（绘图什么真辛苦）
强连通分量：
在有向图 G 中。若两个顶点相互可达，则称两个顶点强连通(strongly
connected)。
假设有向图G的每两个顶点都强连通，称G是一个强连通图。非强连通图有向图的极大强连通子图。称为强连通分量(strongly
connected components)。
比方上面这幅图(
a, b, e ), ( d, c, h ), ( f, g ) 分别为三个 SCC。
tarjan算法伪代码：
该算法由 Robert
Tarjan 发明，原论文：Tarjan1972
时间复杂度是深搜的时间复杂度 O( N
+ E )。
BEGIN
INTERGER i;
PROCEDURE STRONGCONNECT(v);
BEGIN
LOWLINK(v):= NUMBER(v):= i := i + 1;
put v on stack of points;
FOR w in the adjacency list of v DO
BEGIN
IF w is not yet numbered THEN
BEGIN comment( v, w ) is a tree arc;
STRONGCONNECT(w);
LOWLINK(V) := min( LOWLINK(V),
LOWLINK(W));
END;
ELSE IF NUMBER(W) < NUMBER(V) DO
BEGIN comment( v, w ) is a frond or cross-link;
if w is on stack of points THEN
LOWLINK(v) := min( LOWLINK(v),
NUMBER(w));
END;
END;
if( LOWLINK(v) = NUMBER(v) ) THEN
BEGIN comment v is the root of a compont;
start new strongly connected compont;
WHILE w on top of point stack satisfies
NUMBER(w) >= NUMBER(v) DO
BEGIN
delete w from point stack and put w
in current component;
END;
END;
END;
i := 0;
empty stack of points;
FOR w a vertex IF w is not yet numbered THEN STRONGCONNECT(w);
END
tarjan算法的运行动态图：
1.建图。每一个顶点有三个域。第一个是顶点名。第二个空格是发现该点的时间戳 DFN ，第三个空格是该点可以追溯到的最早的栈中节点的次序号
LOW。
2. 深搜，比方搜索路径为 a --> b --> c --> g --> f。 沿途记下自身的 DFN 和 LOW
3.达到 f 点后，f 点仅仅有一个可达顶点 g， 但 g 不是 f 的后继顶点。且 g 在栈中，则更新 f 的 LOW 变为 g 的发现时间。
4.这时候回到 g 点。可是 f 点还得再栈中，仅仅有发现时间 DFN 与 LOW 同样的顶点才干从栈中弹出（和压在上面的节点一起弹出构成SCC）
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5.这时候 g 点的 DFN == LOW
watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvcGFuZG9yYV9tYWRhcmE=/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/SouthEast" width="347" height="250" >
6.于是将 f 和 g 都弹出
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7.如虚线所看到的，他们构成了一个SCC
8.以下就是一样的了，（ 图画的好辛苦 ）
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python代码：
def strongly_connected_components( graph ):
dfn_count = [0]
result = []
stack = []
low = {}
dfn = {}
def stronglyconnected( node ):
dfn[node] = dfn_count[0]
low[node] = dfn_count[0]
dfn_count[0] += 1
stack.append( node )
if node not in graph:
successors = []
else:
successors = graph[node]
for successor in successors:
if successor not in dfn:
stronglyconnected( successor )
low[node] = min( low[successor], low[node] )
elif successor in stack:
low[node] = min( low[node], dfn[successor] )
if low[node] == dfn[node]:
li = []
item = None
while True:
item = stack.pop()
li.append( item )
if item == node: break
result.append( li )
for node in graph:
if node not in low:
stronglyconnected( node )
return result
if __name__ == '__main__':
graph = {
'a': [ 'b' ],
'b': [ 'c', 'e', 'f' ],
'c': [ 'g', 'd' ],
'd': [ 'c', 'h' ],
'e': [ 'a', 'f' ],
'f': [ 'g' ],
'g': [ 'f' ],
'h': [ 'g', 'd' ]
}
print strongly_connected_components( graph )
执行结果：
C++代码：
#include <iostream>
#include <vector>
#include <string.h>
using namespace std;
#define MAX_SIZE 100
bool Graph[MAX_SIZE][MAX_SIZE];
int Stack[MAX_SIZE];
bool nodeIsInStack[MAX_SIZE];
int stack_pointer = 0;
int Lows[MAX_SIZE];
int Dfns[MAX_SIZE];
int node_num = 0; // 图中节点得到个数
int find_time = 0; // 每一个节点的发现时间
int scc_num = 0; // 记录 scc 的个数
void find_scc( int start_node ){
find_time++;
Lows[start_node] = Dfns[start_node] = find_time;
stack_pointer++;
Stack[stack_pointer] = start_node;
nodeIsInStack[start_node] = true;
for( int end_node = 1; end_node <= node_num; ++end_node ){
//若start_node和end_ndoe之间存在边
if( Graph[start_node][end_node] ){
//若是end_node尚未被訪问
if( Dfns[end_node] == 0 ){
find_scc( end_node );
Lows[start_node] = min( Lows[start_node], Lows[end_node] );
}
//若end_node在栈中。也就是start_node -> end_node是返祖边
else if( nodeIsInStack[end_node] ){
Lows[start_node] = min( Lows[start_node], Dfns[end_node] );
}
}
}
//若是start_node的时间戳与Lows相等在构成SCC
if( Dfns[start_node] == Lows[start_node] ){
scc_num++;
cout << "scc_num: " << scc_num << endl;
int pop_node_index = Stack[stack_pointer];
stack_pointer--;
nodeIsInStack[pop_node_index] = false;
cout << pop_node_index << " ";
while( start_node != pop_node_index ){
pop_node_index = Stack[stack_pointer];
nodeIsInStack[pop_node_index] = false;
stack_pointer--;
cout << pop_node_index << " ";
}
cout << endl;
}
}
void init_values(){
memset( Graph, false, sizeof( Graph ) );
memset( Stack, 0, sizeof( Stack ) );
memset( nodeIsInStack, false, sizeof( nodeIsInStack ) );
memset( Lows, 0, sizeof( Lows ) );
memset( Dfns, 0, sizeof( Dfns ) );
}
int main(){
init_values();
// 初始化图
//这里用数字取代节点的名字
int start_node, end_node;
cin >> node_num;
while( true ){
cin >> start_node >> end_node;
//起始点终止点都为 0 的时候结束
if( start_node == 0 && end_node == 0 )
break;
Graph[start_node][end_node] = true;
}
for( int start_node = 1; start_node <= node_num; ++start_node ){
//该节点尚未被訪问到
if( Dfns[start_node] == 0 ){
find_scc( start_node );
}
}
}