Given a directed graph, find out if a vertex v is reachable from another vertex u for all vertex pairs (u, v) in the given graph. Here reachable means that there is a path from vertex u to v. The reach-ability matrix is called transitive closure of a graph.
For example, consider below graph:
Graph
Transitive closure of above graphs is
1 1 1 1
1 1 1 1
1 1 1 1
0 0 0 1
.png)
We have discussed an O(V3) solution for this here. The solution was based on Floyd Warshall Algorithm. In this post, DFS solution is discussed. So for dense graph, it would become O(V3) and for sparse graph, it would become O(V2).
Below are the abstract steps of the algorithm.
- Create a matrix tc[V][V] that would finally have transitive closure of the given graph. Initialize all entries of tc[][] as 0.
- Call DFS for every node of the graph to mark reachable vertices in tc[][]. In recursive calls to DFS, we don't call DFS for an adjacent vertex if it is already marked as reachable in tc[][].
- Below is the implementation of the above idea. The code uses adjacency list representation of input graph and builds a matrix tc[V][V] such that tc[u][v] would be true if v is reachable from u.
Implementation:
C++
// C++ program to print transitive closure of a graph
#include <bits/stdc++.h>
using namespace std;
class Graph {
int V; // No. of vertices
bool** tc; // To store transitive closure
list<int>* adj; // array of adjacency lists
void DFSUtil(int u, int v);
public:
Graph(int V); // Constructor
// function to add an edge to graph
void addEdge(int v, int w) { adj[v].push_back(w); }
// prints transitive closure matrix
void transitiveClosure();
};
Graph::Graph(int V)
{
this->V = V;
adj = new list<int>[V];
tc = new bool*[V];
for (int i = 0; i < V; i++) {
tc[i] = new bool[V];
memset(tc[i], false, V * sizeof(bool));
}
}
// A recursive DFS traversal function that finds
// all reachable vertices for s.
void Graph::DFSUtil(int s, int v) {
// Mark reachability from s to v as true.
tc[s][v] = true;
// Explore all vertices adjacent to v
for (int u : adj[v]) {
// If s is not yet connected to u, explore further
if (!tc[s][u]) {
DFSUtil(s, u);
}
}
}
// The function to find transitive closure. It uses
// recursive DFSUtil()
void Graph::transitiveClosure()
{
// Call the recursive helper function to print DFS
// traversal starting from all vertices one by one
for (int i = 0; i < V; i++)
DFSUtil(i,
i); // Every vertex is reachable from self.
for (int i = 0; i < V; i++) {
for (int j = 0; j < V; j++)
cout << tc[i][j] << " ";
cout << endl;
}
}
// Driver code
int main()
{
// Create a graph given in the above diagram
Graph g(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
cout << "Transitive closure matrix is \n";
g.transitiveClosure();
return 0;
}
Java
// JAVA program to print transitive
// closure of a graph.
import java.util.ArrayList;
import java.util.Arrays;
// A directed graph using
// adjacency list representation
public class Graph {
// No. of vertices in graph
private int vertices;
// adjacency list
private ArrayList<Integer>[] adjList;
// To store transitive closure
private int[][] tc;
// Constructor
public Graph(int vertices) {
// initialise vertex count
this.vertices = vertices;
this.tc = new int[this.vertices][this.vertices];
// initialise adjacency list
initAdjList();
}
// utility method to initialise adjacency list
@SuppressWarnings("unchecked")
private void initAdjList() {
adjList = new ArrayList[vertices];
for (int i = 0; i < vertices; i++) {
adjList[i] = new ArrayList<>();
}
}
// add edge from u to v
public void addEdge(int u, int v) {
// Add v to u's list.
adjList[u].add(v);
}
// The function to find transitive
// closure. It uses
// recursive DFSUtil()
public void transitiveClosure() {
// Call the recursive helper
// function to print DFS
// traversal starting from all
// vertices one by one
for (int i = 0; i < vertices; i++) {
dfsUtil(i, i);
}
for (int i = 0; i < vertices; i++) {
System.out.println(Arrays.toString(tc[i]));
}
}
// A recursive DFS traversal
// function that finds
// all reachable vertices for s
private void dfsUtil(int s, int v) {
// Mark reachability from
// s to v as true.
if(s==v){
tc[s][v] = 1;
}
else
tc[s][v] = 1;
// Find all the vertices reachable
// through v
for (int adj : adjList[v]) {
if (tc[s][adj]==0) {
dfsUtil(s, adj);
}
}
}
// Driver Code
public static void main(String[] args) {
// Create a graph given
// in the above diagram
Graph g = new Graph(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
System.out.println("Transitive closure " +
"matrix is");
g.transitiveClosure();
}
}
// This code is contributed
// by Himanshu Shekhar
C#
// C# program to print transitive
// closure of a graph.
using System;
using System.Collections.Generic;
// A directed graph using
// adjacency list representation
public class Graph {
// No. of vertices in graph
private int vertices;
// adjacency list
private List<int>[] adjList;
// To store transitive closure
private int[, ] tc;
// Constructor
public Graph(int vertices)
{
// initialise vertex count
this.vertices = vertices;
this.tc = new int[this.vertices, this.vertices];
// initialise adjacency list
initAdjList();
}
// utility method to initialise adjacency list
private void initAdjList()
{
adjList = new List<int>[ vertices ];
for (int i = 0; i < vertices; i++) {
adjList[i] = new List<int>();
}
}
// add edge from u to v
public void addEdge(int u, int v)
{
// Add v to u's list.
adjList[u].Add(v);
}
// The function to find transitive
// closure. It uses
// recursive DFSUtil()
public void transitiveClosure()
{
// Call the recursive helper
// function to print DFS
// traversal starting from all
// vertices one by one
for (int i = 0; i < vertices; i++) {
dfsUtil(i, i);
}
for (int i = 0; i < vertices; i++) {
for (int j = 0; j < vertices; j++)
Console.Write(tc[i, j] + " ");
Console.WriteLine();
}
}
// A recursive DFS traversal
// function that finds
// all reachable vertices for s
private void dfsUtil(int s, int v)
{
// Mark reachability from
// s to v as true.
tc[s, v] = 1;
// Find all the vertices reachable
// through v
foreach(int adj in adjList[v])
{
if (tc[s, adj] == 0) {
dfsUtil(s, adj);
}
}
}
// Driver Code
public static void Main(String[] args)
{
// Create a graph given
// in the above diagram
Graph g = new Graph(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
Console.WriteLine("Transitive closure "
+ "matrix is");
g.transitiveClosure();
}
}
// This code is contributed by Rajput-Ji
Javascript
<script>
/* Javascript program to print transitive
closure of a graph*/
class Graph
{
// Constructor
constructor(v)
{
this.V = v;
this.adj = new Array(v);
this.tc = Array.from(Array(v), () => new Array(v).fill(0));
for(let i = 0; i < v; i++)
this.adj[i] = [];
}
// function to add an edge to graph
addEdge(v, w)
{
this.adj[v].push(w);
}
// A recursive DFS traversal function that finds
// all reachable vertices for s.
DFSUtil(s, v)
{
// Mark reachability from s to v as true.
this.tc[s][v] = 1;
// Find all the vertices reachable through v
for(let i of this.adj[v].values())
{
if(this.tc[s][i] == 0)
this.DFSUtil(s, i);
}
}
// The function to find transitive closure. It uses
// recursive DFSUtil()
transitiveClosure()
{
// Call the recursive helper function to print DFS
// traversal starting from all vertices one by one
for(let i = 0; i < this.V; i++)
this.DFSUtil(i, i); // Every vertex is reachable from self
document.write("Transitive closure matrix is<br />")
for(let i=0; i < this.V; i++)
{
for(let j=0; j < this.V; j++)
document.write(this.tc[i][j] + " ");
document.write("<br />")
}
}
};
// driver code
g = new Graph(4);
g.addEdge(0, 1);
g.addEdge(0, 2);
g.addEdge(1, 2);
g.addEdge(2, 0);
g.addEdge(2, 3);
g.addEdge(3, 3);
g.transitiveClosure();
// This code is contributed by cavi4762.
</script>
Python3
# Python program to print transitive
# closure of a graph.
from collections import defaultdict
class Graph:
def __init__(self,vertices):
# No. of vertices
self.V = vertices
# default dictionary to store graph
self.graph = defaultdict(list)
# To store transitive closure
self.tc = [[0 for j in range(self.V)] for i in range(self.V)]
# function to add an edge to graph
def addEdge(self, u, v):
self.graph[u].append(v)
# A recursive DFS traversal function that finds
# all reachable vertices for s
def DFSUtil(self, s, v):
# Mark reachability from s to v as true.
if(s == v):
if( v in self.graph[s]):
self.tc[s][v] = 1
else:
self.tc[s][v] = 1
# Find all the vertices reachable through v
for i in self.graph[v]:
if self.tc[s][i] == 0:
if s==i:
self.tc[s][i]=1
else:
self.DFSUtil(s, i)
# The function to find transitive closure. It uses
# recursive DFSUtil()
def transitiveClosure(self):
# Call the recursive helper function to print DFS
# traversal starting from all vertices one by one
for i in range(self.V):
self.DFSUtil(i, i)
print(self.tc)
# Create a graph given in the above diagram
g = Graph(4)
g.addEdge(0, 1)
g.addEdge(0, 2)
g.addEdge(1, 2)
g.addEdge(2, 0)
g.addEdge(2, 3)
g.addEdge(3, 3)
g.transitiveClosure()
Output
Transitive closure matrix is 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1
Time Complexity : O(V^3) where V is the number of vertexes . For dense graph, it would become O(V3) and for sparse graph, it would become O(V2).
Auxiliary Space: O(V^2) where V is number of vertices.
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