);
        if(vis[v] == 1)
            low[pos] = min(low[pos], low[v]);
    }
    if(dfn[pos] == low[pos]){
        ++scc_num;
        do{
            ++count[scc_num];
            u = sta[--id2];
            low[u] = scc_num;
            vis[u] = 2;
        } while(u != pos);
    }
}

void solve(int n) //Tarjan
{
    int i, j, ok = 0, ans; sec = id2 = scc_num = 0;
    for(i = 1; i <= n; ++i)
        if(!vis[i]) DFS(i);
    for(i = 1; i <= n; ++i)
        for(j = head[i]; j != -1; j = E[j].next)
            if(low[i] != low[E[j].to]){
                out[low[i]] = 1; break;
            }
    for(i = 1; i <= scc_num; ++i)
        if(!out[i]){
            if(++ok > 1) break;
            ans = count[i];
        }
    if(ok != 1) printf("0\n");
    else printf("%d\n", ans);
}

int main()
{
    int n, m;
    while(scanf("%d%d", &n, &m) == 2){
        getMap(n, m);
        solve(n);
    }
    return 0;
}

Garbow解法：与Tarjan思想同样，仅仅是实现方式略有不同，效率更高一些。

#include <stdio.h>
#include <string.h>
#define maxn 10002
#define maxm 50002
//sta2用以维护当前连通分量的根
int head[maxn], id, sta1[maxn], id1, sta2[maxn], id2;
int low[maxn], scc[maxn], sccNum, sec, count[maxn];
struct Node{
    int to, next;
} E[maxm];
bool out[maxn];

void addEdge(int u, int v)
{
    E[id].to = v; 
    E[id].next = head[u];
    head[u] = id++;
}

void getMap(int n, int m)
{
    int i, u, v; id = 0;
    memset(head, -1, sizeof(int) * (n + 1));
    for(i = 0; i < m; ++i){
        scanf("%d%d", &u, &v);
        addEdge(u, v);
    }
}

void Garbow(int pos)
{
    low[pos] = ++sec;
    sta1[id1++] = sta2[id2++] = pos;
    for(int i = head[pos]; i != -1; i = E[i].next){
        if(!low[E[i].to]) Garbow(E[i].to);
        else if(!scc[E[i].to]){
            while(low[sta2[id2-1]] > low[E[i].to]) --id2;
        }
    }
    if(pos == sta2[id2-1]){        
        int v; ++sccNum; --id2;
        do{
            v = sta1[--id1];
            scc[v] = sccNum;
            ++count[sccNum];
        } while(sta1[id1] != pos);
    }
}

void solve(int n)
{
    int i, j; id1 = id2 = sec = sccNum = 0;
    memset(low, 0, sizeof(int) * (n + 1));
    memset(scc, 0, sizeof(int) * (n + 1));
    memset(count, 0, sizeof(int) * (n + 1));
    memset(out, 0, sizeof(bool) * (n + 1));
    for(i = 1; i <= n; ++i)
        if(!low[i]) Garbow(i);
    for(i = 1; i <= n; ++i)
        for(j = head[i]; j != -1; j = E[j].next)
            if(scc[i] != scc[E[j].to]){
                out[scc[i]] = 1; break;
            }
    int tmp = 0, ans;
    for(i = 1; i <= sccNum; ++i)
        if(!out[i]){
            if(++tmp > 1){
                ans = 0; break;
            }
            ans = count[i];
        }
    printf("%d\n", ans);
}

int main()
{
    int n, m;
    while(scanf("%d%d", &n, &m) == 2){
        getMap(n, m);
        solve(n);
    }
    return 0;
}