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Tarjan's Algorithm for Strongly Connected Components
Tarjan’s Algorithm is used to find strongly connected components of a directed graph. It requires only one DFS traversal to implement this algorithm.
Using DFS traversal we can find DFS tree of the forest. From the DFS tree, strongly connected components are found. When the root of such sub-tree is found we can display the whole subtree. That subtree is one strongly connected component.
Input and Output
Input: Adjacency matrix of the graph. 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 Output: The strongly connected components: 4 3 1 2 0
Algorithm
findComponent(u, disc, low, stack, stackItemFlag)
Input: The start node, discovery time, low, the disc will hold the discovery time of the vertex, and low will hold information about subtrees. The stack to hold vertices and another flag array to track which node is in the stack.
Output: Display the SCC.
Begin time := 0 //the value of time will not be initialized for next function calls set disc[u] := time+1 and low[u] := time + 1 time := time + 1 push u into stack stackItemFalg[u] := true for all vertex v which is adjacent with u, do if v is not discovered, then fincComponent(v, disc, low, stack, stackItemFalg) low[u] = minimum of low[u] and low[v] else if stackItemFalg[v] is true, then low[u] := minimum of low[u] and disc[v] done poppedItem := 0 if low[u] = disc[u], then while u is not in the stack top, do poppedItem := top of stack display poppedItem stackItemFlag[poppedItem] := false pop item from stack done poppedItem := top of stack display poppedItem stackItemFlag[poppedItem] := false pop item from stack End
strongConComponent(graph)
Input &,minus; The given Graph.
Output − All the strongly connected components.
Begin initially set all items in the disc array to undiscovered for all elements in low to φ and mark no item is stored into the stack for all node i in the graph, do if disc[i] is undiscovered, then findComponent(i, disc, low, stack, stackItemFlag) End
Example
#include<iostream>
#include<stack>
#define NODE 5
using namespace std;
int graph[NODE][NODE] = {
{0, 0, 1, 1, 0},
{1, 0, 0, 0, 0},
{0, 1, 0, 0, 0},
{0, 0, 0, 0, 1},
{0, 0, 0, 0, 0}
};
int min(int a, int b) {
return (a<b)?a:b;
}
void findComponent(int u, int disc[], int low[], stack<int>&stk, bool stkItem[]) {
static int time = 0;
disc[u] = low[u] = ++time; //inilially discovery time and low value is 1
stk.push(u);
stkItem[u] = true; //flag as u in the stack
for(int v = 0; v<NODE; v++) {
if(graph[u][v]) {
if(disc[v] == -1) { //when v is not visited
findComponent(v, disc, low, stk, stkItem);
low[u] = min(low[u], low[v]);
} else if(stkItem[v]) //when v is in the stack, update low for u
low[u] = min(low[u], disc[v]);
}
}
int poppedItem = 0;
if(low[u] == disc[u]) {
while(stk.top() != u) {
poppedItem = stk.top();
cout << poppedItem << " ";
stkItem[poppedItem] = false; //mark as item is popped
stk.pop();
}
poppedItem = stk.top();
cout << poppedItem <<endl;
stkItem[poppedItem] = false;
stk.pop();
}
}
void strongConComponent() {
int disc[NODE], low[NODE];
bool stkItem[NODE];
stack<int> stk;
for(int i = 0; i<NODE; i++) { //initialize all elements
disc[i] = low[i] = -1;
stkItem[i] = false;
}
for(int i = 0; i<NODE; i++) //initialize all elements
if(disc[i] == -1)
findComponent(i, disc, low, stk, stkItem);
}
int main() {
strongConComponent();
}Output
4 3 1 2 0
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