public static ReductionResult RunTest(Graph graph)
{
var result = new ReductionResult();
List<int> bottlenecks = new List<int>();
for (int i = 0; i < graph.Terminals.Count; i++)
{
var pathToAll = Algorithms.DijkstraPathToAll(graph.Terminals[i], graph);
// Only add the maximum bottleneck of all paths
bottlenecks.Add(pathToAll.SelectMany(x => x.Value.Edges).Max(e => e.Cost));
}
int B = bottlenecks.Max();
List<Edge> redundant = new List<Edge>();
foreach (var edge in graph.Edges.Where(edge => edge.Cost > B))
redundant.Add(edge);
foreach (var edge in redundant)
{
graph.RemoveEdge(edge);
result.RemovedEdges.Add(edge);
}
return result;
}
public static ReductionResult RunTest(Graph graph)
{
var result = new ReductionResult();
var mst = Algorithms.Kruskal(graph.TerminalDistanceGraph);
var S = mst.Edges.Max(edge => edge.Cost);
foreach (var edge in graph.Edges.Where(e => e.Cost > S).ToList())
{
graph.RemoveEdge(edge);
result.RemovedEdges.Add(edge);
}
return result;
}
public static ReductionResult RunTest(Graph graph, int upperBound)
{
var result = new ReductionResult();
// T. Polzin, lemma 25: Vertex removal
int reductionBound = 0;
int allRadiusesExceptTwoMostExpensive = graph.Terminals.Select(graph.GetVoronoiRadiusForTerminal).OrderBy(x => x).Take(graph.Terminals.Count - 2).Sum();
HashSet<Vertex> removeVertices = new HashSet<Vertex>();
foreach (var vertex in graph.Vertices.Except(graph.Required))
{
var nearestTerminals = Algorithms.NearestTerminals(vertex, graph, 2);
int lowerBound = nearestTerminals.Sum(x => x.TotalCost) + allRadiusesExceptTwoMostExpensive;
if (lowerBound > upperBound)
removeVertices.Add(vertex);
else if (lowerBound > reductionBound)
reductionBound = lowerBound;
}
foreach (var removeVertex in removeVertices)
{
graph.RemoveVertex(removeVertex);
result.RemovedVertices.Add(removeVertex);
}
// Check for disconnected components (and remove those that not contain any terminals)
var componentTable = graph.CreateComponentTable();
var terminalComponents = new HashSet<int>();
foreach (var vertex in graph.Terminals)
terminalComponents.Add(componentTable[vertex]);
foreach (var vertex in graph.Vertices.Where(x => !terminalComponents.Contains(componentTable[x])).ToList())
{
graph.RemoveVertex(vertex);
result.RemovedVertices.Add(vertex);
}
// T. Polzin, lemma 26: edge removal
allRadiusesExceptTwoMostExpensive = graph.Terminals.Select(graph.GetVoronoiRadiusForTerminal).OrderBy(x => x).Take(graph.Terminals.Count - 2).Sum();
HashSet<Edge> removeEdges = new HashSet<Edge>();
Dictionary<Vertex, int> distancesToBase = new Dictionary<Vertex, int>();
foreach (var terminal in graph.Terminals)
{
var toAll = Algorithms.DijkstraToAll(terminal, graph);
foreach (var vertex in graph.Vertices)
{
if (vertex == terminal)
continue;
if (!distancesToBase.ContainsKey(vertex))
distancesToBase.Add(vertex, toAll[vertex]);
else if (toAll[vertex] < distancesToBase[vertex])
distancesToBase[vertex] = toAll[vertex];
}
}
foreach (var edge in graph.Edges)
{
var v1 = edge.Either();
var v2 = edge.Other(v1);
if (graph.Terminals.Contains(v1) || graph.Terminals.Contains(v2))
continue;
var v1z1 = distancesToBase[v1];
var v2z2 = distancesToBase[v2];
var lowerBound = edge.Cost + v1z1 + v2z2 + allRadiusesExceptTwoMostExpensive;
if (lowerBound > upperBound)
removeEdges.Add(edge);
else if (lowerBound > reductionBound)
reductionBound = lowerBound;
}
foreach (var removeEdge in removeEdges)
{
graph.RemoveEdge(removeEdge);
result.RemovedEdges.Add(removeEdge);
}
result.ReductionUpperBound = reductionBound;
return result;
}
/// <summary>
/// Runs an approximation of the Special Distance Test to reduce the graph.
/// This test runs much faster and offers only a small difference in performance.
/// </summary>
/// <param name="graph">The graph on which to run the test.</param>
/// <returns>The reduced graph.</returns>
public static ReductionResult RunTest(Graph graph)
{
if (!graph.Terminals.All(graph.ContainsVertex))
Debugger.Break();
var tmst = Algorithms.Kruskal(graph.TerminalDistanceGraph);
var terminalSpecialDistances = new MultiDictionary<Vertex, int>();
for (int i = 0; i < graph.Terminals.Count - 1; i++)
{
var tFrom = graph.Terminals[i];
var toAll = Algorithms.DijkstraPathToAll(tFrom, tmst);
for (int j = i + 1; j < graph.Terminals.Count; j++)
{
var tTo = graph.Terminals[j];
var path = toAll[tTo];
var sd = path.Edges.Max(x => x.Cost);
terminalSpecialDistances.Add(tFrom, tTo, sd);
}
}
var result = new ReductionResult();
// Find all special distances between terminals
int count = 0;
int e = graph.NumberOfEdges;
Dictionary<Vertex, Path> nearest = new Dictionary<Vertex, Path>();
HashSet<Edge> remove = new HashSet<Edge>();
int edgesWithoutRemoval = 0;
foreach (var edge in graph.Edges.OrderByDescending(x => x.Cost))
{
count++;
var vFrom = edge.Either();
var vTo = edge.Other(vFrom);
int SDEstimate = int.MaxValue;
Path pathToNearestFrom = null;
if (nearest.ContainsKey(vFrom))
pathToNearestFrom = nearest[vFrom];
else
{
pathToNearestFrom = Algorithms.NearestTerminal(vFrom, graph);
nearest.Add(vFrom, pathToNearestFrom);
}
var aNearestTerminalFrom = pathToNearestFrom.End;
Path pathToNearestTo = null;
if (nearest.ContainsKey(vTo))
pathToNearestTo = nearest[vTo];
else
{
pathToNearestTo = Algorithms.NearestTerminal(vTo, graph);
nearest.Add(vTo, pathToNearestTo);
}
var bNearestTerminalTo = pathToNearestTo.End;
// SD = Max( dist(v, z_a), dist(w, z_b), sd(z_a, z_b) )
var sd = Math.Max(pathToNearestFrom.TotalCost, pathToNearestTo.TotalCost);
if (aNearestTerminalFrom != bNearestTerminalTo)
{
var sdTerminals = terminalSpecialDistances[aNearestTerminalFrom, bNearestTerminalTo];
sd = Math.Max(sd, sdTerminals);
}
if (sd < SDEstimate)
SDEstimate = sd;
if (edge.Cost > SDEstimate)
{
edgesWithoutRemoval = 0;
remove.Add(edge);
}
else if (++edgesWithoutRemoval >= graph.NumberOfEdges / 100) // Expecting a 1% reduction
{
break;
}
}
foreach (var edge in remove)
{
graph.RemoveEdge(edge);
result.RemovedEdges.Add(edge);
}
return result;
}
private void AddEdgesToMST(Graph mst, List<Edge> edges)
{
foreach (var edge in edges)
{
if (edge.WhereBoth(x => mst.GetDegree(x) > 0)) //Both vertices of this edge are in the MST, introducing this edge creates cycle!
{
var v1 = edge.Either();
var v2 = edge.Other(v1);
var components = mst.CreateComponentTable();
if (components[v1] == components[v2])
{
// Both are in the same component, and a path exists.
// Travel the path to see if adding this edge makes it cheaper
var path = Algorithms.DijkstraPath(v1, v2, mst);
// However, we can remove a set of edges between two nodes
// When going from T1 to T3
// E.g.: T1 - v2 - v3 - v4 - T2 - v5 - v6 - T3
// The edges between T1, v2, v3, v4, T2 cost more than the path between T1 and T3.
// Removing those edges, and adding the new edge from T1 to T3, also connects
// T1 to T2, so still a tree!
var from = path.Start;
var last = path.Start;
int betweenTerminals = 0;
var subtractedPath = new Path(path.Start);
List<Edge> edgesInSubtraction = new List<Edge>();
Dictionary<Edge, List<Edge>> subtractions = new Dictionary<Edge, List<Edge>>();
for (int i = 0; i < path.Edges.Count; i++)
{
var pe = path.Edges[i];
betweenTerminals += pe.Cost;
last = pe.Other(last);
edgesInSubtraction.Add(pe);
if (mst.GetDegree(last) > 2 || mst.Terminals.Contains(last) || i == path.Edges.Count - 1)
{
var subtractedEdge = new Edge(from, last, betweenTerminals);
subtractions.Add(subtractedEdge, edgesInSubtraction);
edgesInSubtraction = new List<Edge>();
subtractedPath.Edges.Add(subtractedEdge);
from = last;
betweenTerminals = 0;
}
}
var mostCostly = subtractedPath.Edges[0];
for (int i = 1; i < subtractedPath.Edges.Count; i++)
{
if (subtractedPath.Edges[i].Cost > mostCostly.Cost)
mostCostly = subtractedPath.Edges[i];
}
if (mostCostly.Cost >= edge.Cost)
{
foreach (var e in subtractions[mostCostly])
mst.RemoveEdge(e, false);
mst.AddEdge(edge);
}
}
else // Connect the two disconnected components!
mst.AddEdge(edge);
}
else
mst.AddEdge(edge);
}
}
/// <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 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;
}
