public static Potential <int> GetUnnamedFamily2Potential(int n, int m)
{
if (n <= 1)
{
throw new ArgumentOutOfRangeException(nameof(n));
}
if (m <= 2)
{
throw new ArgumentOutOfRangeException(nameof(m)); // Not sure about smaller m; they might work
}
var ngonArrows = Enumerable.Range(0, n).Select(k => n - k).Select(k => new Arrow <int>(k % n, k - 1));
var ngonCycle = new DetachedCycle <int>(new Path <int>(ngonArrows));
var cycleDict = new Dictionary <DetachedCycle <int>, int>()
{
{ ngonCycle, -1 }
};
foreach (var k in Enumerable.Range(0, n))
{
var mgonArrows = Enumerable.Range(0, m - 2).Select(j => new Arrow <int>(k + j * n, k + (j + 1) * n))
.AppendElement(new Arrow <int>(k + (m - 2) * n, (k + 1) % n))
.AppendElement(new Arrow <int>((k + 1) % n, k));
var mgonCycle = new DetachedCycle <int>(new Path <int>(mgonArrows));
cycleDict[mgonCycle] = 1;
}
return(new Potential <int>(cycleDict));
}
public void NeqOperator_OnNull()
{
DetachedCycle <int> nullCycle = null;
var cycle = CreateCycle(1, new Arrow <int>(1, 2), new Arrow <int>(2, 3), new Arrow <int>(3, 1));
Assert.That(nullCycle != null, Is.False);
Assert.That(cycle != null, Is.True);
}
public static Potential <int> GetCyclePotential(int n)
{
if (n <= 1)
{
throw new ArgumentOutOfRangeException(nameof(n));
}
var arrows = Enumerable.Range(1, n - 1).Select(k => new Arrow <int>(k, k + 1)).AppendElement(new Arrow <int>(n, 1));
var cycle = new DetachedCycle <int>(new Path <int>(1, arrows));
var cycleDict = new Dictionary <DetachedCycle <int>, int>()
{
{ cycle, 1 },
};
return(new Potential <int>(cycleDict));
}
public static Potential <int> GetUnnamedFamily1Potential(int n)
{
if (n <= 1)
{
throw new ArgumentOutOfRangeException(nameof(n));
}
var ngonArrows = Enumerable.Range(0, n).Select(k => n - k).Select(k => new Arrow <int>(k % n, k - 1));
var ngonCycle = new DetachedCycle <int>(new Path <int>(ngonArrows));
var cycleDict = new Dictionary <DetachedCycle <int>, int>()
{
{ ngonCycle, -1 }
};
foreach (var k in Enumerable.Range(0, n))
{
var triangleArrows = new Arrow <int>[] { new Arrow <int>(k, k + n), new Arrow <int>(k + n, (k + 1) % n), new Arrow <int>((k + 1) % n, k) };
var triangleCycle = new DetachedCycle <int>(new Path <int>(triangleArrows));
cycleDict[triangleCycle] = 1;
}
return(new Potential <int>(cycleDict));
}
static Potential <int> CreatePotential(DetachedCycle <int> cycle, int coefficient)
{
return(new Potential <int>(cycle, coefficient));
}
private IEnumerable <IEnumerable <TVertex> > SearchForFaces <TVertex>(IReadOnlyUndirectedGraph <TVertex> graph, Orientation searchOrientation)
where TVertex : IEquatable <TVertex>, IComparable <TVertex>, IVertexInPlane
{
// Algorithm (outline):
// Keep track of traversed edges *and the direction in which the edge was traversed*
// (i.e., keep track of traversed directed edges).
//
// For every directed edge (not already traversed), make a search:
// Keep track of the path of edges traversed.
// Start by traversing that edge.
// Keep turning "maximally" in the direction given by the search orientation until a vertex is visited twice.
// This gives a full path that decomposes into a path and a cycle.
// The path always consists of only boundary directed edges (boundary w.r.t. the search orientation),
// while the cycle bounds a (bounded) face precisely when its orientation agrees with the search orientation.
// Remark: Sort of bad to use Arrow here (writing a DirectedEdge class with a common interface would be a proper solution),
// but it allows so much code to be reused.
var cycles = new HashSet <DetachedCycle <TVertex> >();
var directedEdges = graph.Edges.SelectMany(e => new Arrow <TVertex>[] { new Arrow <TVertex>(e.Vertex1, e.Vertex2), new Arrow <TVertex>(e.Vertex2, e.Vertex1) });
var remainingEdges = new HashSet <Arrow <TVertex> >(directedEdges);
var remainingEdgesStack = new Stack <Arrow <TVertex> >(directedEdges);
while (remainingEdges.Count > 0)
{
Arrow <TVertex> startEdge;
while (!remainingEdges.Contains(startEdge = remainingEdgesStack.Pop()))
{
;
}
var path = new Path <TVertex>(startEdge.Source, startEdge.Target);
var prevVertex = startEdge.Source;
var curVertex = startEdge.Target;
// Begin the search for this start edge!
while (true)
{
// Terminate the search if we have found a cycle
if (path.TryExtractTrailingCycle(out var closedPath, out int startIndex))
{
var cycleOrientation = graph.GetOrientationOfCycle(closedPath.Vertices);
// If the orientations agree, then the cycle is a bounding cycle for a (bounded) face
// Else, it bounds the unbounded face (which we do not care about)
if (cycleOrientation == searchOrientation)
{
var detachedCycle = new DetachedCycle <TVertex>(closedPath);
// When restarting with a boundary arrow (directed edge), we might get the same cycle again
if (!cycles.Contains(detachedCycle))
{
cycles.Add(new DetachedCycle <TVertex>(closedPath));
}
}
foreach (var edge in path.Arrows)
{
remainingEdges.Remove(edge);
}
break;
}
// Get next successor (next in the angle sense)
// If no successor, terminate the search (all arrows are direction-boundary arrows)
// Else, update path, prevVertex and curVertex and do next iteration
var neighbors = graph.AdjacencyLists[curVertex]; // Note that this includes prevVertex
if (neighbors.Count == 1) // The last edge was a dead end!
{
foreach (var arrow in path.Arrows)
{
remainingEdges.Remove(arrow);
}
break;
}
var baseVertex = curVertex;
var basePos = baseVertex.Position;
// Sort the vertices by angle so that the vertex following the predecessor vertex (prevVertex) is
// the vertex corresponding to a "maximal" turn.
var neighborsSortedByAngle = searchOrientation == Orientation.Clockwise ?
neighbors.OrderBy(vertex => vertex.Position, new AngleBasedPointComparer(basePos)) :
neighbors.OrderByDescending(vertex => vertex.Position, new AngleBasedPointComparer(basePos));
var neighborsSortedByAngleList = new CircularList <TVertex>(neighborsSortedByAngle);
int predecessorIndex = neighborsSortedByAngleList.IndexOf(prevVertex);
neighborsSortedByAngleList.RotateLeft(predecessorIndex);
var nextVertex = neighborsSortedByAngleList.Skip(1).First();
path = path.AppendVertex(nextVertex);
prevVertex = curVertex;
curVertex = nextVertex;
}
}
return(cycles.Select(cycle => cycle.CanonicalPath.Vertices));
}