/// <summary>
/// kruskal's algorithm
/// </summary>
/// <param name="g"></param>
public KruskalMST(EdgeWeightedGraph g)
{
//more efficient to build heap by passing array of edges
MinHeap <Edge> pq = new MinHeap <Edge>();
foreach (Edge e in g.Edges())
{
pq.Insert(e);
}
//rum greedy algorithm
UF uf = new UF(g.V);
while (!pq.IsEmpty() && mst.Count < g.V - 1)
{
Edge e = pq.Pop();
int v = e.Either();
int w = e.Other(v);
if (!uf.IsConnected(v, w))
{
//v-w does not create a cycle
uf.Union(v, w);
mst.Enqueue(e);
weight += e.Weight();
}
}
}
/// <summary>
/// check optimality conditions (takes time proportional EVlgV)
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
public bool Check(EdgeWeightedGraph g)
{
//check total weight
double total = 0.0;
foreach (var edge in Edges())
{
total += edge.Weight();
}
double EPSILON = 1E-12;
if (System.Math.Abs(total - Weight()) > EPSILON)
{
//weight of edges must equal to weight()
return(false);
}
//check that it is acyclic
UF uf = new UF(g.V);
foreach (var edge in Edges())
{
int v = edge.Either();
int w = edge.Other(v);
if (uf.IsConnected(v, w))
{
return(false);
}
uf.Union(w, v);
}
//check that it is a spanning forest
foreach (var edge in Edges())
{
int v = edge.Either();
int w = edge.Other(v);
if (!uf.IsConnected(w, v))
{
return(false);
}
}
//check that is is a minimal spanning forest(cut optimality conditions0
foreach (var edge in Edges())
{
int v = edge.Either();
int w = edge.Other(v);
//all edges in mst except e
uf = new UF(g.V);
foreach (var f in mst)
{
int x = f.Either();
int y = f.Other(x);
if (f != edge)
{
uf.Union(x, y);
}
}
foreach (var f in g.Edges())
{
int x = f.Either();
int y = f.Other(x);
if (!uf.IsConnected(x, y))
{
if (f.Weight() < edge.Weight())
{
return(false);
}
}
}
}
return(true);
}
