private readonly IndexMinPQ<Double> _pq; // priority queue of vertices
#endregion Fields
#region Constructors
/// <summary>
/// Computes a shortest-paths tree from the source vertex <tt>s</tt> to every
/// other vertex in the edge-weighted graph <tt>G</tt>.
/// </summary>
/// <param name="g">g the edge-weighted graph</param>
/// <param name="s">s the source vertex</param>
/// <exception cref="ArgumentException">if an edge weight is negative</exception>
/// <exception cref="ArgumentException">unless 0 <= <tt>s</tt> <= <tt>V</tt> - 1</exception>
public DijkstraUndirectedSP(EdgeWeightedGraph g, int s)
{
foreach (var e in g.Edges())
{
if (e.Weight < 0)
throw new ArgumentException($"edge {e} has negative weight");
}
_distTo = new double[g.V];
_edgeTo = new EdgeW[g.V];
for (var v = 0; v < g.V; v++)
_distTo[v] = double.PositiveInfinity;
_distTo[s] = 0.0;
// relax vertices in order of distance from s
_pq = new IndexMinPQ<Double>(g.V);
_pq.Insert(s, _distTo[s]);
while (!_pq.IsEmpty())
{
var v = _pq.DelMin();
foreach (var e in g.Adj(v))
Relax(e, v);
}
// check optimality conditions
//assert check(G, s);
}
/// <summary>
/// Compute a minimum spanning tree (or forest) of an edge-weighted graph.
/// </summary>
/// <param name="g">g the edge-weighted graph</param>
public KruskalMST(EdgeWeightedGraph g)
{
// more efficient to build heap by passing array of edges
var pq = new MinPQ<EdgeW>();
foreach (var e in g.Edges())
{
pq.Insert(e);
}
// run greedy algorithm
var uf = new UF(g.V);
while (!pq.IsEmpty() && _mst.Size() < g.V - 1)
{
var e = pq.DelMin();
var v = e.Either();
var w = e.Other(v);
if (!uf.Connected(v, w))
{ // v-w does not create a cycle
uf.Union(v, w); // merge v and w components
_mst.Enqueue(e); // add edge e to mst
_weight += e.Weight;
}
}
// check optimality conditions
//assert check(G);
}
private readonly IndexMinPQ <Double> _pq; // priority queue of vertices
/// <summary>
/// Computes a shortest-paths tree from the source vertex <tt>s</tt> to every
/// other vertex in the edge-weighted graph <tt>G</tt>.
/// </summary>
/// <param name="g">g the edge-weighted graph</param>
/// <param name="s">s the source vertex</param>
/// <exception cref="ArgumentException">if an edge weight is negative</exception>
/// <exception cref="ArgumentException">unless 0 <= <tt>s</tt> <= <tt>V</tt> - 1</exception>
public DijkstraUndirectedSP(EdgeWeightedGraph g, int s)
{
foreach (var e in g.Edges())
{
if (e.Weight < 0)
{
throw new ArgumentException($"edge {e} has negative weight");
}
}
_distTo = new double[g.V];
_edgeTo = new EdgeW[g.V];
for (var v = 0; v < g.V; v++)
{
_distTo[v] = double.PositiveInfinity;
}
_distTo[s] = 0.0;
// relax vertices in order of distance from s
_pq = new IndexMinPQ <Double>(g.V);
_pq.Insert(s, _distTo[s]);
while (!_pq.IsEmpty())
{
var v = _pq.DelMin();
foreach (var e in g.Adj(v))
{
Relax(e, v);
}
}
// check optimality conditions
//assert check(G, s);
}
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);
}
/// <summary>
/// check optimality conditions:
/// (i) for all edges e = v-w: distTo[w] <= distTo[v] + e.weight()
/// (ii) for all edge e = v-w on the SPT: distTo[w] == distTo[v] + e.weight()
/// </summary>
/// <param name="g"></param>
/// <param name="s"></param>
/// <returns></returns>
public bool Check(EdgeWeightedGraph g, int s)
{
// check that edge weights are nonnegative
if (g.Edges().Any(e => e.Weight < 0))
{
Console.Error.WriteLine("negative edge weight detected");
return false;
}
// check that distTo[v] and edgeTo[v] are consistent
if (Math.Abs(_distTo[s]) > 1E12 || _edgeTo[s] != null)
{
Console.Error.WriteLine("distTo[s] and edgeTo[s] inconsistent");
return false;
}
for (var v = 0; v < g.V; v++)
{
if (v == s) continue;
if (_edgeTo[v] == null && !double.IsPositiveInfinity(_distTo[v]))
{
Console.Error.WriteLine("distTo[] and edgeTo[] inconsistent");
return false;
}
}
// check that all edges e = v-w satisfy distTo[w] <= distTo[v] + e.weight()
for (var v = 0; v < g.V; v++)
{
foreach (var e in g.Adj(v))
{
var w = e.Other(v);
if (_distTo[v] + e.Weight < _distTo[w])
{
Console.Error.WriteLine($"edge {e} not relaxed");
return false;
}
}
}
// check that all edges e = v-w on SPT satisfy distTo[w] == distTo[v] + e.weight()
for (var w = 0; w < g.V; w++)
{
if (_edgeTo[w] == null) continue;
var e = _edgeTo[w];
if (w != e.Either() && w != e.Other(e.Either())) return false;
int v = e.Other(w);
if (Math.Abs(_distTo[v] + e.Weight - _distTo[w]) > 1E12)
{
Console.Error.WriteLine($"edge {e} on shortest path not tight");
return false;
}
}
return true;
}
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);
}
/// <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;
}
/// <summary>
/// check optimality conditions:
/// (i) for all edges e = v-w: distTo[w] <= distTo[v] + e.weight()
/// (ii) for all edge e = v-w on the SPT: distTo[w] == distTo[v] + e.weight()
/// </summary>
/// <param name="g"></param>
/// <param name="s"></param>
/// <returns></returns>
public bool Check(EdgeWeightedGraph g, int s)
{
// check that edge weights are nonnegative
if (g.Edges().Any(e => e.Weight < 0))
{
Console.Error.WriteLine("negative edge weight detected");
return(false);
}
// check that distTo[v] and edgeTo[v] are consistent
if (Math.Abs(_distTo[s]) > 1E12 || _edgeTo[s] != null)
{
Console.Error.WriteLine("distTo[s] and edgeTo[s] inconsistent");
return(false);
}
for (var v = 0; v < g.V; v++)
{
if (v == s)
{
continue;
}
if (_edgeTo[v] == null && !double.IsPositiveInfinity(_distTo[v]))
{
Console.Error.WriteLine("distTo[] and edgeTo[] inconsistent");
return(false);
}
}
// check that all edges e = v-w satisfy distTo[w] <= distTo[v] + e.weight()
for (var v = 0; v < g.V; v++)
{
foreach (var e in g.Adj(v))
{
var w = e.Other(v);
if (_distTo[v] + e.Weight < _distTo[w])
{
Console.Error.WriteLine($"edge {e} not relaxed");
return(false);
}
}
}
// check that all edges e = v-w on SPT satisfy distTo[w] == distTo[v] + e.weight()
for (var w = 0; w < g.V; w++)
{
if (_edgeTo[w] == null)
{
continue;
}
var e = _edgeTo[w];
if (w != e.Either() && w != e.Other(e.Either()))
{
return(false);
}
int v = e.Other(w);
if (Math.Abs(_distTo[v] + e.Weight - _distTo[w]) > 1E12)
{
Console.Error.WriteLine($"edge {e} on shortest path not tight");
return(false);
}
}
return(true);
}
/// <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);
}