POJ 2762 Going from u to v or from v to u? 图的单连通性 tarjan or kosaraju

虽然本题是求单连通的，但是我们需要先求强连通分量，因为，强连通分量中存在双向路径，因此可以缩点，缩点后就好处理多了。

如果要满足题意，缩点后的树必须是一条链，而且所有边的方向都是一样的，如果出现分支，很容易证明会出现不可到达的一对点。

那么剩下的就是求最长链的顶点数是否等于强连通分量的个数了。

那么就可以使用拓扑排序，或者直接DFS。

拓扑排序中，不能出现同时有两个点的入度为0。

DFS从入度为0的顶点开始搜，是求出深搜的层数，因为单链的话层数是等于顶点数的

下面贴出用Kosaraju算法实现的代码

/*
ID: sdj22251
PROG: subset
LANG: C++
*/
#include <iostream>
#include <vector>
#include <list>
#include <map>
#include <set>
#include <deque>
#include <queue>
#include <stack>
#include <bitset>
#include <algorithm>
#include <functional>
#include <numeric>
#include <utility>
#include <sstream>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <cctype>
#include <string>
#include <cstring>
#include <cmath>
#include <ctime>
#define LOCA
#define MAXN 10005
#define INF 100000000
#define eps 1e-7
using namespace std;
struct Edge
{
int v, next;
}edge[10 * MAXN], revedge[10 * MAXN];
int head[MAXN], revhead[MAXN], e, visited[MAXN];
int order[MAXN], cnt, id[MAXN], tmp;
int uu[5 * MAXN], vv[5 * MAXN], in[MAXN];
int n, m;
void init()
{
e = 0;
memset(head, -1, sizeof(head));
memset(revhead, -1, sizeof(revhead));
memset(in, 0 , sizeof(in));
}
void insert(const int &x, const int &y)
{
edge[e].v = y;
edge[e].next = head[x];
head[x] = e;
revedge[e].v = x;
revedge[e].next = revhead[y];
revhead[y] = e;
e++;
}
void readdata()
{
for(int i = 0; i < m; i++)
{
scanf("%d%d", &uu[i], &vv[i]);
insert(uu[i], vv[i]);
}
}
void dfs(int u)   //搜索原图
{
visited[u] = 1;
for(int i = head[u]; i != -1; i = edge[i].next)
{
int v = edge[i].v;
if(!visited[v])
dfs(v);
}
order[cnt++] = u;
}
void dfs_rev(int u)  //搜索逆图
{
visited[u] = 1;
id[u] = cnt;
for(int i = revhead[u]; i != -1; i = revedge[i].next)
{
int v = revedge[i].v;
if(!visited[v])
dfs_rev(v);
}
}
void search(int u, int deep)  //缩点后的搜索
{
visited[u] = 1;
tmp = max(tmp, deep);
for(int i = head[u]; i != -1; i = edge[i].next)
{
int v = edge[i].v;
if(!visited[v])
{
if(id[u] == id[v])
search(v, deep);
else search(v, deep + 1);
}
}
}
void Kosaraju()
{
init();
readdata();
memset(visited, 0, sizeof(visited));
cnt = 0;
for(int i = 1; i <= n; i++)
{
if(!visited[i])
dfs(i);
}
memset(visited, 0, sizeof(visited));
cnt = 0;
for(int i = n - 1; i >= 0; i--)
{
if(!visited[order[i]])
{
cnt++;
dfs_rev(order[i]);
}
}
for(int i = 0; i < m; i++)
{
int u = id[uu[i]];
int v = id[vv[i]];
if(u != v) in[v]++;
}
tmp = 0;
memset(visited, 0, sizeof(visited));
for(int i = 1; i <= n; i++)
{
if(in[id[i]] == 0)
{
search(i, 1);
break;
}
}
}
int main()
{
int T;
scanf("%d", &T);
while(T--)
{
scanf("%d%d", &n, &m);
Kosaraju();
int ans = 0;
if(tmp != cnt) printf("No\n");
else printf("Yes\n");
}
return 0;
}

tarjan代码

#include <iostream>
#include <map>
#include <cstdio>
#include <stack>
#include <cstring>
#include <algorithm>
#define MAXN 10005
#define MAXM 100005
#define INF 1000000000
using namespace std;
int n, m;
int scc;//强连通分量
int index;//每个节点的dfs访问次序编号
int dfn[MAXN];//标记结点i的dfs访问次序
int low[MAXN];//记录节点u或u的子树中的所有节点的最小标号
int fa[MAXN];//属于哪个分支
bool instack[MAXN];//是否在栈中
int in[MAXN], head[MAXN], e;
int out[MAXN];//出度
stack <int>s;
int tmp;
int vis[MAXN];
struct Edge
{
int v, next;
}edge[MAXM];
void insert(int x, int y)
{
edge[e].v = y;
edge[e].next = head[x];
head[x] = e++;
}
void tarjan(int u)
{
dfn[u] = low[u] = ++index;
s.push(u);
instack[u] = true;
for (int j = head[u]; j != -1; j = edge[j].next)
{
int v = edge[j].v;
if(dfn[v] == 0)//未曾访问过
{
tarjan(v);
low[u] = min(low[u], low[v]);
}
else if(instack[v])
low[u] = min(low[u], dfn[v]);
}
if(dfn[u] == low[u])
{
scc++;
while(1)
{
int tmp = s.top();
s.pop();
instack[tmp] = 0;
fa[tmp] = scc;
if(tmp == u) break;
}
}
}
void init()
{
scc = index = 0;
memset(dfn, 0, sizeof(dfn));
memset(instack, 0, sizeof(instack));
e = 0;
memset(head, -1, sizeof(head));
memset(in, 0, sizeof(in));
memset(out, 0, sizeof(out));

}
void dfs(int v, int deep)
{
tmp = max(tmp, deep);
vis[v] = 1;
for(int i = head[v]; i != -1; i = edge[i].next)
if(!vis[edge[i].v])
{
if(fa[edge[i].v] == fa[v]) dfs(edge[i].v, deep);
else dfs(edge[i].v, deep + 1);
}
}
void solve()
{
for (int i = 1;i <= n; i++)
{
if (!dfn[i])
tarjan(i);
}
for (int i = 1;i <= n; i++)
{
for(int j = head[i]; j != -1; j = edge[j].next)
{
int u = fa[i];
int v = fa[edge[j].v];
if(u != v)
{
out[u]++;
in[v]++;
}
}
}
tmp = 0;
memset(vis, 0, sizeof(vis));
for(int i = 1; i <= n; i++)
if(in[fa[i]] == 0)
{
dfs(i, 1);
break;
}
printf(tmp == scc ? "Yes\n" : "No\n");
}

int main()
{
int T, x, y;
scanf("%d", &T);
while(T--)
{
init();
scanf("%d%d", &n, &m);
while(m--)
{
scanf("%d%d", &x, &y);
insert(x, y);
}
solve();
}
return 0;
}