private LinkedQueue <Edge> mst = new LinkedQueue <Edge>(); // edges in MST
/// <summary>
/// Compute a minimum spanning tree (or forest) of an edge-weighted graph.</summary>
/// <param name="G">the edge-weighted graph</param>
///
public KruskalMST(EdgeWeightedGraph G)
{
// more efficient to build heap by passing array of edges
MinPQ <Edge> pq = new MinPQ <Edge>();
foreach (Edge e in G.Edges())
{
pq.Insert(e);
}
// run greedy algorithm
UF uf = new UF(G.V);
while (!pq.IsEmpty && mst.Count < G.V - 1)
{
Edge e = pq.DelMin();
int v = e.Either;
int w = e.Other(v);
if (!uf.Connected(v, w))
{ // v-w does not create a cycle
uf.Union(v, w); // merge v and w components
mst.Enqueue(e); // add edge e to mst
weight += e.Weight;
}
}
// check optimality conditions
Debug.Assert(check(G));
}
private double weight; // weight of MST
/// <summary>
/// Compute a minimum spanning tree (or forest) of an edge-weighted graph.</summary>
/// <param name="G">the edge-weighted graph</param>
///
public BoruvkaMST(EdgeWeightedGraph G)
{
UF uf = new UF(G.V);
// repeat at most log V times or until we have V-1 edges
for (int t = 1; t < G.V && mst.Count < G.V - 1; t = t + t)
{
// foreach tree in forest, find closest edge
// if edge weights are equal, ties are broken in favor of first edge in G.edges()
Edge[] closest = new Edge[G.V];
foreach (Edge e in G.Edges())
{
int v = e.Either, w = e.Other(v);
int i = uf.Find(v), j = uf.Find(w);
if (i == j)
{
continue; // same tree
}
if (closest[i] == null || less(e, closest[i]))
{
closest[i] = e;
}
if (closest[j] == null || less(e, closest[j]))
{
closest[j] = e;
}
}
// add newly discovered edges to MST
for (int i = 0; i < G.V; i++)
{
Edge e = closest[i];
if (e != null)
{
int v = e.Either, w = e.Other(v);
// don't add the same edge twice
if (!uf.Connected(v, w))
{
mst.Add(e);
weight += e.Weight;
uf.Union(v, w);
}
}
}
}
// check optimality conditions
Debug.Assert(check(G));
}
public static void MainTest(string[] args)
{
TextInput StdIn = new TextInput();
int N = StdIn.ReadInt();
UF uf = new UF(N);
while (!StdIn.IsEmpty)
{
int p = StdIn.ReadInt();
int q = StdIn.ReadInt();
if (uf.Connected(p, q))
{
continue;
}
uf.Union(p, q);
Console.WriteLine(p + " " + q);
}
Console.WriteLine(uf.Count + " components");
}
// check optimality conditions (takes time proportional to E V lg* V)
private bool check(EdgeWeightedGraph G)
{
// check total weight
double total = 0.0;
foreach (Edge e in Edges())
{
total += e.Weight;
}
if (Math.Abs(total - Weight) > FLOATING_POINT_EPSILON)
{
Console.Error.Write("Weight of edges does not equal Weight: {0} vs. {0}\n", total, Weight);
return(false);
}
// check that it is acyclic
UF uf = new UF(G.V);
foreach (Edge e in Edges())
{
int v = e.Either, w = e.Other(v);
if (uf.Connected(v, w))
{
Console.Error.WriteLine("Not a forest");
return(false);
}
uf.Union(v, w);
}
// check that it is a spanning forest
foreach (Edge e in G.Edges())
{
int v = e.Either, w = e.Other(v);
if (!uf.Connected(v, w))
{
Console.Error.WriteLine("Not a spanning forest");
return(false);
}
}
// check that it is a minimal spanning forest (cut optimality conditions)
foreach (Edge e in Edges())
{
// all edges in MST except e
uf = new UF(G.V);
foreach (Edge f in mst)
{
int x = f.Either, y = f.Other(x);
if (f != e)
{
uf.Union(x, y);
}
}
// check that e is min weight edge in crossing cut
foreach (Edge f in G.Edges())
{
int x = f.Either, y = f.Other(x);
if (!uf.Connected(x, y))
{
if (f.Weight < e.Weight)
{
Console.Error.WriteLine("Edge " + f + " violates cut optimality conditions");
return(false);
}
}
}
}
return(true);
}
