本文给出了C++程序和Python程序。
tarjan算法是由Robert Tarjan提出的求解有向图强连通分量的线性时间的算法。
程序来源：Tarjan’s Algorithm to find Strongly Connected Components。
百度百科：tarjan算法。维基百科：Tarjan's strongly connected components algorithm。
C++程序：
// A C++ program to find strongly connected components in a given
// directed graph using Tarjan's algorithm (single DFS)
#include<iostream>
#include <list>
#include <stack>
#define NIL -1
using namespace std;
// A class that represents an directed graph
class Graph
{
int V; // No. of vertices
list<int> *adj; // A dynamic array of adjacency lists
// A Recursive DFS based function used by SCC()
void SCCUtil(int u, int disc[], int low[],
stack<int> *st, bool stackMember[]);
public:
Graph(int V); // Constructor
void addEdge(int v, int w); // function to add an edge to graph
void SCC(); // prints strongly connected components
};
Graph::Graph(int V)
{
this->V = V;
adj = new list<int>[V];
}
void Graph::addEdge(int v, int w)
{
adj[v].push_back(w);
}
// A recursive function that finds and prints strongly connected
// components using DFS traversal
// u --> The vertex to be visited next
// disc[] --> Stores discovery times of visited vertices
// low[] -- >> earliest visited vertex (the vertex with minimum
// discovery time) that can be reached from subtree
// rooted with current vertex
// *st -- >> To store all the connected ancestors (could be part
// of SCC)
// stackMember[] --> bit/index array for faster check whether
// a node is in stack
void Graph::SCCUtil(int u, int disc[], int low[], stack<int> *st,
bool stackMember[])
{
// A static variable is used for simplicity, we can avoid use
// of static variable by passing a pointer.
static int time = 0;
// Initialize discovery time and low value
disc[u] = low[u] = ++time;
st->push(u);
stackMember[u] = true;
// Go through all vertices adjacent to this
list<int>::iterator i;
for (i = adj[u].begin(); i != adj[u].end(); ++i)
{
int v = *i; // v is current adjacent of 'u'
// If v is not visited yet, then recur for it
if (disc[v] == -1)
{
SCCUtil(v, disc, low, st, stackMember);
// Check if the subtree rooted with 'v' has a
// connection to one of the ancestors of 'u'
// Case 1 (per above discussion on Disc and Low value)
low[u] = min(low[u], low[v]);
}
// Update low value of 'u' only of 'v' is still in stack
// (i.e. it's a back edge, not cross edge).
// Case 2 (per above discussion on Disc and Low value)
else if (stackMember[v] == true)
low[u] = min(low[u], disc[v]);
}
// head node found, pop the stack and print an SCC
int w = 0; // To store stack extracted vertices
if (low[u] == disc[u])
{
while (st->top() != u)
{
w = (int) st->top();
cout << w << " ";
stackMember[w] = false;
st->pop();
}
w = (int) st->top();
cout << w << "
";
stackMember[w] = false;
st->pop();
}
}
// The function to do DFS traversal. It uses SCCUtil()
void Graph::SCC()
{
int *disc = new int[V];
int *low = new int[V];
bool *stackMember = new bool[V];
stack<int> *st = new stack<int>();
// Initialize disc and low, and stackMember arrays
for (int i = 0; i < V; i++)
{
disc[i] = NIL;
low[i] = NIL;
stackMember[i] = false;
}
// Call the recursive helper function to find strongly
// connected components in DFS tree with vertex 'i'
for (int i = 0; i < V; i++)
if (disc[i] == NIL)
SCCUtil(i, disc, low, st, stackMember);
}
// Driver program to test above function
int main()
{
cout << "
SCCs in first graph
";
Graph g1(5);
g1.addEdge(1, 0);
g1.addEdge(0, 2);
g1.addEdge(2, 1);
g1.addEdge(0, 3);
g1.addEdge(3, 4);
g1.SCC();
cout << "
SCCs in second graph
";
Graph g2(4);
g2.addEdge(0, 1);
g2.addEdge(1, 2);
g2.addEdge(2, 3);
g2.SCC();
cout << "
SCCs in third graph
";
Graph g3(7);
g3.addEdge(0, 1);
g3.addEdge(1, 2);
g3.addEdge(2, 0);
g3.addEdge(1, 3);
g3.addEdge(1, 4);
g3.addEdge(1, 6);
g3.addEdge(3, 5);
g3.addEdge(4, 5);
g3.SCC();
cout << "
SCCs in fourth graph
";
Graph g4(11);
g4.addEdge(0,1);g4.addEdge(0,3);
g4.addEdge(1,2);g4.addEdge(1,4);
g4.addEdge(2,0);g4.addEdge(2,6);
g4.addEdge(3,2);
g4.addEdge(4,5);g4.addEdge(4,6);
g4.addEdge(5,6);g4.addEdge(5,7);g4.addEdge(5,8);g4.addEdge(5,9);
g4.addEdge(6,4);
g4.addEdge(7,9);
g4.addEdge(8,9);
g4.addEdge(9,8);
g4.SCC();
cout << "
SCCs in fifth graph
";
Graph g5(5);
g5.addEdge(0,1);
g5.addEdge(1,2);
g5.addEdge(2,3);
g5.addEdge(2,4);
g5.addEdge(3,0);
g5.addEdge(4,2);
g5.SCC();
return 0;
}程序运行输出：
SCCs in first graph 4 3 1 2 0 SCCs in second graph 3 2 1 0 SCCs in third graph 5 3 4 6 2 1 0 SCCs in fourth graph 8 9 7 5 4 6 3 2 1 0 10 SCCs in fifth graph 4 3 2 1 0
Python程序：
# Python program to find strongly connected components in a given
# directed graph using Tarjan's algorithm (single DFS)
#Complexity : O(V+E)
from collections import defaultdict
#This class represents an directed graph
# using adjacency list representation
class Graph:
def __init__(self,vertices):
#No. of vertices
self.V= vertices
# default dictionary to store graph
self.graph = defaultdict(list)
self.Time = 0
# function to add an edge to graph
def addEdge(self,u,v):
self.graph[u].append(v)
'''A recursive function that find finds and prints strongly connected
components using DFS traversal
u --> The vertex to be visited next
disc[] --> Stores discovery times of visited vertices
low[] -- >> earliest visited vertex (the vertex with minimum
discovery time) that can be reached from subtree
rooted with current vertex
st -- >> To store all the connected ancestors (could be part
of SCC)
stackMember[] --> bit/index array for faster check whether
a node is in stack
'''
def SCCUtil(self,u, low, disc, stackMember, st):
# Initialize discovery time and low value
disc[u] = self.Time
low[u] = self.Time
self.Time += 1
stackMember[u] = True
st.append(u)
# Go through all vertices adjacent to this
for v in self.graph[u]:
# If v is not visited yet, then recur for it
if disc[v] == -1 :
self.SCCUtil(v, low, disc, stackMember, st)
# Check if the subtree rooted with v has a connection to
# one of the ancestors of u
# Case 1 (per above discussion on Disc and Low value)
low[u] = min(low[u], low[v])
elif stackMember[v] == True:
'''Update low value of 'u' only if 'v' is still in stack
(i.e. it's a back edge, not cross edge).
Case 2 (per above discussion on Disc and Low value) '''
low[u] = min(low[u], disc[v])
# head node found, pop the stack and print an SCC
w = -1 #To store stack extracted vertices
if low[u] == disc[u]:
while w != u:
w = st.pop()
print w,
stackMember[w] = False
print""
#The function to do DFS traversal.
# It uses recursive SCCUtil()
def SCC(self):
# Mark all the vertices as not visited
# and Initialize parent and visited,
# and ap(articulation point) arrays
disc = [-1] * (self.V)
low = [-1] * (self.V)
stackMember = [False] * (self.V)
st =[]
# Call the recursive helper function
# to find articulation points
# in DFS tree rooted with vertex 'i'
for i in range(self.V):
if disc[i] == -1:
self.SCCUtil(i, low, disc, stackMember, st)
# Create a graph given in the above diagram
g1 = Graph(5)
g1.addEdge(1, 0)
g1.addEdge(0, 2)
g1.addEdge(2, 1)
g1.addEdge(0, 3)
g1.addEdge(3, 4)
print "SSC in first graph "
g1.SCC()
g2 = Graph(4)
g2.addEdge(0, 1)
g2.addEdge(1, 2)
g2.addEdge(2, 3)
print "
SSC in second graph "
g2.SCC()
g3 = Graph(7)
g3.addEdge(0, 1)
g3.addEdge(1, 2)
g3.addEdge(2, 0)
g3.addEdge(1, 3)
g3.addEdge(1, 4)
g3.addEdge(1, 6)
g3.addEdge(3, 5)
g3.addEdge(4, 5)
print "
SSC in third graph "
g3.SCC()
g4 = Graph(11)
g4.addEdge(0, 1)
g4.addEdge(0, 3)
g4.addEdge(1, 2)
g4.addEdge(1, 4)
g4.addEdge(2, 0)
g4.addEdge(2, 6)
g4.addEdge(3, 2)
g4.addEdge(4, 5)
g4.addEdge(4, 6)
g4.addEdge(5, 6)
g4.addEdge(5, 7)
g4.addEdge(5, 8)
g4.addEdge(5, 9)
g4.addEdge(6, 4)
g4.addEdge(7, 9)
g4.addEdge(8, 9)
g4.addEdge(9, 8)
print "
SSC in fourth graph "
g4.SCC();
g5 = Graph (5)
g5.addEdge(0, 1)
g5.addEdge(1, 2)
g5.addEdge(2, 3)
g5.addEdge(2, 4)
g5.addEdge(3, 0)
g5.addEdge(4, 2)
print "
SSC in fifth graph "
g5.SCC();
#This code is contributed by Neelam Yadav