Check if the given permutation is a valid DFS of graph
Given a graph with N nodes numbered from 1 to N and M edges and an array of numbers from 1 to N. Check if it is possible to obtain any permutation of array by applying DFS (Depth First Traversal) on given graph.
Prerequisites: DFS | Map in CPP
Examples:
Input: N = 3, M = 2 Edges are: 1) 1-2 2) 2-3 P = {1, 2, 3} Output: YES Explanation: Since there are edges between 1-2 and 2-3, therefore we can have DFS in the order 1-2-3 Input: N = 3, M = 2 Edges are: 1) 1-2 2) 2-3 P = {1, 3, 2} Output: NO Explanation: Since there is no edge between 1 and 3, the DFS traversal is not possible in the order of given permutation.
Approach: We assume that the input graph is represented as adjacency list. The idea is to first sort all adjacency lists according to input order, then traverse the given graph starting from the first node in the given permutation. If we visit all vertices in the same order, then given permutation is a valid DFS.
- Store the indexes of each number in the given permutation in a Hash map.
- Sort every adjacency list according to the indexes of permutation since there is a need to maintain the order.
- Perform the Depth First Traversal Search with source node as 1st number of given permutation.
- Keep a counter variable and at every recursive call, check if the counter has reached the number of nodes, i.e. N and set the flag as 1. If the flag is 0 after complete DFS, answer is ‘NO’ otherwise ‘YES’
Below is the implementation of above approach:
C++
// C++ program to check if given // permutation can be obtained // upon DFS traversal on given graph #include <bits/stdc++.h> using namespace std; // To track of DFS is valid or not. bool flag = false ; // HashMap to store the indexes // of given permutation map< int , int > mp; // Comparator function for sort bool cmp( int a, int b) { // Sort according ascending // order of indexes return mp[a] < mp[b]; } // Graph class represents an undirected // using adjacency list representation class Graph { int V; // No. of vertices int counter; // Counter variable public : // Pointer to an array containing // adjacency lists list< int >* adj; Graph( int V); // Constructor // function to add an edge to graph void addEdge( int u, int v); // DFS traversal of the vertices // reachable from v void DFS( int v, int Perm[]); }; Graph::Graph( int V) { this ->V = V; this ->counter = 0; adj = new list< int >[V + 1]; } void Graph::addEdge( int u, int v) { // Add v to u’s list. adj[u].push_back(v); // Add u to v's list adj[v].push_back(u); } // DFS traversal of the // vertices reachable from v. void Graph::DFS( int v, int Perm[]) { // Increment counter for // every node being traversed counter++; // Check if counter has // reached number of vertices if (counter == V) { // Set flag to 1 flag = 1; return ; } // Recur for all vertices adjacent // to this vertices only if it // lies in the given permutation list< int >::iterator i; for (i = adj[v].begin(); i != adj[v].end(); i++) { // if the current node equals to // current element of permutation if (*i == Perm[counter]) DFS(*i, Perm); } } // Returns true if P[] is a valid DFS of given // graph. In other words P[] can be obtained by // doing a DFS of the graph. bool checkPermutation( int N, int M, vector<pair< int , int > > V, int P[]) { // Create the required graph with // N vertices and M edges Graph G(N); // Add Edges to Graph G for ( int i = 0; i < M; i++) G.addEdge(V[i].first, V[i].second); for ( int i = 0; i < N; i++) mp[P[i]] = i; // Sort every adjacency // list according to HashMap for ( int i = 1; i <= N; i++) G.adj[i].sort(cmp); // Call DFS with source node as P[0] G.DFS(P[0], P); // If Flag has been set to 1, means // given permutation is obtained // by DFS on given graph return flag; } // Driver code int main() { // Number of vertices // and number of edges int N = 3, M = 2; // Vector of pair to store edges vector<pair< int , int > > V; V.push_back(make_pair(1, 2)); V.push_back(make_pair(2, 3)); int P[] = { 1, 2, 3 }; // Return the answer if (checkPermutation(N, M, V, P)) cout << "YES" << endl; else cout << "NO" << endl; return 0; } |
Java
// Java program to check if given // permutation can be obtained // upon DFS traversal on given graph import java.util.ArrayList; import java.util.Comparator; import java.util.HashMap; public class GFG { // To track of DFS is valid or not. static boolean flag = false ; // HashMap to store the indexes // of given permutation static HashMap<Integer, Integer> mp = new HashMap<Integer, Integer>(); // Graph class represents an undirected // using adjacency list representation static class Graph { int V; // No. of vertices int counter; // Counter variable // Pointer to an array containing // adjacency lists ArrayList<Integer> adj[]; @SuppressWarnings ( "unchecked" ) Graph( int V) { // Constructor this .V = V; adj = new ArrayList[V + 1 ]; for ( int i = 0 ; i <= V; i++) adj[i] = new ArrayList<Integer>(); } // function to add an edge to graph void addEdge( int u, int v) { // Add v to u’s list. adj[u].add(v); // Add u to v's list adj[v].add(u); } // DFS traversal of the vertices // reachable from v void DFS( int v, int Perm[]) { // Increment counter for // every node being traversed counter++; // Check if counter has // reached number of vertices if (counter == V) { // Set flag to 1 flag = true ; return ; } // Recur for all vertices adjacent // to this vertices only if it // lies in the given permutation for (Integer i : adj[v]) { // if the current node equals to // current element of permutation if (i == Perm[counter]) DFS(i, Perm); } } } // Returns true if P[] is a valid DFS of given // graph. In other words P[] can be obtained by // doing a DFS of the graph. static boolean checkPermutation( int N, int M, int [][] V, int P[]) { // Create the required graph with // N vertices and M edges Graph G = new Graph(N); // Add Edges to Graph G for ( int i = 0 ; i < M; i++) G.addEdge(V[i][ 0 ], V[i][ 1 ]); for ( int i = 0 ; i < N; i++) mp.put(P[i], i); // Sort every adjacency // list according to HashMap for ( int i = 1 ; i <= N; i++) G.adj[i].sort((a, b) -> a - b); // Call DFS with source node as P[0] G.DFS(P[ 0 ], P); // If Flag has been set to 1, means // given permutation is obtained // by DFS on given graph return flag; } // Driver code public static void main(String[] args) { // Number of vertices // and number of edges int N = 3 , M = 2 ; // Vector of pair to store edges int [][] V = { { 1 , 2 }, { 2 , 3 } }; int P[] = { 1 , 2 , 3 }; // Return the answer if (checkPermutation(N, M, V, P)) System.out.println( "YES" ); else System.out.println( "NO" ); } } // This code is contributed by jainlovely450 |
Python3
# Python3 program to check if given # permutation can be obtained # upon DFS traversal on given graph flag = True def addEdge(adj, u, v): # Add v to u’s list. adj[u].append(v) # Add u to v's list adj[v].append(u) return adj # DFS traversal of the # vertices reachable from v. def DFS(adj, v, Perm): global mp,counter, flag # Increment counter for # every node being traversed counter + = 1 # Check if counter has # reached number of vertices if (counter = = V): # Set flag to 1 flag = 1 return # Recur for all vertices adjacent # to this vertices only if it # lies in the given permutation for i in adj[v]: # If the current node equals to # current element of permutation if (counter< len (Perm) and i = = Perm[counter]): DFS(adj, i, Perm) # Returns true if P[] is a valid DFS of given # graph. In other words P[] can be obtained by # doing a DFS of the graph. def checkPermutation(N, M, V, P): global mp # Create the required graph with # N vertices and M edges G = [[] for i in range (N + 1 )] # Add Edges to Graph G for i in range (M): G = addEdge(G, V[i][ 0 ], V[i][ 1 ]) for i in range (N): mp[P[i]] = i # Sort every adjacency # list according to HashMap for i in range ( 1 , N + 1 ): G[i] = sorted (G[i]) # Call DFS with source node as P[0] DFS(G, P[ 0 ], P) # If Flag has been set to 1, means # given permutation is obtained # by DFS on given graph return flag # Driver code if __name__ = = '__main__' : mp = {} # Number of vertices # and number of edges N, M, counter = 3 , 2 , 0 # Vector of pair to store edges V = [] V.append([ 1 , 2 ]) V.append([ 2 , 3 ]) P = [ 1 , 2 , 3 ] # Return the answer if (checkPermutation(N, M, V, P)): print ( "YES" ) else : print ( "NO" ) # This code is contributed by mohit kumar 29 |
Javascript
<script> // JavaScript program to check if given // permutation can be obtained // upon DFS traversal on given graph let flag = true function addEdge(adj, u, v){ // Add v to u’s list. adj[u].push(v) // Add u to v's list adj[v].push(u) return adj } // DFS traversal of the // vertices reachable from v. function DFS(adj, v, Perm){ // Increment counter for // every node being traversed counter += 1 // Check if counter has // reached number of vertices if (counter == V){ // Set flag to 1 flag = 1 return } // Recur for all vertices adjacent // to this vertices only if it // lies in the given permutation for (let i of adj[v]){ // If the current node equals to // current element of permutation if (counter<Perm.length && i == Perm[counter]) DFS(adj, i, Perm) } } // Returns true if P[] is a valid DFS of given // graph. In other words P[] can be obtained by // doing a DFS of the graph. function checkPermutation(N, M, V, P){ // Create the required graph with // N vertices and M edges let G = new Array(N+1).fill(0).map(()=> new Array()) // Add Edges to Graph G for (let i=0;i<M;i++) G = addEdge(G, V[i][0], V[i][1]) for (let i=0;i<N;i++) mp.set(P[i] , i) // Sort every adjacency // list according to HashMap for (let i=1;i< N + 1;i++){ G[i].sort() } // Call DFS with source node as P[0] DFS(G, P[0], P) // If Flag has been set to 1, means // given permutation is obtained // by DFS on given graph return flag } // Driver code let mp = new Map() // Number of vertices // and number of edges let N = 3,M = 2,counter = 0 // Vector of pair to store edges let V = [] V.push([1, 2]) V.push([2, 3]) let P = [1, 2, 3] // Return the answer if (checkPermutation(N, M, V, P)) document.write( "YES" , "</br>" ) else document.write( "NO" , "</br>" ) // This code is contributed by shinjanpatra </script> |
Output:
YES