Paths to travel each nodes using each edge (Seven Bridges of Königsberg)
There are n nodes and m bridges in between these nodes. Print the possible path through each node using each edges (if possible), traveling through each edges only once.
Examples :
Input : [[0, 1, 0, 0, 1],
[1, 0, 1, 1, 0],
[0, 1, 0, 1, 0],
[0, 1, 1, 0, 0],
[1, 0, 0, 0, 0]]
Output : 5 -> 1 -> 2 -> 4 -> 3 -> 2
Input : [[0, 1, 0, 1, 1],
[1, 0, 1, 0, 1],
[0, 1, 0, 1, 1],
[1, 1, 1, 0, 0],
[1, 0, 1, 0, 0]]
Output : "No Solution"It is one of the famous problems in Graph Theory and known as problem of “Seven Bridges of Königsberg”. This problem was solved by famous mathematician Leonhard Euler in 1735. This problem is also considered as the beginning of Graph Theory.
The problem back then was that: There was 7 bridges connecting 4 lands around the city of Königsberg in Prussia. Was there any way to start from any of the land and go through each of the bridges once and only once? Please see these wikipedia images for more clarity.
Euler first introduced graph theory to solve this problem. He considered each of the lands as a node of a graph and each bridge in between as an edge in between. Now he calculated if there is any Eulerian Path in that graph. If there is an Eulerian path then there is a solution otherwise not.
Problem here, is a generalized version of the problem in 1735.
Below is the implementation :
C++
// A C++ program print Eulerian Trail in a// given Eulerian or Semi-Eulerian Graph#include <iostream>#include <string.h>#include <algorithm>#include <list>using namespace std;// A class that represents an undirected graphclass Graph{// No. of vertices int V; // A dynamic array of adjacency lists list<int> *adj;public: // Constructor and destructor Graph(int V) { this->V = V; adj = new list<int>[V]; } ~Graph() { delete [] adj; } // functions to add and remove edge void addEdge(int u, int v) { adj[u].push_back(v); adj[v].push_back(u); } void rmvEdge(int u, int v); // Methods to print Eulerian tour void printEulerTour(); void printEulerUtil(int s); // This function returns count of vertices // reachable from v. It does DFS int DFSCount(int v, bool visited[]); // Utility function to check if edge u-v // is a valid next edge in Eulerian trail or circuit bool isValidNextEdge(int u, int v);};/* The main function that print Eulerian Trail.It first finds an odd degree vertex (if there is any)and then calls printEulerUtil() to print the path */void Graph::printEulerTour(){ // Find a vertex with odd degree int u = 0; for (int i = 0; i < V; i++) if (adj[i].size() & 1) { u = i; break; } // Print tour starting from oddv printEulerUtil(u); cout << endl;}// Print Euler tour starting from vertex uvoid Graph::printEulerUtil(int u){ // Recur for all the vertices adjacent to // this vertex list<int>::iterator i; for (i = adj[u].begin(); i != adj[u].end(); ++i) { int v = *i; // If edge u-v is not removed and it's a a // valid next edge if (v != -1 && isValidNextEdge(u, v)) { cout << u << "-" << v << " "; rmvEdge(u, v); printEulerUtil(v); } }}// The function to check if edge u-v can be considered// as next edge in Euler Toutbool Graph::isValidNextEdge(int u, int v){ // The edge u-v is valid in one of the following // two cases: // 1) If v is the only adjacent vertex of u int count = 0; // To store count of adjacent vertices list<int>::iterator i; for (i = adj[u].begin(); i != adj[u].end(); ++i) if (*i != -1) count++; if (count == 1) return true; // 2) If there are multiple adjacents, then u-v // is not a bridge // Do following steps to check if u-v is a bridge // 2.a) count of vertices reachable from u bool visited[V]; memset(visited, false, V); int count1 = DFSCount(u, visited); // 2.b) Remove edge (u, v) and after removing // the edge, count vertices reachable from u rmvEdge(u, v); memset(visited, false, V); int count2 = DFSCount(u, visited); // 2.c) Add the edge back to the graph addEdge(u, v); // 2.d) If count1 is greater, then edge (u, v) // is a bridge return (count1 > count2)? false: true;}// This function removes edge u-v from graph.// It removes the edge by replacing adjacent// vertex value with -1.void Graph::rmvEdge(int u, int v){ // Find v in adjacency list of u and replace // it with -1 list<int>::iterator iv = find(adj[u].begin(), adj[u].end(), v); *iv = -1; // Find u in adjacency list of v and replace // it with -1 list<int>::iterator iu = find(adj[v].begin(), adj[v].end(), u); *iu = -1;}// A DFS based function to count reachable// vertices from vint Graph::DFSCount(int v, bool visited[]){ // Mark the current node as visited visited[v] = true; int count = 1; // Recur for all vertices adjacent to this vertex list<int>::iterator i; for (i = adj[v].begin(); i != adj[v].end(); ++i) if (*i != -1 && !visited[*i]) count += DFSCount(*i, visited); return count;}// Driver program to test above functionint main(){ // Let us first create and test // graphs shown in above figure Graph g1(4); g1.addEdge(0, 1); g1.addEdge(0, 2); g1.addEdge(1, 2); g1.addEdge(2, 3); g1.printEulerTour(); Graph g3(4); g3.addEdge(0, 1); g3.addEdge(1, 0); g3.addEdge(0, 2); g3.addEdge(2, 0); g3.addEdge(2, 3); g3.addEdge(3, 1); // comment out this line and you will see that // it gives TLE because there is no possible // output g3.addEdge(0, 3); g3.printEulerTour(); return 0;} |
Java
// A java program print Eulerian Trail in a// given Eulerian or Semi-Eulerian Graphimport java.util.*;public class GFG{ // A class that represents an undirected graph static class Graph { // No. of vertices int V; // A dynamic array of adjacency lists ArrayList<ArrayList<Integer>> adj; // Constructor Graph(int V) { this.V = V; adj = new ArrayList<ArrayList<Integer>>(); for(int i=0; i<V; i++){ adj.add(new ArrayList<Integer>()); } } // functions to add and remove edge void addEdge(int u, int v) { adj.get(u).add(v); adj.get(v).add(u); } // This function removes edge u-v from graph. // It removes the edge by replacing adjacent // vertex value with -1. void rmvEdge(int u, int v) { // Find v in adjacency list of u and replace // it with -1 int iv = find(adj.get(u), v); adj.get(u).set(iv, -1); // Find u in adjacency list of v and replace // it with -1 int iu = find(adj.get(v), u); adj.get(v).set(iu, -1); } int find(ArrayList<Integer> al, int v){ for(int i=0; i<al.size(); i++){ if(al.get(i) == v){ return i; } } return -1; } // Methods to print Eulerian tour /* The main function that print Eulerian Trail. It first finds an odd degree vertex (if there is any) and then calls printEulerUtil() to print the path */ void printEulerTour() { // Find a vertex with odd degree int u = 0; for (int i = 0; i < V; i++){ if (adj.get(i).size() % 2 == 1) { u = i; break; } } // Print tour starting from oddv printEulerUtil(u); System.out.println(); } // Print Euler tour starting from vertex u void printEulerUtil(int u) { // Recur for all the vertices adjacent to // this vertex for (int i = 0; i<adj.get(u).size(); ++i) { int v = adj.get(u).get(i); // If edge u-v is not removed and it's a a // valid next edge if (v != -1 && isValidNextEdge(u, v)) { System.out.print(u + "-" + v + " "); rmvEdge(u, v); printEulerUtil(v); } } } // This function returns count of vertices // reachable from v. It does DFS // A DFS based function to count reachable // vertices from v int DFSCount(int v, boolean visited[]) { // Mark the current node as visited visited[v] = true; int count = 1; // Recur for all vertices adjacent to this vertex for (int i = 0; i<adj.get(v).size(); ++i){ int u = adj.get(v).get(i); if (u != -1 && !visited[u]){ count += DFSCount(u, visited); } } return count; } // Utility function to check if edge u-v // is a valid next edge in Eulerian trail or circuit // The function to check if edge u-v can be considered // as next edge in Euler Tout boolean isValidNextEdge(int u, int v) { // The edge u-v is valid in one of the following // two cases: // 1) If v is the only adjacent vertex of u int count = 0; // To store count of adjacent vertices for (int i = 0; i<adj.get(u).size(); ++i) if (adj.get(u).get(i) != -1) count++; if (count == 1) return true; // 2) If there are multiple adjacents, then u-v // is not a bridge // Do following steps to check if u-v is a bridge // 2.a) count of vertices reachable from u boolean visited[] = new boolean[V]; Arrays.fill(visited, false); int count1 = DFSCount(u, visited); // 2.b) Remove edge (u, v) and after removing // the edge, count vertices reachable from u rmvEdge(u, v); Arrays.fill(visited, false); int count2 = DFSCount(u, visited); // 2.c) Add the edge back to the graph addEdge(u, v); // 2.d) If count1 is greater, then edge (u, v) // is a bridge return (count1 > count2)? false: true; } } // Driver program to test above function public static void main(String args[]) { // Let us first create and test // graphs shown in above figure Graph g1 = new Graph(4); g1.addEdge(0, 1); g1.addEdge(0, 2); g1.addEdge(1, 2); g1.addEdge(2, 3); g1.printEulerTour(); Graph g3 = new Graph(4); g3.addEdge(0, 1); g3.addEdge(1, 0); g3.addEdge(0, 2); g3.addEdge(2, 0); g3.addEdge(2, 3); g3.addEdge(3, 1); // comment out this line and you will see that // it gives TLE because there is no possible // output g3.addEdge(0, 3); g3.printEulerTour(); }}// This code is contributed by adityapande88. |
Output:
2-0 0-1 1-2 2-3 1-0 0-2 2-3 3-1 1-0 0-2
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