private readonly Collections.Queue <EdgeW> _mst = new Collections.Queue <EdgeW>(); // edges in MST
/// <summary>
/// Compute a minimum spanning tree (or forest) of an edge-weighted graph.
/// </summary>
/// <param name="g">g the edge-weighted graph</param>
public KruskalMST(EdgeWeightedGraph g)
{
// more efficient to build heap by passing array of edges
var pq = new MinPQ <EdgeW>();
foreach (var e in g.Edges())
{
pq.Insert(e);
}
// run greedy algorithm
var uf = new UF(g.V);
while (!pq.IsEmpty() && _mst.Size() < g.V - 1)
{
var e = pq.DelMin();
var v = e.Either();
var 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
//assert check(G);
}
private readonly double _weight; // weight of MST
/// <summary>
/// Compute a minimum spanning tree (or forest) of an edge-weighted graph.
/// </summary>
/// <param name="g">g the edge-weighted graph</param>
public BoruvkaMST(EdgeWeightedGraph g)
{
var uf = new UF(g.V);
// repeat at most log V times or until we have V-1 edges
for (var t = 1; t < g.V && _mst.Size() < 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()
var closest = new EdgeW[g.V];
foreach (var 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 (var i = 0; i < g.V; i++)
{
var 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
//assert check(G);
}
private UF Union(Tuple <int, int>[] links, int N)
{
UF uf = new UF(N);
foreach (var pair in links)
{
int p = pair.Item1;
int q = pair.Item2;
if (uf.Connected(p, q))
{
continue;
}
uf.Union(pair.Item1, pair.Item2);
}
return(uf);
}
public KruskalMST(EdgeWeightedGraph g)
{
_mst = new Queue <Edge>();
var pq = new MinPQ <Edge>(g.E);
foreach (var e in g.Edges())
{
pq.Insert(e);
}
var uf = new UF(g.V);
while (!pq.IsEmpty && _mst.Count < g.V - 1)
{
var e = (Edge)pq.DeleteMin();
int v = e.Either(), w = e.Other(v);
if (uf.Connected(v, w))
{
continue;
}
uf.Union(v, w);
_mst.Enqueue(e);
}
}
public KruskalMST(IEdgeWeightGraph G)
{
mst = new Chapter1.Queue <IEdge>();
MinPQ <IEdge> pq = new MinPQ <IEdge>();
foreach (var e in G.Edges())
{
pq.Insert(e);
}
UF uf = new UF(G.V);
while (!pq.IsEmpty && mst.Size < G.V - 1)
{
IEdge e = pq.Delete(); //找到权重最小的边
int v = e.Either, w = e.Other(v);
if (uf.Connected(v, w)) //忽略失效的边
{
continue;
}
uf.Union(v, w);
mst.Enqueue(e);
}
}
/// <summary>
/// check optimality conditions (takes time proportional to E V lg* V)
/// </summary>
/// <param name="g"></param>
/// <returns></returns>
public bool Check(EdgeWeightedGraph g)
{
// check total weight
var total = Edges().Sum(e => e.Weight);
if (Math.Abs(total - Weight()) > FLOATING_POINT_EPSILON)
{
Console.Error.WriteLine($"Weight of edges does not equal weight(): {total} vs. {Weight()}{Environment.NewLine}");
return(false);
}
// check that it is acyclic
var uf = new UF(g.V);
foreach (var e in Edges())
{
int v = e.Either(), w = e.Other(v);
if (uf.Connected(v, w))
{
Console.Error.WriteLine("Not a forest");
return(false);
}
uf.Union(v, w);
}
// check that it is a spanning forest
foreach (var e in g.Edges())
{
int v = e.Either(), w = e.Other(v);
if (!uf.Connected(v, w))
{
Console.Error.WriteLine("Not a spanning forest");
return(false);
}
}
// check that it is a minimal spanning forest (cut optimality conditions)
foreach (var e in Edges())
{
// all edges in MST except e
uf = new UF(g.V);
foreach (var f in _mst)
{
int x = f.Either(), y = f.Other(x);
if (f != e)
{
uf.Union(x, y);
}
}
// check that e is min weight edge in crossing cut
foreach (var f in g.Edges())
{
int x = f.Either(), y = f.Other(x);
if (!uf.Connected(x, y))
{
if (f.Weight < e.Weight)
{
Console.Error.WriteLine($"Edge {f} violates cut optimality conditions");
return(false);
}
}
}
}
return(true);
}
