/// <summary>
/// We can get the min cut using the max-flow min-cut theorm, which states that the max flow in a n/w = min-cut of the network.
/// Steps:
/// 1. Run FordFulkerson algo to get the residual graph.
/// 2. Find the set of vertices that can be reached from the source vertex.
/// 3. Get all the edges from vertices in this set to the vertices which are not present in this set.
/// this will give all the edges of the min cut.
///
/// The running time of this algo is same as the ford fulkerson's max flow algo with edward karp optimization
/// O(V(E^2))
/// </summary>
/// <param name="graph">graph object</param>
/// <param name="startId">id of the source vertex</param>
/// <param name="endId">id of the sink vertex</param>
/// <returns></returns>
public List <string> GetMinCut(GraphWithFwdEdges graph, string startId, string endId)
{
List <string> setOfEdges = new List <string>();
// 1. Run FordFulkerson algo to get the residual graph.
FordFulkersonEdmondKarp ffkp = new FordFulkersonEdmondKarp();
ffkp.GetTheMaxFlow(graph, startId, endId);
// 2. Find the set of vertices that can be reached from the source vertex.
Dictionary <GraphVertex, bool> setOfVerticesFromSource = GetAllVerticesThatCanBeReachedFromSrc(graph, startId);
// 3.Get all the forward edges from vertices in this set to the vertices which are not present in this set.
foreach (GraphVertex vertex in setOfVerticesFromSource.Keys)
{
foreach (GraphVertex neighbour in vertex.Neighbours)
{
string edgeId = vertex.Id + "#" + neighbour.Id;
if (graph.IsEdgeFwd[edgeId] && !setOfVerticesFromSource.ContainsKey(neighbour))
{
// the graph edge should not have 0 flow and also should not be present in the current set(found after the cut)
setOfEdges.Add(edgeId);
}
}
}
return(setOfEdges);
}
/// <summary>
/// Create a source vertex which has edge with unit weight to all the employees.
/// Note we are using unit edge weight cause each employee can take up only one job.
/// Also create a sink vertex, all the jobs will have a directed edge towards the sink and the edge weight is 1.
/// The unit edge weight is used cause each job must be done by one employee.
/// </summary>
/// <param name="employeeJobPreference">dictionary having key as the employeeid and value as the list of job ids which are prefered</param>
public MaximumBipartiteMatching(Dictionary <string, List <string> > employeeJobPreference)
{
// initialize the graph
Graph = new GraphWithFwdEdges();
foreach (string employeeId in employeeJobPreference.Keys)
{
Graph.AddEdge(StartId, employeeId, 1); // Add fwd edge
Graph.AddEdge(employeeId, StartId, 0, false); // Add backedge
foreach (string jobId in employeeJobPreference[employeeId])
{
Graph.AddEdge(employeeId, jobId, 1); // Add fwdEdge
Graph.AddEdge(jobId, employeeId, 0, false); // Add back edge
// Add the directed edge towards sink from the job id
Graph.AddEdge(jobId, SinkId, 1); // Add fwdEdge
Graph.AddEdge(SinkId, jobId, 0, false); // Add back edge
}
}
}
public static void TestMinCut()
{
MinCut mc = new MinCut();
/*
* (A)<------3--------(B)
| X /x
| \ / |
| 3 4 |
| \ / |
| 3 \ x |
| (C) 1
| / \ |
| / \ |
| 1 2 |
| x x |
| (D)-------2------>(E)
| |
| |
| |
| 6 1
| |
| x X
| (F)-------9------->(G)
*/
GraphWithFwdEdges graph = new GraphWithFwdEdges();
graph.AddEdge("A", "B", 3);
graph.AddEdge("B", "A", 0, false);
graph.AddEdge("A", "D", 3);
graph.AddEdge("D", "A", 0, false);
graph.AddEdge("D", "E", 2);
graph.AddEdge("E", "D", 0, false);
graph.AddEdge("E", "B", 1);
graph.AddEdge("B", "E", 0, false);
graph.AddEdge("C", "A", 3);
graph.AddEdge("A", "C", 0, false);
graph.AddEdge("B", "C", 4);
graph.AddEdge("C", "B", 0, false);
graph.AddEdge("C", "D", 1);
graph.AddEdge("D", "C", 0, false);
graph.AddEdge("C", "E", 2);
graph.AddEdge("E", "C", 0, false);
graph.AddEdge("D", "F", 6);
graph.AddEdge("F", "D", 0, false);
graph.AddEdge("F", "G", 9);
graph.AddEdge("G", "F", 0, false);
graph.AddEdge("E", "G", 1);
graph.AddEdge("G", "E", 0, false);
List <string> setOfEdges = mc.GetMinCut(graph, "A", "G");
PrintEdges(setOfEdges);
graph = new GraphWithFwdEdges();
graph.AddEdge("0", "1", 16);
graph.AddEdge("1", "2", 10);
graph.AddEdge("0", "2", 13);
graph.AddEdge("2", "1", 4);
graph.AddEdge("1", "3", 12);
graph.AddEdge("3", "2", 9);
graph.AddEdge("2", "4", 14);
graph.AddEdge("4", "3", 7);
graph.AddEdge("4", "5", 4);
graph.AddEdge("3", "5", 20);
graph.AddEdge("1", "0", 0, false);
graph.AddEdge("2", "0", 0, false);
graph.AddEdge("3", "1", 0, false);
graph.AddEdge("2", "3", 0, false);
graph.AddEdge("4", "2", 0, false);
graph.AddEdge("3", "4", 0, false);
graph.AddEdge("5", "4", 0, false);
graph.AddEdge("5", "3", 0, false);
setOfEdges = mc.GetMinCut(graph, "0", "5");
PrintEdges(setOfEdges);
}