Given positive weighted undirected graph, find minimum weight cycle in it.
Examples:
Minimum weighted cycle is :
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
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