private static IEnumerable<List<Edge>> AllTriangles(Graph graph)
{
List<Vertex> vertices = new List<Vertex>(graph.Vertices);
vertices.Sort((x, y) => graph.GetDegree(x).CompareTo(graph.GetDegree(y)));
vertices.Reverse(); // From highest to lowest
Dictionary<Vertex, int> index = new Dictionary<Vertex, int>();
Dictionary<Vertex, List<Vertex>> A = new Dictionary<Vertex, List<Vertex>>();
for (int i = 0; i < vertices.Count; i++)
{
index.Add(vertices[i], i);
A.Add(graph.Vertices[i], new List<Vertex>());
}
foreach (var firstVertex in vertices)
{
foreach (var firstEdge in graph.GetEdgesForVertex(firstVertex))
{
var secondVertex = firstEdge.Other(firstVertex);
if (index[firstVertex] < index[secondVertex])
{
foreach (var thirdVertex in A[firstVertex].Intersect(A[secondVertex]))
{
var secondEdge =
graph.GetEdgesForVertex(firstVertex).Single(x => x.Other(firstVertex) == thirdVertex);
var thirdEdge =
graph.GetEdgesForVertex(secondVertex).Single(x => x.Other(secondVertex) == thirdVertex);
yield return new List<Edge>(new [] { firstEdge, secondEdge, thirdEdge });
}
A[secondVertex].Add(firstVertex);
}
}
}
}
/// <summary>
/// Runs the Degree Test to reduce the graph.
/// </summary>
/// <param name="graph">The graph on which to run the test.</param>
/// <returns>The reduced graph.</returns>
public static ReductionResult RunTest(Graph graph)
{
// Use some simple rules to reduce the problem.
// The rules are: - if a vertex v has degree 1 and is not part of the required nodes, remove the vertex.
// - if a vertex v has degree 1 and is part of the required nodes, the one edge it is
// connected to has to be in the solution, and as a consequence, the node at the other side
// is either a Steiner node or also required.
// - (not implemented yet) if a vertex v is required and has degree 2, and thus is connected to 2 edges,
// namely e1 and e2, and cost(e1) < cost(e2) and e1 = (u, v) and u is
// also a required vertex, then every solution must contain e1
var result = new ReductionResult();
// Remove leaves as long as there are any
var leaves = graph.Vertices.Where(v => graph.GetDegree(v) == 1 && !graph.Terminals.Contains(v)).ToList();
while (leaves.Count > 0)
{
foreach (var leaf in leaves)
graph.RemoveVertex(leaf);
result.RemovedVertices.AddRange(leaves);
leaves = graph.Vertices.Where(v => graph.GetDegree(v) == 1 && !graph.Terminals.Contains(v)).ToList();
}
// When a leaf is required, add the node on the other side of its one edge to required nodes
var requiredLeaves = graph.Vertices.Where(v => graph.GetDegree(v) == 1 && graph.Terminals.Contains(v)).ToList();
foreach (var requiredLeaf in requiredLeaves)
{
// Find the edge this leaf is connected to
var alsoRequired = graph.GetEdgesForVertex(requiredLeaf).Single().Other(requiredLeaf);
if (!graph.Terminals.Contains(alsoRequired))
graph.RequiredSteinerNodes.Add(alsoRequired);
}
return result;
}
public static int Dijkstra(Vertex from, Vertex to, Graph graph)
{
HashSet<Vertex> visited = new HashSet<Vertex>();
Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node> nodes = new Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node>();
FibonacciHeap<int, Vertex> labels = new FibonacciHeap<int, Vertex>();
// Initialize labels.
foreach (var vertex in graph.Vertices)
{
var n = labels.Add(vertex == from ? 0 : int.MaxValue, vertex);
nodes.Add(vertex, n);
}
int currentLabel = int.MaxValue;
while (!visited.Contains(to))
{
var currentNode = labels.ExtractMin();
var current = currentNode.Value;
currentLabel = currentNode.Key;
// Consider all edges ending in unvisited neighbours
var edges =
graph.GetEdgesForVertex(current).Where(x => !visited.Contains(x.Other(current)));
// Update labels on the other end.
foreach (var edge in edges)
{
if (currentNode.Key + edge.Cost < nodes[edge.Other(current)].Key)
labels.DecreaseKey(nodes[edge.Other(current)], currentNode.Key + edge.Cost);
}
visited.Add(current);
}
return currentLabel;
}
public static Path DijkstraPath(Vertex from, Vertex to, Graph graph)
{
HashSet<Vertex> visited = new HashSet<Vertex>();
Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node> nodes =
new Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node>();
Dictionary<Vertex, Edge> comingFrom = new Dictionary<Vertex, Edge>();
FibonacciHeap<int, Vertex> labels = new FibonacciHeap<int, Vertex>();
// Initialize labels.
foreach (var vertex in graph.Vertices)
{
var n = labels.Add(vertex == from ? 0 : int.MaxValue, vertex);
nodes.Add(vertex, n);
comingFrom.Add(vertex, null);
}
while (!visited.Contains(to))
{
var currentNode = labels.ExtractMin();
var current = currentNode.Value;
// Consider all edges ending in unvisited neighbours
var edges =
graph.GetEdgesForVertex(current).Where(x => !visited.Contains(x.Other(current)));
// Update labels on the other end.
foreach (var edge in edges)
{
if (currentNode.Key + edge.Cost < nodes[edge.Other(current)].Key)
{
labels.DecreaseKey(nodes[edge.Other(current)], currentNode.Key + edge.Cost);
comingFrom[edge.Other(current)] = edge;
}
}
visited.Add(current);
}
// Now travel back, to find the actual path
List<Edge> pathEdges = new List<Edge>();
Vertex pathVertex = to;
while (pathVertex != from)
{
pathEdges.Add(comingFrom[pathVertex]);
pathVertex = comingFrom[pathVertex].Other(pathVertex);
}
pathEdges.Reverse();
Path path = new Path(from);
path.Edges.AddRange(pathEdges);
return path;
}
private void ReconnectTerminals(Graph workingSolution, Graph problemInstance, Graph problemInstanceDistance)
{
Stopwatch reconnectStopwatch = new Stopwatch();
reconnectStopwatch.Start();
var components = workingSolution.CreateComponentTable();
var componentsToConnect =
components.Where(x => problemInstance.Terminals.Contains(x.Key))
.Select(x => x.Value)
.Distinct()
.ToList();
if (componentsToConnect.Count <= 1) return;
MultiDictionary<int, Tuple<Vertex, Vertex>> componentConnectingPathDictionary = new MultiDictionary<int, Tuple<Vertex, Vertex>>();
List<Vertex> componentGraphVertices = new List<Vertex>();
foreach (var i in componentsToConnect)
componentGraphVertices.Add(new Vertex(i));
Graph componentGraph = new Graph(componentGraphVertices);
for (int i = 0; i < componentsToConnect.Count; i++)
{
int fromComponent = componentsToConnect[i];
for (int j = i + 1; j < componentsToConnect.Count; j++)
{
int toComponent = componentsToConnect[j];
int minDistance = int.MaxValue;
foreach (var fromVertex in components.Where(x => x.Value == fromComponent)
.Select(x => x.Key))
{
var distanceEdges = problemInstanceDistance.GetEdgesForVertex(fromVertex);
foreach (var toVertexEdge in distanceEdges)
{
var toVertex = toVertexEdge.Other(fromVertex);
if (components[toVertex] != toComponent)
continue;
int distance = toVertexEdge.Cost;
if (!componentConnectingPathDictionary.ContainsKey(fromComponent, toComponent))
{
componentConnectingPathDictionary.Add(fromComponent, toComponent, new Tuple<Vertex, Vertex>(fromVertex, toVertex));
componentGraph.AddEdge(new Edge(componentGraphVertices[i], componentGraphVertices[j], minDistance));
}
if (distance < minDistance)
{
minDistance = distance;
componentConnectingPathDictionary[fromComponent, toComponent] = new Tuple<Vertex, Vertex>(fromVertex, toVertex);
componentGraph.GetEdgesForVertex(componentGraphVertices[i])
.Single(x => x.Other(componentGraphVertices[i]) == componentGraphVertices[j])
.Cost = minDistance;
}
}
}
}
}
componentGraph = Algorithms.Kruskal(componentGraph);
foreach (var edge in componentGraph.Edges)
{
var v1 = edge.Either();
var v2 = edge.Other(v1);
var vertices = componentConnectingPathDictionary[v1.VertexName, v2.VertexName];
var path = Algorithms.DijkstraPath(vertices.Item1, vertices.Item2, problemInstance);
foreach (var pathEdge in path.Edges)
workingSolution.AddEdge(pathEdge);
}
reconnectStopwatch.Stop();
}
/// <summary>
/// Runs the Special Distance Test to reduce the graph.
/// </summary>
/// <param name="graph">The graph on which to run the test.</param>
/// <returns>The reduced graph.</returns>
public static ReductionResult RunTest(Graph graph)
{
List<Edge> redundant = new List<Edge>();
var result = new ReductionResult();
var distanceGraph = graph.CreateDistanceGraph();
var specialDistanceGraph = CreateInitialSpecialDistanceGraph(graph);
for (int i = 0; i < graph.NumberOfVertices; i++)
{
Console.Write("SD Test: {0}/{1} \r", i, graph.NumberOfVertices);
// Step 1. L := { i }
// for all j set delta_j = d_ij
// In each iteration, delta_j means the current special distance from the start vertex to j.
var vFrom = graph.Vertices[i];
List<Vertex> L = new List<Vertex>(new [] { vFrom }); // Initially only the start veretx is labeled.
// Initially set special distance equal to distance
foreach (var edge in specialDistanceGraph.GetEdgesForVertex(vFrom))
{
edge.Cost =
distanceGraph.GetEdgesForVertex(vFrom).Single(x => x.Other(vFrom) == edge.Other(vFrom)).Cost;
//edge.Cost = Math.Min(edge.Cost,
// distanceGraph.GetEdgesForVertex(vFrom).Single(x => x.Other(vFrom) == edge.Other(vFrom)).Cost);
}
List<Vertex> unhandledTerminals = null; // K \ L
while ((unhandledTerminals = graph.Terminals.Where(x => !L.Contains(x)).ToList()).Count > 0)
{ // While K \ L is not empty
// Find the terminal which minimizes delta(j) for all j in K \ L
int currentMinimum = int.MaxValue;
Vertex k = null;
var edgesFrom = specialDistanceGraph.GetEdgesForVertex(vFrom);
foreach (var terminal in unhandledTerminals)
{
var deltaEdge = edgesFrom.First(x => x.Other(vFrom) == terminal);
if (deltaEdge.Cost < currentMinimum)
{
currentMinimum = deltaEdge.Cost;
k = terminal;
}
}
L.Add(k);
// Re-lable all vertices that haven't gotten a definitive label yet.
var delta_k = edgesFrom.First(x => x.Other(vFrom) == k).Cost;
foreach (var unlabeled in graph.Vertices.Where(x => !L.Contains(x)))
{
var d_kj = distanceGraph.GetEdgesForVertex(k).First(x => x.Other(k) == unlabeled).Cost;
var deltaEdge = edgesFrom.First(x => x.Other(vFrom) == unlabeled);
deltaEdge.Cost = Math.Min(deltaEdge.Cost, Math.Max(delta_k, d_kj));
}
}
var specialEdges = specialDistanceGraph.GetEdgesForVertex(vFrom);
var distanceEdges = distanceGraph.GetEdgesForVertex(vFrom);
var edges = graph.GetEdgesForVertex(vFrom);
foreach (var redundantEdge in specialEdges.Where(x => x.Cost < distanceEdges.First(y => y.Other(vFrom) == x.Other(vFrom)).Cost))
{
// Special distance is smaller than distance. Edge is redundant.
var edge = edges.FirstOrDefault(x => x.Other(vFrom) == redundantEdge.Other(vFrom));
if (edge != null)
redundant.Add(edge);
}
}
foreach (var edge in redundant)
{
graph.RemoveEdge(edge);
result.RemovedEdges.Add(edge);
}
Console.Write(" \r");
return result;
}
public static Dictionary<Vertex, Path> DijkstraPathToAll(Vertex from, Graph graph, bool onlyTerminals)
{
Dictionary<Vertex, Edge> comingFrom = new Dictionary<Vertex, Edge>();
HashSet<Vertex> visited = new HashSet<Vertex>();
Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node> nodes =
new Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node>();
Dictionary<Vertex, Path> paths = new Dictionary<Vertex, Path>();
FibonacciHeap<int, Vertex> labels = new FibonacciHeap<int, Vertex>();
// Initialize labels.
foreach (var vertex in graph.Vertices)
{
var n = labels.Add(vertex == from ? 0 : int.MaxValue, vertex);
nodes.Add(vertex, n);
}
while (paths.Count < (onlyTerminals ? graph.Terminals.Count - 1 : graph.NumberOfVertices - 1))
{
var currentNode = labels.ExtractMin();
var current = currentNode.Value;
// Consider all edges ending in unvisited neighbours
var edges =
graph.GetEdgesForVertex(current).Where(x => !visited.Contains(x.Other(current)));
// Update labels on the other end.
foreach (var edge in edges)
{
if (currentNode.Key + edge.Cost < nodes[edge.Other(current)].Key)
{
labels.DecreaseKey(nodes[edge.Other(current)], currentNode.Key + edge.Cost);
comingFrom[edge.Other(current)] = edge;
}
}
visited.Add(current);
if (current != from && (!onlyTerminals || graph.Terminals.Contains(current)))
{
// Travel back the path
List<Edge> pathEdges = new List<Edge>();
Vertex pathVertex = current;
while (pathVertex != from)
{
pathEdges.Add(comingFrom[pathVertex]);
pathVertex = comingFrom[pathVertex].Other(pathVertex);
}
pathEdges.Reverse();
Path path = new Path(from);
path.Edges.AddRange(pathEdges);
paths[current] = path;
}
}
return paths;
}
private static bool ExistsPath(Vertex v1, Vertex v2, Graph graph, List<Vertex> visited)
{
foreach (var edge in graph.GetEdgesForVertex(v1))
{
if (edge.Other(v1) == v2)
{
return true;
}
if (!visited.Contains(edge.Other(v1)))
{
var visited2 = visited.ToList();
visited2.Add(v1);
if (ExistsPath(edge.Other(v1), v2, graph, visited2))
return true;
}
}
return false;
}
public static List<Path> NearestTerminals(Vertex from, Graph graph, int n)
{
List<Path> foundPaths = new List<Path>();
HashSet<Vertex> visited = new HashSet<Vertex>();
Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node> nodes = new Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node>();
FibonacciHeap<int, Vertex> labels = new FibonacciHeap<int, Vertex>();
Dictionary<Vertex, Edge> comingFrom = new Dictionary<Vertex, Edge>();
if (graph.Terminals.Contains(from))
foundPaths.Add(new Path(from));
// Initialize labels.
foreach (var vertex in graph.Vertices)
{
var node = labels.Add(vertex == from ? 0 : int.MaxValue, vertex);
nodes.Add(vertex, node);
comingFrom.Add(vertex, null);
}
while (!labels.IsEmpty() && foundPaths.Count < n)
{
var currentNode = labels.ExtractMin();
var current = currentNode.Value;
// Consider all edges ending in unvisited neighbours
var edges =
graph.GetEdgesForVertex(current).Where(x => !visited.Contains(x.Other(current)));
// Update labels on the other end.
foreach (var edge in edges)
{
if (currentNode.Key + edge.Cost < nodes[edge.Other(current)].Key)
{
labels.DecreaseKey(nodes[edge.Other(current)], currentNode.Key + edge.Cost);
comingFrom[edge.Other(current)] = edge;
}
}
visited.Add(current);
if (graph.Terminals.Contains(current) && current != from)
{
// Now travel back, to find the actual path
List<Edge> pathEdges = new List<Edge>();
Vertex pathVertex = current;
while (pathVertex != from)
{
pathEdges.Add(comingFrom[pathVertex]);
pathVertex = comingFrom[pathVertex].Other(pathVertex);
}
pathEdges.Reverse();
Path path = new Path(from);
path.Edges.AddRange(pathEdges);
foundPaths.Add(path);
}
}
return foundPaths;
}
private static Dictionary<int, double> CalculateComponentFunctionValues(Graph graph,
Dictionary<Vertex, int> components)
{
Dictionary<int, double> fValues = new Dictionary<int, double>();
foreach (var componentNumber in components.Values.Distinct())
{
//Get all edges in this component
HashSet<Edge> edgesInComponent = new HashSet<Edge>();
var verticesInComponent = components.Where(x => x.Value == componentNumber).Select(x => x.Key);
var nrOfTerminals = verticesInComponent.Count(x => graph.Terminals.Contains(x));
foreach (var v in verticesInComponent)
{
foreach (var e in graph.GetEdgesForVertex(v))
edgesInComponent.Add(e);
}
if (edgesInComponent.Count > 0)
{
double fVal = edgesInComponent.Sum(x => x.Cost)/Math.Pow(nrOfTerminals, 2);
fValues.Add(componentNumber, fVal);
}
else
fValues.Add(componentNumber, double.MaxValue);
}
return fValues;
}
public static Graph RunSolver(Graph graph)
{
var solution = new Graph(graph.Vertices);
DijkstraState state = new DijkstraState();
// Create the states needed for every execution of the Dijkstra algorithm
foreach (var terminal in graph.Terminals)
state.AddVertexToInterleavingDijkstra(terminal, graph);
// Initialize
Vertex currentVertex = state.GetNextVertex();
FibonacciHeap<int, Vertex> labels = state.GetLabelsFibonacciHeap();
HashSet<Vertex> visited = state.GetVisitedHashSet();
Dictionary<Vertex, Path> paths = state.GetPathsFound();
Dictionary<Vertex, FibonacciHeap<int, Vertex>.Node> nodes = state.GetNodeMapping();
Dictionary<Vertex, Edge> comingFrom = state.GetComingFromDictionary();
Dictionary<Vertex, int> components = solution.CreateComponentTable();
Dictionary<Vertex, double> terminalFValues = CreateInitialFValuesTable(graph);
int maxLoopsNeeded = graph.Terminals.Count * graph.NumberOfVertices;
int loopsDone = 0;
int updateInterval = 100;
int longestPath = graph.Terminals.Max(x => Algorithms.DijkstraToAll(x, graph).Max(y => y.Value));
while (state.GetLowestLabelVertex() != null)
{
if (loopsDone % updateInterval == 0)
Console.Write("\rRunning IDA... {0:0.0}% \r", 100.0 * loopsDone / maxLoopsNeeded);
loopsDone++;
if (state.GetLowestLabelVertex() != currentVertex)
{
// Interleave. Switch to the Dijkstra procedure of the vertex which currently has the lowest distance.
state.SetLabelsFibonacciHeap(labels);
state.SetVisitedHashSet(visited);
state.SetPathsFound(paths);
state.SetComingFromDictionary(comingFrom);
currentVertex = state.GetNextVertex();
labels = state.GetLabelsFibonacciHeap();
visited = state.GetVisitedHashSet();
paths = state.GetPathsFound();
nodes = state.GetNodeMapping();
comingFrom = state.GetComingFromDictionary();
}
// Do one loop in Dijkstra algorithm
var currentNode = labels.ExtractMin();
var current = currentNode.Value;
if (currentNode.Key > longestPath / 2)
break; //Travelled across the half of longest distance. No use in going further.
// Consider all edges ending in unvisited neighbours
var edges = graph.GetEdgesForVertex(current).Where(x => !visited.Contains(x.Other(current)));
// Update labels on the other end
foreach (var edge in edges)
{
if (currentNode.Key + edge.Cost < nodes[edge.Other(current)].Key)
{
labels.DecreaseKey(nodes[edge.Other(current)], currentNode.Key + edge.Cost);
comingFrom[edge.Other(current)] = edge;
}
}
visited.Add(current);
if (current != currentVertex)
{
// Travel back the new path
List<Edge> pathEdges = new List<Edge>();
Vertex pathVertex = current;
while (pathVertex != currentVertex)
{
pathEdges.Add(comingFrom[pathVertex]);
pathVertex = comingFrom[pathVertex].Other(pathVertex);
}
pathEdges.Reverse();
Path path = new Path(currentVertex);
path.Edges.AddRange(pathEdges);
paths[current] = path;
}
// Find matching endpoints from two different terminals
var mutualEnd = state.FindPathsEndingInThisVertex(current);
if (mutualEnd.Count() > 1)
{
var terminals = mutualEnd.Select(x => x.Start).ToList();
// Step 1. Calculate new heuristic function value for this shared point.
// f(x) = (Cost^2)/(NumberOfTerminals^3)
var f1 = Math.Pow(mutualEnd.Sum(p => p.TotalCost), 2) / Math.Pow(terminals.Count, 3);
var f2 = Math.Pow(mutualEnd.Sum(p => p.TotalCost), 1) / Math.Pow(terminals.Count, 2);
var f3 = Math.Pow(mutualEnd.Sum(p => p.TotalCost), 3) / Math.Pow(terminals.Count, 2);
var terminalsAvgF = terminals.Select(x => terminalFValues[x]).Average();
var terminalsMinF = terminals.Select(x => terminalFValues[x]).Min();
var f = (new[] { f1, f2, f3 }).Max();
Debug.WriteLine("F value: {0}, Fmin: {3} - Connecting terminals: {1} via {2}", f, string.Join(", ", terminals.Select(x => x.VertexName)), current.VertexName, terminalsMinF);
// Do not proceed if f > avgF AND working in same component
if (terminals.Select(x => components[x]).Distinct().Count() == 1 && f > terminalsMinF)
continue;
Debug.WriteLine("Proceeding with connection...");
// Step 2. Disconnect terminals in mutual component.
foreach (var group in terminals.GroupBy(x => components[x]))
{
if (group.Count() <= 1)
continue;
HashSet<Edge> remove = new HashSet<Edge>();
var sameComponentTerminals = group.ToList();
for (int i = 0; i < sameComponentTerminals.Count-1; i++)
{
for (int j = i+1; j< sameComponentTerminals.Count; j++)
{
var removePath = Algorithms.DijkstraPath(sameComponentTerminals[i], sameComponentTerminals[j], solution);
foreach (var e in removePath.Edges)
remove.Add(e);
}
}
foreach (var e in remove)
solution.RemoveEdge(e, false);
}
components = solution.CreateComponentTable();
// Step 3. Reconnect all now disconnected terminals via shared endpoint
foreach (var t in terminals)
{
var path = Algorithms.DijkstraPath(t, current, graph);
foreach (var edge in path.Edges)
solution.AddEdge(edge);
// Update f value
terminalFValues[t] = f;
}
components = solution.CreateComponentTable();
}
}
// If this solution is connected, take MST
if (graph.Terminals.Select(x => components[x]).Distinct().Count() == 1)
{
// Clean up!
foreach (var vertex in solution.Vertices.Where(x => solution.GetDegree(x) == 0).ToList())
solution.RemoveVertex(vertex);
int componentNumber = graph.Terminals.Select(x => components[x]).Distinct().Single();
foreach (var vertex in components.Where(x => x.Value != componentNumber).Select(x => x.Key).ToList())
solution.RemoveVertex(vertex);
solution = Algorithms.Kruskal(solution);
return solution;
}
// If the solution is not connected, it is not a good solution.
return null;
}
