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Find minimum weight cycle in an undirected graph
  • Difficulty Level : Hard
  • Last Updated : 22 Dec, 2020

Given a positive weighted undirected graph, find the 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 the graph, then we find the shortest path between two corner vertices of it. We add an edge back before we process the 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 the weight of the 
        removed edge to the shortest path.
    e). update min_weight_cycle  if needed 
3). return minimum weighted cycle

Below is the implementation of the above idea 

C++




// c++ program to find shortest 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 store 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 the shortest path from source to sink using
// Dijkstra’s shortest path algorithm [ Time complexity
// O(E logV )]
int Graph :: ShortestPath ( int u, int v )
{
    // Create a set to store vertices that are being
    // preprocessed
    set< pair<int, int> > setds;
 
    // Create a vector for distances and initialize all
    // distances as infinite (INF)
    vector<int> dist(V, INF);
 
    // Insert source itself in Set and initialize its
    // distance as 0.
    setds.insert(make_pair(0, u));
    dist[u] = 0;
 
    /* Looping till all shortest distance are finalized
    then setds will become empty */
    while (!setds.empty())
    {
        // The first vertex in Set is the minimum distance
        // vertex, extract it from set.
        pair<int, int> tmp = *(setds.begin());
        setds.erase(setds.begin());
 
        // vertex label is stored in second of pair (it
        // has to be done this way to keep the vertices
        // sorted distance (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 shorter path to v through u.
            if (dist[v] > dist[u] + weight)
            {
                /* If the distance of v is not INF then it must be in
                    our set, so removing it and inserting again
                    with updated less distance.
                    Note : We extract only those vertices from Set
                    for which distance is finalized. So for them,
                    we would never reach here. */
                if (dist[v] != INF)
                    setds.erase(setds.find(make_pair(dist[v], v)));
 
                // Updating distance of v
                dist[v] = dist[u] + weight;
                setds.insert(make_pair(dist[v], v));
            }
        }
    }
 
    // return shortest path from current source to sink
    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 shortest path
        // between these two vertex using Dijkstra’s
        // shortest path algorithm .
        removeEdge( e.u, e.v, e.weight ) ;
 
        // minimum distance between these two vertices
        int distance = ShortestPath( e.u, e.v );
 
        // to make a cycle we have to add weight of
        // currently removed edge if this is the shortest
        // cycle then update min_cycle
        min_cycle = min( min_cycle, distance + e.weight );
 
        // add current edge back to the graph
        addEdge( e.u, e.v, e.weight );
    }
 
    // return shortest cycle
    return min_cycle ;
}
 
// driver program to test above function
int main()
{
    int V = 9;
    Graph g(V);
 
    // making above shown 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;
}

Python3




# Python3 program to find shortest weighted
# cycle in undirected graph
from sys import maxsize
 
INF = int(0x3f3f3f3f)
 
class Edge:
     
    def __init__(self, u: int,
                       v: int,
                  weight: int) -> None:
                       
        self.u = u
        self.v = v
        self.weight = weight
 
# Weighted undirected Graph
class Graph:
     
    def __init__(self, V: int) -> None:
         
        self.V = V
        self.adj = [[] for _ in range(V)]
 
        # Used to store all edge information
        self.edge = []
 
    # Function add edge to graph
    def addEdge(self, u: int,
                      v: int,
                      w: int) -> None:
         
        self.adj[u].append((v, w))
        self.adj[v].append((u, w))
 
        # Add Edge to edge list
        e = Edge(u, v, w)
        self.edge.append(e)
 
    # Function remove edge from undirected graph
    def removeEdge(self, u: int,
                         v: int, w: int) -> None:
         
        self.adj[u].remove((v, w))
        self.adj[v].remove((u, w))
 
    # Find the shortest path from source
    # to sink using Dijkstra’s shortest
    # path algorithm [ Time complexity
    # O(E logV )]
    def ShortestPath(self, u: int, v: int) -> int:
         
        # Create a set to store vertices that
        # are being preprocessed
        setds = set()
 
        # Create a vector for distances and
        # initialize all distances as infinite (INF)
        dist = [INF] * self.V
 
        # Insert source itself in Set and
        # initialize its distance as 0.
        setds.add((0, u))
        dist[u] = 0
 
        # Looping till all shortest distance are
        # finalized then setds will become empty
        while (setds):
             
            # The first vertex in Set is the minimum
            # distance vertex, extract it from set.
            tmp = setds.pop()
 
            # Vertex label is stored in second of
            # pair (it has to be done this way to
            # keep the vertices sorted distance
            # (distance must be first item in pair)
            uu = tmp[1]
 
            # 'i' is used to get all adjacent
            # vertices of a vertex
            for i in self.adj[uu]:
                 
                # Get vertex label and weight of
                # current adjacent of u.
                vv = i[0]
                weight = i[1]
 
                # If there is shorter path to v through u.
                if (dist[vv] > dist[uu] + weight):
                     
                    # If the distance of v is not INF then
                    # it must be in our set, so removing it
                    # and inserting again with updated less
                    # distance. Note : We extract only those
                    # vertices from Set for which distance
                    # is finalized. So for them, we would
                    # never reach here.
                    if (dist[vv] != INF):
                        if ((dist[vv], vv) in setds):
                            setds.remove((dist[vv], vv))
 
                    # Updating distance of v
                    dist[vv] = dist[uu] + weight
                    setds.add((dist[vv], vv))
 
        # Return shortest path from
        # current source to sink
        return dist[v]
 
    # Function return minimum weighted cycle
    def FindMinimumCycle(self) -> int:
         
        min_cycle = maxsize
        E = len(self.edge)
         
        for i in range(E):
             
            # Current Edge information
            e = self.edge[i]
 
            # Get current edge vertices which we currently
            # remove from graph and then find shortest path
            # between these two vertex using Dijkstra’s
            # shortest path algorithm .
            self.removeEdge(e.u, e.v, e.weight)
 
            # Minimum distance between these two vertices
            distance = self.ShortestPath(e.u, e.v)
 
            # To make a cycle we have to add weight of
            # currently removed edge if this is the
            # shortest cycle then update min_cycle
            min_cycle = min(min_cycle,
                            distance + e.weight)
 
            # Add current edge back to the graph
            self.addEdge(e.u, e.v, e.weight)
 
        # Return shortest cycle
        return min_cycle
 
# Driver Code
if __name__ == "__main__":
 
    V = 9
 
    g = Graph(V)
 
    # Making above shown 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)
 
    print(g.FindMinimumCycle())
 
# This code is contributed by sanjeev2552

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 the algorithm to find a 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.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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