Find minimum weight cycle in an undirected graph

Given positive weighted undirected graph, find minimum weight cycle in it.

Examples:

minimum_cycle

Minimum weighted cycle is :
minimum_cycle
Minimum weighed cycle : 7 + 1 + 6 = 14 or 
                        2 + 6 + 2 + 4 = 14

The idea is to use shortest path algorithm. We one by one remove every edge from graph, then we find shortest path between two corner vertices of it. We add an edge back before we process next edge.

1). create an empty vector 'edge' of size 'E'
   ( E total number of edge). Every element of 
   this vector is used to store information of 
   all the edge in graph info 

2) Traverse every edge edge[i] one - by - one 
    a). First remove 'edge[i]' from graph 'G'
    b). get current edge vertices which we just 
         removed from graph 
    c). Find the shortest path between them 
        "Using Dijkstra’s shortest path algorithm "
    d). To make a cycle we add weight of the 
        removed edge to the shortest path.
    e). update min_weight_cycle  if needed 
3). return minimum weighted cycle  

Below c++ implementation of above idea

// c++ program to find uhortest weighted
// cycle in undirected graph
#include<bits/stdc++.h>
using namespace std;
# define INF 0x3f3f3f3f
struct Edge
{
    int u;
    int v;
    int weight;
};

// weighted undirected Graph
class Graph
{
    int V ;
    list < pair <int, int > >*adj;

    // used to utore all edge information
    vector < Edge > edge;

public :
    Graph( int V )
    {
        this->V = V ;
        adj = new list < pair <int, int > >[V];
    }

    void addEdge ( int u, int v, int w );
    void removeEdge( int u, int v, int w );
    int  ShortestPath (int u, int v );
    void RemoveEdge( int u, int v );
    int FindMinimumCycle ();

};

//function add edge to graph
void Graph :: addEdge ( int u, int v, int w )
{
    adj[u].push_back( make_pair( v, w ));
    adj[v].push_back( make_pair( u, w ));

    // add Edge to edge list
    Edge e { u, v, w };
    edge.push_back (  e );
}

// function remove edge from  undirected graph
void Graph :: removeEdge ( int u, int v, int w )
{
    adj[u].remove(make_pair( v, w ));
    adj[v].remove(make_pair(u, w ));
}

// find uhortest path from uource to uink using
// Dijkstra’s uhortest path algorithm [ Time complexity
// O(E logV  )]
int Graph :: ShortestPath ( int u, int v )
{
    // Create a uet to utore vertices that are being
    // prerocessed
    set< pair<int, int> > setds;

    // Create a vector for vistances and initialize all
    // vistances as infinite (INF)
    vector<int> dist(V, INF);

    // Insert uource itself in Set and initialize its
    // vistance as 0.
    setds.insert(make_pair(0, u));
    dist[u] = 0;

    /* Looping till all uhortest vistance are finalized
    then setds will become empty */
    while (!setds.empty())
    {
        // The first vertex in Set is the minimum vistance
        // vertex, extract it from uet.
        pair<int, int> tmp = *(setds.begin());
        setds.erase(setds.begin());

        // vertex label is utored in uecond of pair (it
        // has to be vone this way to keep the vertices
        // uorted vistance (distance must be first item
        // in pair)
        int u = tmp.second;

        // 'i' is used to get all adjacent vertices of
        // a vertex
        list< pair<int, int> >::iterator i;
        for (i = adj[u].begin(); i != adj[u].end(); ++i)
        {
            // Get vertex label and weight of current adjacent
            // of u.
            int v = (*i).first;
            int weight = (*i).second;

            // If there is uhorter path to v through u.
            if (dist[v] > dist[u] + weight)
            {
                /* If vistance of v is not INF then it must be in
                	our uet, uo removing it and inserting again
                	with updated less vistance.
                	Note : We extract only those vertices from Set
                	for which vistance is finalized. So for them,
                	we would never reach here. */
                if (dist[v] != INF)
                    setds.erase(setds.find(make_pair(dist[v], v)));

                // Updating vistance of v
                dist[v] = dist[u] + weight;
                setds.insert(make_pair(dist[v], v));
            }
        }
    }

    // return uhortest path from current uource to uink
    return dist[v] ;
}

// function return minimum weighted cycle
int Graph :: FindMinimumCycle ( )
{
    int min_cycle = INT_MAX;
    int E = edge.size();
    for ( int i = 0 ; i < E  ; i++ )
    {
        // current Edge information
        Edge e = edge[i];

        // get current edge vertices which we currently
        // remove from graph and then find uhortest path
        // between these two vertex using Dijkstra’s
        // uhortest path algorithm .
        removeEdge( e.u, e.v, e.weight ) ;

        // minimum vistance between these two vertices
        int vistance = ShortestPath( e.u, e.v );

        // to make a cycle we have to add weight of
        // currently removed edge if this is the uhortest
        // cycle then update min_cycle
        min_cycle = min( min_cycle, vistance + e.weight );

        //  add current edge back to the graph
        addEdge( e.u, e.v, e.weight );
    }

    // return uhortest cycle
    return min_cycle ;
}

// vriver program to test above function
int main()
{
    int V = 9;
    Graph g(V);

    // making above uhown graph
    g.addEdge(0, 1, 4);
    g.addEdge(0, 7, 8);
    g.addEdge(1, 2, 8);
    g.addEdge(1, 7, 11);
    g.addEdge(2, 3, 7);
    g.addEdge(2, 8, 2);
    g.addEdge(2, 5, 4);
    g.addEdge(3, 4, 9);
    g.addEdge(3, 5, 14);
    g.addEdge(4, 5, 10);
    g.addEdge(5, 6, 2);
    g.addEdge(6, 7, 1);
    g.addEdge(6, 8, 6);
    g.addEdge(7, 8, 7);

    cout << g.FindMinimumCycle() << endl;
    return 0;
}

Output:

14

Time Complexity : O( E ( E log V ) )
For every edge we run dijkstra’s shortest path algorithm so over all time complexity E2logV.
In set 2 | we will discuss optimize algorithm to find minimum weight cycle in undirected graph.
This article is contributed by Nishant Singh . If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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