/// <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);
}
/// <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);
}
/// <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;
}