Hamiltonian Cycle | Backtracking-6

• Difficulty Level : Hard
• Last Updated : 06 Jan, 2022

Hamiltonian Path in an undirected graph is a path that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian Path such that there is an edge (in the graph) from the last vertex to the first vertex of the Hamiltonian Path. Determine whether a given graph contains Hamiltonian Cycle or not. If it contains, then prints the path. Following are the input and output of the required function.
Input:
A 2D array graph[V][V] where V is the number of vertices in graph and graph[V][V] is adjacency matrix representation of the graph. A value graph[i][j] is 1 if there is a direct edge from i to j, otherwise graph[i][j] is 0.
Output:
An array path[V] that should contain the Hamiltonian Path. path[i] should represent the ith vertex in the Hamiltonian Path. The code should also return false if there is no Hamiltonian Cycle in the graph.
For example, a Hamiltonian Cycle in the following graph is {0, 1, 2, 4, 3, 0}.

(0)--(1)--(2)
|   / \   |
|  /   \  |
| /     \ |
(3)-------(4)

And the following graph doesn’t contain any Hamiltonian Cycle.

(0)--(1)--(2)
|   / \   |
|  /   \  |
| /     \ |
(3)      (4)

Naive Algorithm
Generate all possible configurations of vertices and print a configuration that satisfies the given constraints. There will be n! (n factorial) configurations.

while there are untried conflagrations
{
generate the next configuration
if ( there are edges between two consecutive vertices of this
configuration and there is an edge from the last vertex to
the first ).
{
print this configuration;
break;
}
}

Backtracking Algorithm
Create an empty path array and add vertex 0 to it. Add other vertices, starting from the vertex 1. Before adding a vertex, check for whether it is adjacent to the previously added vertex and not already added. If we find such a vertex, we add the vertex as part of the solution. If we do not find a vertex then we return false.

Implementation of Backtracking solution
Following are implementations of the Backtracking solution.

C++

 /* C++ program for solution of HamiltonianCycle problem using backtracking */#include using namespace std; // Number of vertices in the graph#define V 5 void printSolution(int path[]); /* A utility function to check ifthe vertex v can be added at index 'pos'in the Hamiltonian Cycle constructedso far (stored in 'path[]') */bool isSafe(int v, bool graph[V][V],            int path[], int pos){    /* Check if this vertex is an adjacent    vertex of the previously added vertex. */    if (graph [path[pos - 1]][ v ] == 0)        return false;     /* Check if the vertex has already been included.    This step can be optimized by creating    an array of size V */    for (int i = 0; i < pos; i++)        if (path[i] == v)            return false;     return true;} /* A recursive utility functionto solve hamiltonian cycle problem */bool hamCycleUtil(bool graph[V][V],                  int path[], int pos){    /* base case: If all vertices are    included in Hamiltonian Cycle */    if (pos == V)    {        // And if there is an edge from the        // last included vertex to the first vertex        if (graph[path[pos - 1]][path] == 1)            return true;        else            return false;    }     // Try different vertices as a next candidate    // in Hamiltonian Cycle. We don't try for 0 as    // we included 0 as starting point in hamCycle()    for (int v = 1; v < V; v++)    {        /* Check if this vertex can be added        // to Hamiltonian Cycle */        if (isSafe(v, graph, path, pos))        {            path[pos] = v;             /* recur to construct rest of the path */            if (hamCycleUtil (graph, path, pos + 1) == true)                return true;             /* If adding vertex v doesn't lead to a solution,            then remove it */            path[pos] = -1;        }    }     /* If no vertex can be added to    Hamiltonian Cycle constructed so far,    then return false */    return false;} /* This function solves the Hamiltonian Cycle problemusing Backtracking. It mainly uses hamCycleUtil() tosolve the problem. It returns false if there is noHamiltonian Cycle possible, otherwise return trueand prints the path. Please note that there may bemore than one solutions, this function prints oneof the feasible solutions. */bool hamCycle(bool graph[V][V]){    int *path = new int[V];    for (int i = 0; i < V; i++)        path[i] = -1;     /* Let us put vertex 0 as the first vertex in the path.    If there is a Hamiltonian Cycle, then the path can be    started from any point of the cycle as the graph is undirected */    path = 0;    if (hamCycleUtil(graph, path, 1) == false )    {        cout << "\nSolution does not exist";        return false;    }     printSolution(path);    return true;} /* A utility function to print solution */void printSolution(int path[]){    cout << "Solution Exists:"            " Following is one Hamiltonian Cycle \n";    for (int i = 0; i < V; i++)        cout << path[i] << " ";     // Let us print the first vertex again    // to show the complete cycle    cout << path << " ";    cout << endl;} // Driver Codeint main(){    /* Let us create the following graph        (0)--(1)--(2)        | / \ |        | / \ |        | / \ |        (3)-------(4) */    bool graph1[V][V] = {{0, 1, 0, 1, 0},                        {1, 0, 1, 1, 1},                        {0, 1, 0, 0, 1},                        {1, 1, 0, 0, 1},                        {0, 1, 1, 1, 0}};         // Print the solution    hamCycle(graph1);         /* Let us create the following graph    (0)--(1)--(2)    | / \ |    | / \ |    | / \ |    (3) (4) */    bool graph2[V][V] = {{0, 1, 0, 1, 0},                         {1, 0, 1, 1, 1},                         {0, 1, 0, 0, 1},                         {1, 1, 0, 0, 0},                         {0, 1, 1, 0, 0}};     // Print the solution    hamCycle(graph2);     return 0;} // This is code is contributed by rathbhupendra

C

 /* C program for solution of Hamiltonian Cycle problem   using backtracking */#include // Number of vertices in the graph#define V 5 void printSolution(int path[]); /* A utility function to check if the vertex v can be added at   index 'pos' in the Hamiltonian Cycle constructed so far (stored   in 'path[]') */bool isSafe(int v, bool graph[V][V], int path[], int pos){    /* Check if this vertex is an adjacent vertex of the previously       added vertex. */    if (graph [ path[pos-1] ][ v ] == 0)        return false;     /* Check if the vertex has already been included.      This step can be optimized by creating an array of size V */    for (int i = 0; i < pos; i++)        if (path[i] == v)            return false;     return true;} /* A recursive utility function to solve hamiltonian cycle problem */bool hamCycleUtil(bool graph[V][V], int path[], int pos){    /* base case: If all vertices are included in Hamiltonian Cycle */    if (pos == V)    {        // And if there is an edge from the last included vertex to the        // first vertex        if ( graph[ path[pos-1] ][ path ] == 1 )           return true;        else          return false;    }     // Try different vertices as a next candidate in Hamiltonian Cycle.    // We don't try for 0 as we included 0 as starting point in hamCycle()    for (int v = 1; v < V; v++)    {        /* Check if this vertex can be added to Hamiltonian Cycle */        if (isSafe(v, graph, path, pos))        {            path[pos] = v;             /* recur to construct rest of the path */            if (hamCycleUtil (graph, path, pos+1) == true)                return true;             /* If adding vertex v doesn't lead to a solution,               then remove it */            path[pos] = -1;        }    }     /* If no vertex can be added to Hamiltonian Cycle constructed so far,       then return false */    return false;} /* This function solves the Hamiltonian Cycle problem using Backtracking.  It mainly uses hamCycleUtil() to solve the problem. It returns false  if there is no Hamiltonian Cycle possible, otherwise return true and  prints the path. Please note that there may be more than one solutions,  this function prints one of the feasible solutions. */bool hamCycle(bool graph[V][V]){    int *path = new int[V];    for (int i = 0; i < V; i++)        path[i] = -1;     /* Let us put vertex 0 as the first vertex in the path. If there is       a Hamiltonian Cycle, then the path can be started from any point       of the cycle as the graph is undirected */    path = 0;    if ( hamCycleUtil(graph, path, 1) == false )    {        printf("\nSolution does not exist");        return false;    }     printSolution(path);    return true;} /* A utility function to print solution */void printSolution(int path[]){    printf ("Solution Exists:"            " Following is one Hamiltonian Cycle \n");    for (int i = 0; i < V; i++)        printf(" %d ", path[i]);     // Let us print the first vertex again to show the complete cycle    printf(" %d ", path);    printf("\n");} // driver program to test above functionint main(){   /* Let us create the following graph      (0)--(1)--(2)       |   / \   |       |  /   \  |       | /     \ |      (3)-------(4)    */   bool graph1[V][V] = {{0, 1, 0, 1, 0},                      {1, 0, 1, 1, 1},                      {0, 1, 0, 0, 1},                      {1, 1, 0, 0, 1},                      {0, 1, 1, 1, 0},                     };     // Print the solution    hamCycle(graph1);    /* Let us create the following graph      (0)--(1)--(2)       |   / \   |       |  /   \  |       | /     \ |      (3)       (4)    */    bool graph2[V][V] = {{0, 1, 0, 1, 0},                      {1, 0, 1, 1, 1},                      {0, 1, 0, 0, 1},                      {1, 1, 0, 0, 0},                      {0, 1, 1, 0, 0},                     };     // Print the solution    hamCycle(graph2);     return 0;}

Java

 /* Java program for solution of Hamiltonian Cycle problem   using backtracking */class HamiltonianCycle{    final int V = 5;    int path[];     /* A utility function to check if the vertex v can be       added at index 'pos'in the Hamiltonian Cycle       constructed so far (stored in 'path[]') */    boolean isSafe(int v, int graph[][], int path[], int pos)    {        /* Check if this vertex is an adjacent vertex of           the previously added vertex. */        if (graph[path[pos - 1]][v] == 0)            return false;         /* Check if the vertex has already been included.           This step can be optimized by creating an array           of size V */        for (int i = 0; i < pos; i++)            if (path[i] == v)                return false;         return true;    }     /* A recursive utility function to solve hamiltonian       cycle problem */    boolean hamCycleUtil(int graph[][], int path[], int pos)    {        /* base case: If all vertices are included in           Hamiltonian Cycle */        if (pos == V)        {            // And if there is an edge from the last included            // vertex to the first vertex            if (graph[path[pos - 1]][path] == 1)                return true;            else                return false;        }         // Try different vertices as a next candidate in        // Hamiltonian Cycle. We don't try for 0 as we        // included 0 as starting point in hamCycle()        for (int v = 1; v < V; v++)        {            /* Check if this vertex can be added to Hamiltonian               Cycle */            if (isSafe(v, graph, path, pos))            {                path[pos] = v;                 /* recur to construct rest of the path */                if (hamCycleUtil(graph, path, pos + 1) == true)                    return true;                 /* If adding vertex v doesn't lead to a solution,                   then remove it */                path[pos] = -1;            }        }         /* If no vertex can be added to Hamiltonian Cycle           constructed so far, then return false */        return false;    }     /* This function solves the Hamiltonian Cycle problem using       Backtracking. It mainly uses hamCycleUtil() to solve the       problem. It returns false if there is no Hamiltonian Cycle       possible, otherwise return true and prints the path.       Please note that there may be more than one solutions,       this function prints one of the feasible solutions. */    int hamCycle(int graph[][])    {        path = new int[V];        for (int i = 0; i < V; i++)            path[i] = -1;         /* Let us put vertex 0 as the first vertex in the path.           If there is a Hamiltonian Cycle, then the path can be           started from any point of the cycle as the graph is           undirected */        path = 0;        if (hamCycleUtil(graph, path, 1) == false)        {            System.out.println("\nSolution does not exist");            return 0;        }         printSolution(path);        return 1;    }     /* A utility function to print solution */    void printSolution(int path[])    {        System.out.println("Solution Exists: Following" +                           " is one Hamiltonian Cycle");        for (int i = 0; i < V; i++)            System.out.print(" " + path[i] + " ");         // Let us print the first vertex again to show the        // complete cycle        System.out.println(" " + path + " ");    }     // driver program to test above function    public static void main(String args[])    {        HamiltonianCycle hamiltonian =                                new HamiltonianCycle();        /* Let us create the following graph           (0)--(1)--(2)            |   / \   |            |  /   \  |            | /     \ |           (3)-------(4)    */        int graph1[][] = {{0, 1, 0, 1, 0},            {1, 0, 1, 1, 1},            {0, 1, 0, 0, 1},            {1, 1, 0, 0, 1},            {0, 1, 1, 1, 0},        };         // Print the solution        hamiltonian.hamCycle(graph1);         /* Let us create the following graph           (0)--(1)--(2)            |   / \   |            |  /   \  |            | /     \ |           (3)       (4)    */        int graph2[][] = {{0, 1, 0, 1, 0},            {1, 0, 1, 1, 1},            {0, 1, 0, 0, 1},            {1, 1, 0, 0, 0},            {0, 1, 1, 0, 0},        };         // Print the solution        hamiltonian.hamCycle(graph2);    }}// This code is contributed by Abhishek Shankhadhar

Python3

 # Python program for solution of# hamiltonian cycle problem class Graph():    def __init__(self, vertices):        self.graph = [[0 for column in range(vertices)]                            for row in range(vertices)]        self.V = vertices     ''' Check if this vertex is an adjacent vertex        of the previously added vertex and is not        included in the path earlier '''    def isSafe(self, v, pos, path):        # Check if current vertex and last vertex        # in path are adjacent        if self.graph[ path[pos-1] ][v] == 0:            return False         # Check if current vertex not already in path        for vertex in path:            if vertex == v:                return False         return True     # A recursive utility function to solve    # hamiltonian cycle problem    def hamCycleUtil(self, path, pos):         # base case: if all vertices are        # included in the path        if pos == self.V:            # Last vertex must be adjacent to the            # first vertex in path to make a cycle            if self.graph[ path[pos-1] ][ path ] == 1:                return True            else:                return False         # Try different vertices as a next candidate        # in Hamiltonian Cycle. We don't try for 0 as        # we included 0 as starting point in hamCycle()        for v in range(1,self.V):             if self.isSafe(v, pos, path) == True:                 path[pos] = v                 if self.hamCycleUtil(path, pos+1) == True:                    return True                 # Remove current vertex if it doesn't                # lead to a solution                path[pos] = -1         return False     def hamCycle(self):        path = [-1] * self.V         ''' Let us put vertex 0 as the first vertex            in the path. If there is a Hamiltonian Cycle,            then the path can be started from any point            of the cycle as the graph is undirected '''        path = 0         if self.hamCycleUtil(path,1) == False:            print ("Solution does not exist\n")            return False         self.printSolution(path)        return True     def printSolution(self, path):        print ("Solution Exists: Following",                 "is one Hamiltonian Cycle")        for vertex in path:            print (vertex, end = " ")        print (path, "\n") # Driver Code ''' Let us create the following graph    (0)--(1)--(2)    | / \ |    | / \ |    | /     \ |    (3)-------(4) '''g1 = Graph(5)g1.graph = [ [0, 1, 0, 1, 0], [1, 0, 1, 1, 1],            [0, 1, 0, 0, 1,],[1, 1, 0, 0, 1],            [0, 1, 1, 1, 0], ] # Print the solutiong1.hamCycle(); ''' Let us create the following graph    (0)--(1)--(2)    | / \ |    | / \ |    | /     \ |    (3)     (4) '''g2 = Graph(5)g2.graph = [ [0, 1, 0, 1, 0], [1, 0, 1, 1, 1],        [0, 1, 0, 0, 1,], [1, 1, 0, 0, 0],        [0, 1, 1, 0, 0], ] # Print the solutiong2.hamCycle(); # This code is contributed by Divyanshu Mehta

C#

 // C# program for solution of Hamiltonian// Cycle problem using backtrackingusing System; public class HamiltonianCycle{    readonly int V = 5;    int []path;     /* A utility function to check    if the vertex v can be added at    index 'pos'in the Hamiltonian Cycle    constructed so far (stored in 'path[]') */    bool isSafe(int v, int [,]graph,                int []path, int pos)    {        /* Check if this vertex is        an adjacent vertex of the        previously added vertex. */        if (graph[path[pos - 1], v] == 0)            return false;         /* Check if the vertex has already        been included. This step can be        optimized by creating an array        of size V */        for (int i = 0; i < pos; i++)            if (path[i] == v)                return false;         return true;    }     /* A recursive utility function    to solve hamiltonian cycle problem */    bool hamCycleUtil(int [,]graph, int []path, int pos)    {        /* base case: If all vertices        are included in Hamiltonian Cycle */        if (pos == V)        {            // And if there is an edge from the last included            // vertex to the first vertex            if (graph[path[pos - 1],path] == 1)                return true;            else                return false;        }         // Try different vertices as a next candidate in        // Hamiltonian Cycle. We don't try for 0 as we        // included 0 as starting point in hamCycle()        for (int v = 1; v < V; v++)        {            /* Check if this vertex can be            added to Hamiltonian Cycle */            if (isSafe(v, graph, path, pos))            {                path[pos] = v;                 /* recur to construct rest of the path */                if (hamCycleUtil(graph, path, pos + 1) == true)                    return true;                 /* If adding vertex v doesn't                lead to a solution, then remove it */                path[pos] = -1;            }        }         /* If no vertex can be added to Hamiltonian Cycle        constructed so far, then return false */        return false;    }     /* This function solves the Hamiltonian    Cycle problem using Backtracking. It    mainly uses hamCycleUtil() to solve the    problem. It returns false if there    is no Hamiltonian Cycle possible,    otherwise return true and prints the path.    Please note that there may be more than    one solutions, this function prints one    of the feasible solutions. */    int hamCycle(int [,]graph)    {        path = new int[V];        for (int i = 0; i < V; i++)            path[i] = -1;         /* Let us put vertex 0 as the first        vertex in the path. If there is a        Hamiltonian Cycle, then the path can be        started from any point of the cycle        as the graph is undirected */        path = 0;        if (hamCycleUtil(graph, path, 1) == false)        {            Console.WriteLine("\nSolution does not exist");            return 0;        }         printSolution(path);        return 1;    }     /* A utility function to print solution */    void printSolution(int []path)    {        Console.WriteLine("Solution Exists: Following" +                        " is one Hamiltonian Cycle");        for (int i = 0; i < V; i++)            Console.Write(" " + path[i] + " ");         // Let us print the first vertex again        //  to show the complete cycle        Console.WriteLine(" " + path + " ");    }     // Driver code    public static void Main(String []args)    {        HamiltonianCycle hamiltonian =                                new HamiltonianCycle();        /* Let us create the following graph        (0)--(1)--(2)            | / \ |            | / \ |            | /     \ |        (3)-------(4) */        int [,]graph1= {{0, 1, 0, 1, 0},            {1, 0, 1, 1, 1},            {0, 1, 0, 0, 1},            {1, 1, 0, 0, 1},            {0, 1, 1, 1, 0},        };         // Print the solution        hamiltonian.hamCycle(graph1);         /* Let us create the following graph        (0)--(1)--(2)            | / \ |            | / \ |            | /     \ |        (3)     (4) */        int [,]graph2 = {{0, 1, 0, 1, 0},            {1, 0, 1, 1, 1},            {0, 1, 0, 0, 1},            {1, 1, 0, 0, 0},            {0, 1, 1, 0, 0},        };         // Print the solution        hamiltonian.hamCycle(graph2);    }} // This code contributed by Rajput-Ji



Javascript



Output:

Solution Exists: Following is one Hamiltonian Cycle
0  1  2  4  3  0

Solution does not exist

Note that the above code always prints a cycle starting from 0. The starting point should not matter as the cycle can be started from any point. If you want to change the starting point, you should make two changes to the above code.
Change “path = 0;” to “path = s;” where s is your new starting point. Also change loop “for (int v = 1; v < V; v++)” in hamCycleUtil() to “for (int v = 0; v < V; v++)”.