/// <summary>
/// Writes the edges of the passed graph to the console.
/// </summary>
/// <param name="graph">
/// Graph to print.
/// </param>
private static void PrintGraph(IWeightedGraph<IntVertex, FredmanTarjanEdge> graph)
{
foreach (var edge in
graph.Edges.Where(edge => edge.Source < edge.Target))
{
Debug.WriteLine("{0} ---> {1}: {2}", edge.Source, edge.Target, edge.Weight);
}
}
public static IWeightedGraph <TV, TW> BuildResidualGraph <TV, TW>(IWeightedGraph <TV, TW> graph,
Func <TW, TW, TW> substractFlowValues,
TW zeroValue) where TV : IEquatable <TV> where TW : IComparable
{
Dictionary <TV, Vertex <TV> > verteces = new Dictionary <TV, Vertex <TV> >();
foreach (var vertex in graph.GetAllVerteces())
{
verteces.Add(vertex.Value, new Vertex <TV>(vertex.Value));
}
GraphBuilder <TV, TW> builder = new GraphBuilder <TV, TW>(true).AddVerteces(verteces.Values);
foreach (IWeightedEdge <TV, TW> edge in graph.GetAllEdges())
{
IWeightedDirectedEdge <TV, TW> directedEdge = (IWeightedDirectedEdge <TV, TW>)edge;
TW currentFlow = directedEdge.GetAttribute <TW>(ATTR_FLOW);
TW maxFlow = directedEdge.Weight;
TW residualFlow = substractFlowValues(maxFlow, currentFlow);
// Add forward edge with residual flow value as capacity
if (residualFlow.CompareTo(zeroValue) > 0)
{
Edge <TV, TW> forwardEdge = new Edge <TV, TW>(verteces[directedEdge.OriginVertex.Value],
verteces[directedEdge.TargetVertex.Value],
residualFlow);
forwardEdge.SetAttribute(ATTR_DIRECTION, EdgeDirection.Forward);
builder.AddEdge(forwardEdge);
}
// Add backward edge with current flow as capacity
if (currentFlow.CompareTo(zeroValue) > 0)
{
Edge <TV, TW> backwardEdge = new Edge <TV, TW>(verteces[directedEdge.TargetVertex.Value],
verteces[directedEdge.OriginVertex.Value], currentFlow);
backwardEdge.SetAttribute(ATTR_DIRECTION, EdgeDirection.Backward);
builder.AddEdge(backwardEdge);
}
}
return(builder.Build());
}
public static ImmutableArray <TNode> UniformCostSearch <TNode>(
this IWeightedGraph <TNode> graph,
TNode start,
TNode goal) where TNode : notnull, IEquatable <TNode>
{
var frontier = new PriorityQueue <TNode, float>();
frontier.Enqueue(start, 0);
var cameFrom = new Dictionary <TNode, TNode>()
{
[start] = start
};
var costSoFar = new Dictionary <TNode, float>()
{
[start] = 0
};
while (frontier.TryDequeue(out var current, out _))
{
if (current.Equals(goal))
{
break;
}
foreach (var next in graph.Neigbours(current))
{
var newCost = costSoFar[current] + graph.Cost(current, next);
if (costSoFar.TryGetValue(next, out var nextCost) is false || newCost < nextCost)
{
costSoFar[next] = newCost;
var priority = newCost;
frontier.Enqueue(next, priority);
cameFrom[next] = current;
}
}
}
return(ReconstructPath(start, goal, cameFrom));
}
/// <summary>
/// Finds the cheapest path between two nodes of the specified graph.
/// </summary>
/// <typeparam name="T">The type of the graph's nodes.</typeparam>
/// <param name="graph">A graph of nodes.</param>
/// <param name="start">The starting node.</param>
/// <param name="goal">The goal node to find a path to.</param>
/// <param name="heuristic">An optional heuristic to determine the cost of moving from one node to another.</param>
/// <returns>
/// The minimum cost to traverse the graph from <paramref name="start"/> to <paramref name="goal"/>.
/// </returns>
public static long AStar <T>(IWeightedGraph <T> graph, T start, T goal, Func <T, T, long>?heuristic = default)
where T : notnull
{
heuristic ??= (x, y) => graph.Cost(x, y);
var frontier = new PriorityQueue <T, long>();
frontier.Enqueue(start, 0);
var costSoFar = new Dictionary <T, long>()
{
[start] = 0
};
while (frontier.Count > 0)
{
T current = frontier.Dequeue();
if (current.Equals(goal))
{
break;
}
foreach (T next in graph.Neighbors(current))
{
long newCost = costSoFar[current] + heuristic(current, next);
if (!costSoFar.TryGetValue(next, out long otherCost) || newCost < otherCost)
{
costSoFar[next] = newCost;
long goalCost = heuristic(next, goal);
long priority = newCost + goalCost;
frontier.Enqueue(next, priority);
}
}
}
return(costSoFar.GetValueOrDefault(goal, long.MaxValue));
}
public LazyPrimeMST(IWeightedGraph <Weight> graph)
{
this.g = graph;
this.marked = new bool[graph.V()];
this.ipq = new MinHeap <Edge <Weight> >(graph.E()); // 边的个数
this.path = new List <Edge <Weight> >();
//Lazy prime算法
visit(0); //从0开始遍历
while (!ipq.IsEmpty())
{
Edge <Weight> minEdge = ipq.ExtractMin();
// 如果两个点在同一分区， 则边不符合跨区
if (marked[minEdge.V()] == marked[minEdge.W()])
{
continue;
}
// 把符合的边加入到最小生成树MST
path.Add(minEdge);
if (!marked[minEdge.V()])
{
visit(minEdge.V());
}
else
{
visit(minEdge.W());
}
}
dynamic wt = default(Weight);
for (int i = 0; i < path.Count; i++)
{
wt += path[i].Wt();
}
this.wt = (Weight)wt;
}
private static void UpdatePseudoBalance <TV, TW>(IWeightedGraph <TV, TW> graph,
IVertex <TV> vertex,
Func <TW, TW, TW> combineValues,
Func <TW, TW, TW> substractValues,
TW zeroValue)
where TV : IEquatable <TV> where TW : IComparable
{
TW pseudoBalance = zeroValue;
// Add flow values of outgoing edges
foreach (IWeightedEdge <TV, TW> edge in graph.GetEdgesOfVertex(vertex))
{
if (edge is IWeightedDirectedEdge <TV, TW> directedEdge)
{
pseudoBalance = combineValues(pseudoBalance, edge.GetAttribute <TW>(Constants.FLOW));
}
else
{
throw new GraphNotDirectedException();
}
}
// Add flow values of incoming edges
foreach (IWeightedEdge <TV, TW> edge in graph.GetAllEdges())
{
if (edge is IWeightedDirectedEdge <TV, TW> directedEdge)
{
if (directedEdge.TargetVertex == vertex)
{
pseudoBalance = substractValues(pseudoBalance, edge.GetAttribute <TW>(Constants.FLOW));
}
}
else
{
throw new GraphNotDirectedException();
}
}
vertex.SetAttribute(Constants.PSEUDO_BALANCE, pseudoBalance);
}
public static IWeightedGraph <TV, TW> BuildResidualGraph <TV, TW>(IWeightedGraph <TV, TW> graph,
Func <TW, TW, TW> substractValues,
Func <TW, TW> negateValue,
TW zeroValue)
where TV : IEquatable <TV> where TW : IComparable
{
// Build residual graph as done in Edmonds-Karp
IWeightedGraph <TV, TW> residualGraph = EdmondsKarp.BuildResidualGraph(graph, substractValues, zeroValue);
// Add costs
foreach (IWeightedEdge <TV, TW> edge in graph.GetAllEdges())
{
if (edge is IWeightedDirectedEdge <TV, TW> directedEdge)
{
TW costs = edge.GetAttribute <TW>(Constants.COSTS);
// Add costs to forward edges
if (residualGraph.AreVertecesConnected(directedEdge.OriginVertex.Value, directedEdge.TargetVertex.Value))
{
residualGraph.GetEdgeBetweenVerteces(directedEdge.OriginVertex.Value, directedEdge.TargetVertex.Value)
.SetAttribute(Constants.COSTS, costs);
}
// Add negative costs to backward edges
if (residualGraph.AreVertecesConnected(directedEdge.TargetVertex.Value, directedEdge.OriginVertex.Value))
{
residualGraph.GetEdgeBetweenVerteces(directedEdge.TargetVertex.Value, directedEdge.OriginVertex.Value)
.SetAttribute(Constants.COSTS, negateValue(costs));
}
}
else
{
throw new GraphNotDirectedException();
}
}
// Done - return residual graph
return(residualGraph);
}
public static bool search <T>(IWeightedGraph <T> graph, T start, T goal, out Dictionary <T, T> cameFrom)
{
var foundPath = false;
cameFrom = new Dictionary <T, T>();
cameFrom.Add(start, start);
var costSoFar = new Dictionary <T, int>();
var frontier = new PriorityQueue <AStarNode <T> >(1000);
frontier.Enqueue(new AStarNode <T>(start), 0);
costSoFar[start] = 0;
while (frontier.Count > 0)
{
var current = frontier.Dequeue();
if (current.data.Equals(goal))
{
foundPath = true;
break;
}
foreach (var next in graph.getNeighbors(current.data))
{
var newCost = costSoFar[current.data] + graph.cost(current.data, next);
if (!costSoFar.ContainsKey(next) || newCost < costSoFar[next])
{
costSoFar[next] = newCost;
var priority = newCost + graph.heuristic(next, goal);
frontier.Enqueue(new AStarNode <T>(next), priority);
cameFrom[next] = current.data;
}
}
}
return(foundPath);
}
private static int BreadthFirstSearch(
IWeightedGraph graph,
Vertex source,
Vertex target,
IWeightedGraph legalFlows,
out IDictionary<Vertex, Vertex> path)
{
path = new Dictionary<Vertex, Vertex>();
IDictionary<Vertex, int> pathCapacity = new Dictionary<Vertex, int>();
path[source] = null; // make sure source is not rediscovered
pathCapacity[source] = int.MaxValue;
Queue<Vertex> queue = new Queue<Vertex>();
queue.Enqueue(source);
while(queue.Count > 0)
{
Vertex u = queue.Dequeue();
foreach (Vertex v in graph.Neighbors(u))
{
// if there is available capacity between u and v,
// ... and v is not seen before in search
if (graph.GetCapacity(u, v) - legalFlows.GetCapacity(u, v) > 0 &&
path.ContainsKey(v) == false)
{
path[v] = u;
pathCapacity[v] = Math.Min(
pathCapacity[u],
graph.GetCapacity(u, v) - legalFlows.GetCapacity(u ,v));
if (!v.Equals(target)) queue.Enqueue(v);
else return pathCapacity[target];
}
}
}
return 0;
}
public PrimMST(IWeightedGraph <Weight> g)
{
this.g = g;
marked = new bool[g.V()];
ipq = new IndexMinHeap <Weight>(g.V());
edgeTo = new List <Edge <Weight> >(g.V());
for (int i = 0; i < g.V(); i++)
{
edgeTo.Insert(i, null);
}
mst = new List <Edge <Weight> >();
// Prim 算法
visit(0);
while (!ipq.IsEmpty())
{
// 去除最小的边
int minV = ipq.GetMinIndex();
ipq.ExtractMin();
Edge <Weight> minEdge = edgeTo[minV];
if (minEdge != null)
{
mst.Add(minEdge);
visit(minV);
}
}
dynamic weight = mst[0].Wt();
for (int i = 1; i < mst.Count; i++)
{
weight += mst[i].Wt();
}
this.mstWeight = (Weight)weight;
}
public static bool Search <T>(IWeightedGraph <T> graph, T start, T goal, out Dictionary <T, T> cameFrom)
{
var foundPath = false;
cameFrom = new Dictionary <T, T>();
cameFrom.Add(start, start);
var costSoFar = new Dictionary <T, float>();
var frontier = new SimplePriorityQueue <T>();
costSoFar[start] = 0;
frontier.Enqueue(start, 0);
while (frontier.Count > 0)
{
var current = frontier.Dequeue();
if (current.Equals(goal))
{
foundPath = true;
break;
}
foreach (var next in graph.GetNeighbors(current))
{
var newCost = costSoFar[current] + graph.Cost(current, next);
if (!costSoFar.ContainsKey(next) || newCost < costSoFar[next])
{
costSoFar[next] = newCost;
frontier.Enqueue(next, newCost);
cameFrom[next] = current;
}
}
}
return(foundPath);
}
private void TestTspAlgorithm(IWeightedGraph <int, double> graph,
TspAlgorithm algorithm,
int expectedVerteces,
int expectedEdges,
double optimalTourCosts = 0.0,
double precision = 0.01)
{
IWeightedGraph <int, double> tour = null;
// Compute tour with the chosen algorithm
switch (algorithm)
{
case TspAlgorithm.NearestNeighbor:
tour = NearestNeighbor.FindTour(graph);
break;
case TspAlgorithm.DoubleTree:
tour = DoubleTree.FindTour(graph);
break;
case TspAlgorithm.BruteForce:
tour = BruteForce.FindOptimalTour(graph, 0.0, double.MaxValue, (w1, w2) => w1 + w2);
break;
default:
throw new NotSupportedException($"Testing TSP with the {algorithm} algorithm is currently not supported.");
}
// Check route for component count
Assert.AreEqual(expectedVerteces, tour.VertexCount);
Assert.AreEqual(expectedEdges, tour.GetAllEdges().Count());
// For algorithms that find the optimal tour, check the optmial tour costs
if (algorithm == TspAlgorithm.BruteForce)
{
AssertDoublesNearlyEqual(optimalTourCosts, tour.GetAllEdges().Sum(e => e.Weight), precision);
}
}
private TimeSpan RunTest(string file, IWeightedGraph <int, double> graph, ALGORITHM algorithm)
{
Stopwatch sw = new Stopwatch();
switch (algorithm)
{
case ALGORITHM.KRUSKAL:
sw.Start();
Algorithms.MinimumSpanningTree.Kruskal.FindMinimumSpanningTree(graph);
sw.Stop();
break;
case ALGORITHM.PRIM:
sw.Start();
Algorithms.MinimumSpanningTree.Prim.FindMinimumSpanningTree(graph, 0, double.MaxValue);
sw.Stop();
break;
}
Logger.LogDebug($" - Computed MST for '{file}' using {algorithm}'s algorithm in {sw.Elapsed}.");
return(sw.Elapsed);
}
public static IWeightedGraph <TV, TW> FindTour <TV, TW>(IWeightedGraph <TV, TW> graph) where TV : IEquatable <TV>
{
GraphBuilder <TV, TW> builder = new GraphBuilder <TV, TW>();
builder.AddVerteces(graph.GetAllVerteces().Select(v => v.Value));
// Find minimum spanning tree
IWeightedGraph <TV, TW> msp = Kruskal.FindMinimumSpanningTree(graph);
// Connect verteces in the order of the MSP
IVertex <TV> lastVertex = null;
DepthFirstSearch.Traverse(msp, currentVertex =>
{
if (lastVertex != null)
{
try
{
builder.AddEdge(lastVertex.Value, currentVertex.Value, graph.GetEdgeBetweenVerteces(lastVertex.Value, currentVertex.Value).Weight);
}
catch (VertecesNotConnectedException <TV> exception)
{
throw new GraphNotCompleteException("The graph is not complete.", exception);
}
}
lastVertex = currentVertex;
});
// Add closing edge
IVertex <TV> firstVertex = msp.GetFirstVertex();
IWeightedEdge <TV, TW> closingEdge = graph.GetEdgeBetweenVerteces(lastVertex.Value, firstVertex.Value);
builder.AddEdge(lastVertex.Value, firstVertex.Value, closingEdge.Weight);
// Done!
return(builder.Build());
}
public AStarSearch(IWeightedGraph <ICoordinates> graph, ICoordinates start, ICoordinates goal)
{
PriorityQueue <ICoordinates> frontier = new PriorityQueue <ICoordinates> ();
frontier.enqueue(start, 0);
cameFrom [start] = start;
costSoFar [start] = 0;
while (frontier.getCount() > 0)
{
ICoordinates currentPosition = frontier.dequeue();
if (currentPosition.Equals(goal)) // Break loop when find goal
{
break;
}
foreach (var nextPosition in graph.getNeighbors(currentPosition))
{
if (nextPosition == null)
{
continue;
}
double cost = costSoFar[currentPosition]
+ graph.getCost(currentPosition, nextPosition);
if (!costSoFar.ContainsKey(nextPosition) ||
cost < costSoFar[nextPosition]) // Enqueue when find new nearest path to nextPosition
{
double priority = cost + heuristicFunction(currentPosition, nextPosition);
frontier.enqueue(nextPosition, priority);
cameFrom [nextPosition] = currentPosition;
costSoFar [nextPosition] = cost;
}
}
}
}
/// <summary>
/// Finds the maximum flow between a source and target node and returns a new weighted graph
/// representing the legal flows between vertices to generate the maximum flow.
/// </summary>
/// <remarks>
/// Edmonds-Karp Max Flow Algorithm, based on my C# port (git.io/KewoqQ)
/// of the Wikipedia psuedocode found here: (en.wikipedia.org/wiki/Edmonds%E2%80%93Karp_algorithm)
/// </remarks>
/// <param name="graph">a weighted graph representing the flow network</param>
/// <param name="source">the source node</param>
/// <param name="target">the target node</param>
/// <param name="legalFlows">an output graph representing legal flows</param>
/// <returns>the maximum flow between source and target</returns>
public static int FindMaxFlow(
this IWeightedGraph graph,
Vertex source,
Vertex target,
out IWeightedGraph legalFlows)
{
Graphs.Weighted.Mutable.IWeightedMutableGraph _legalFlows =
new Graphs.Weighted.Mutable.AdjacencyMatrixGraph(graph.Vertices().Count());
legalFlows = _legalFlows;
foreach (Vertex v in graph.Vertices())
{
_legalFlows.AddVertex(v);
}
int f = 0; // initial flow is 0
while (true)
{
IDictionary<Vertex, Vertex> path;
int capacity = BreadthFirstSearch(graph, source, target, legalFlows, out path);
if (capacity == 0) break;
f += capacity;
// backtrack search, and write flow
Vertex v = target;
while (!v.Equals(source))
{
Vertex u = path[v];
_legalFlows.SetEdge(u, v, _legalFlows.GetCapacity(u, v) + capacity);
_legalFlows.SetEdge(v, u, _legalFlows.GetCapacity(v, u) - capacity);
v = u;
}
}
return f;
}
public static IEnumerable <IEdge <TV> > FindMaxMatching <TV>(IGraph <TV> graph,
ISet <IVertex <TV> > set1,
ISet <IVertex <TV> > set2,
TV superSourceValue,
TV superTargetValue) where TV : IEquatable <TV>
{
// Build directed graph with super nodes and capacity 1
IWeightedGraph <TV, int> graphWithSuperNodes = BuildGraphWithSuperNodesAndCapacity(graph, superSourceValue, superTargetValue, set1, set2, 1);
IVertex <TV> sourceNode = graphWithSuperNodes.GetFirstMatchingVertex(v => v.Value.Equals(superSourceValue));
IVertex <TV> targetNode = graphWithSuperNodes.GetFirstMatchingVertex(v => v.Value.Equals(superTargetValue));
// Compute max flow in graph with super nodes
IWeightedGraph <TV, int> maxFlow = EdmondsKarp.FindMaxFlow(graphWithSuperNodes,
sourceNode,
targetNode,
(x, y) => x + y,
(x, y) => x - y,
0);
// Add edges with flow to matching
List <IEdge <TV> > matchings = new List <IEdge <TV> >();
foreach (IVertex <TV> vertex in set1)
{
foreach (IEdge <TV> edge in graph.GetEdgesOfVertex(vertex))
{
if (maxFlow.GetEdgeBetweenVerteces(vertex.Value, edge.ConnectedVertex(vertex).Value).Weight == 1)
{
matchings.Add(edge);
}
}
}
// Done - return matching!
return(matchings);
}
private void TestShortestPathsAlgorithm(IWeightedGraph <int, double> graph,
IVertex <int> source,
IVertex <int> target,
double pathCosts,
ShortestPathAlgorithm algorithm,
double precision = 0.1)
{
IWeightedGraph <int, double> shortestPath = null;
switch (algorithm)
{
case ShortestPathAlgorithm.Dijkstra:
shortestPath = Dijkstra.FindShortestPath(graph, source, target, 0.0, double.MaxValue, (x, y) => x + y);
break;
case ShortestPathAlgorithm.BellmanFordMoore:
shortestPath = BellmanFordMoore.FindShortestPath(graph, source, target, 0.0, double.MaxValue, (x, y) => x + y);
break;
default:
throw new NotSupportedException($"Testing shortest path for the {algorithm} algorithm is currently not supported.");
}
AssertDoublesNearlyEqual(pathCosts, shortestPath.GetAllEdges().Sum(e => e.Weight), precision);
}
public AStarSearch(IWeightedGraph <GraphPoint> graph, GraphPoint start, GraphPoint goal)
{
start.wType = GraphPoint.WType.start;
goal.wType = GraphPoint.WType.end;
var box = new Tools.PriorityQueue <GraphPoint>();
box.Enqueue(start, 0);
cameFrom[start] = start;
costSoFar[start] = 0;
while (box.Count > 0)
{
var current = box.Dequeue();
if (current.x == goal.x && current.y == goal.y)
{
cameFrom[goal] = current;
break;
}
foreach (var next in graph.Neighbors(current))
{
double newCost = costSoFar[current]
+ graph.Cost(current, next);
if (!costSoFar.ContainsKey(next) ||
newCost < costSoFar[next])
{
costSoFar[next] = newCost;
double priority = newCost + Heuristic(next, goal);
box.Enqueue(next, priority);
cameFrom[next] = current;
}
}
}
}
public ShortestPathSearch(IWeightedGraph graph)
{
this.graph = graph;
}
public static IWeightedGraph <TV, TW> FindTour <TV, TW>(IWeightedGraph <TV, TW> graph) where TV : IEquatable <TV>
{
return(FindTour(graph, graph.GetFirstVertex()));
}
public DepthFirstSearch(IWeightedGraph graph)
{
this.graph = graph;
}
public BreadthFirstSearch(IWeightedGraph graph)
{
this.graph = graph;
}
/// <summary>
/// Uses Bellman-Ford's algorithm to compute the shortest paths in a weighted graph. Returns a <see cref="WeightedGraphShortestPaths{TVertex, TWeight}"/> object containing the shortest paths. TWeight being the type of <see cref="short"/>.
/// </summary>
/// <typeparam name="TVertex">The data type of the vertices. TVertex implements <see cref="IComparable{T}"/>.</typeparam>
/// <param name="graph">The graph structure that implements <see cref="IWeightedGraph{TVertex, TWeight}"/> with TWeight being of type <see cref="short"/>.</param>
/// <param name="source">The source vertex for which the shortest paths are computed.</param>
/// <returns>Returns a <see cref="WeightedGraphShortestPaths{TVertex, TWeight}"/> containing the shortest paths for the given source vertex. TWeight being the type of <see cref="short"/>.</returns>
public static WeightedGraphShortestPaths <TVertex, short> BellmanFordShortestPaths <TVertex>(this IWeightedGraph <TVertex, short> graph, TVertex source)
where TVertex : IComparable <TVertex>
{
return(BellmanFordShortestPaths(graph, source, (x, y) => (short)(x + y)));
}
/// <summary>
/// Tries to compute the shortest paths in a weighted graph with the Bellman-Ford's algorithm. Returns true if successful; otherwise false.
/// </summary>
/// <typeparam name="TVertex">The data type of the vertices. TVertex implements <see cref="IComparable{T}"/>.</typeparam>
/// <param name="graph">The graph structure that implements <see cref="IWeightedGraph{TVertex, TWeight}"/> with TWeight being of type <see cref="short"/>.</param>
/// <param name="source">The source vertex for which the shortest paths are computed.</param>
/// <param name="shortestPaths">The <see cref="WeightedGraphShortestPaths{TVertex, TWeight}"/> object (TWeight being the type of <see cref="short"/>) that contains the shortest paths of the graph if the algorithm was successful; otherwise null.</param>
/// <returns>Returns true if the shortest paths were successfully computed; otherwise false. Also returns false if the given source vertex does not belong to the graph.</returns>
public static bool TryGetBellmanFordShortestPaths <TVertex>(this IWeightedGraph <TVertex, short> graph, TVertex source, out WeightedGraphShortestPaths <TVertex, short> shortestPaths)
where TVertex : IComparable <TVertex>
{
return(TryGetBellmanFordShortestPaths(graph, source, out shortestPaths, (x, y) => (short)(x + y)));
}
/// <summary>
/// Uses Bellman-Ford's algorithm to compute the shortest paths in a weighted graph. Returns a <see cref="WeightedGraphShortestPaths{TVertex, TWeight}"/> object containing the shortest paths.
/// </summary>
/// <typeparam name="TVertex">The data type of the vertices. TVertex implements <see cref="IComparable{T}"/>.</typeparam>
/// <typeparam name="TWeight">The data type of weight of the edges. TWeight implements <see cref="IComparable{T}"/>.</typeparam>
/// <param name="graph">The graph structure that implements <see cref="IWeightedGraph{TVertex, TWeight}"/>.</param>
/// <param name="source">The source vertex for which the shortest paths are computed.</param>
/// <param name="weightAdder">The method corresponding to the <see cref="AddWeights{TWeight}"/> delegate used for calculating the sum of two edge weights.</param>
/// <returns>Returns a <see cref="WeightedGraphShortestPaths{TVertex, TWeight}"/> containing the shortest paths for the given source vertex.</returns>
public static WeightedGraphShortestPaths <TVertex, TWeight> BellmanFordShortestPaths <TVertex, TWeight>(this IWeightedGraph <TVertex, TWeight> graph, TVertex source, AddWeights <TWeight> weightAdder)
where TVertex : IComparable <TVertex>
where TWeight : IComparable <TWeight>
{
if (!graph.ContainsVertex(source))
{
throw new ArgumentException("Vertex does not belong to the graph!");
}
WeightedGraphShortestPaths <TVertex, TWeight> shortestPaths;
if (TryGetBellmanFordShortestPaths(graph, source, out shortestPaths, weightAdder))
{
return(shortestPaths);
}
else
{
throw new InvalidOperationException("Negative weight cycle found!");
}
}
/// <summary>
/// Tries to compute the shortest paths in a weighted graph with the Bellman-Ford's algorithm. Returns true if successful; otherwise false.
/// </summary>
/// <typeparam name="TVertex">The data type of the vertices. TVertex implements <see cref="IComparable{T}"/>.</typeparam>
/// <typeparam name="TWeight">The data type of weight of the edges. TWeight implements <see cref="IComparable{T}"/>.</typeparam>
/// <param name="graph">The graph structure that implements <see cref="IWeightedGraph{TVertex, TWeight}"/>.</param>
/// <param name="source">The source vertex for which the shortest paths are computed.</param>
/// <param name="shortestPaths">The <see cref="WeightedGraphShortestPaths{TVertex, TWeight}"/> object that contains the shortest paths of the graph if the algorithm was successful; otherwise null.</param>
/// <param name="weightAdder">The method corresponding to the <see cref="AddWeights{TWeight}"/> delegate used for calculating the sum of two edge weights.</param>
/// <returns>Returns true if the shortest paths were successfully computed; otherwise false. Also returns false if the given source vertex does not belong to the graph.</returns>
public static bool TryGetBellmanFordShortestPaths <TVertex, TWeight>(this IWeightedGraph <TVertex, TWeight> graph, TVertex source, out WeightedGraphShortestPaths <TVertex, TWeight> shortestPaths, AddWeights <TWeight> weightAdder)
where TVertex : IComparable <TVertex>
where TWeight : IComparable <TWeight>
{
shortestPaths = null;
if (!graph.ContainsVertex(source))
{
return(false);
}
// A dictionary holding a vertex as key and as value a key-value pair of its previous vertex in the path(being the key) and the weight of the edge connecting them(being the value).
var previousVertices = new Dictionary <TVertex, KeyValuePair <TVertex, TWeight> >();
// The dictionary holding a vertex as key and as value a key-value pair holding the weight of the path from the source vertex(being the key) and the distance from the source vertex(being the value).
var weightAndDistance = new Dictionary <TVertex, KeyValuePair <TWeight, int> >();
// Add source vertex to computed weights and distances
weightAndDistance.Add(source, new KeyValuePair <TWeight, int>(default(TWeight), 0));
// Get all edges and sort them by their source and then by their destination vertex to create a consistent shortest paths output.
var allEdges = graph.Edges.ToList();
if (allEdges.Count > 0)
{
allEdges.QuickSort((x, y) =>
{
int cmp = x.Source.CompareTo(y.Source);
if (cmp == 0)
{
cmp = x.Destination.CompareTo(y.Destination);
}
return(cmp);
});
}
// We relax all edges n - 1 times and if at the n-th step a relaxation is needed we have a negative cycle
int lastStep = graph.VerticesCount - 1;
for (int i = 0; i < graph.VerticesCount; i++)
{
foreach (var edge in allEdges)
{
var edgeSource = edge.Source;
var edgeDestination = edge.Destination;
var edgeWeight = edge.Weight;
// If we have computed the path to the the source vertex of the edge
if (weightAndDistance.ContainsKey(edgeSource))
{
var wd = weightAndDistance[edgeSource];
TWeight curWeight = wd.Key;
int curDistance = wd.Value;
// If the edge destination is the source we continue with the next edge
if (object.Equals(source, edgeDestination))
{
continue;
}
// Compute path weight
TWeight pathWeight;
if (curWeight == null)
{
pathWeight = edgeWeight;
}
else
{
pathWeight = weightAdder(curWeight, edgeWeight);
}
// If this path is longer than an already computed path we continue with the next edge
if (weightAndDistance.ContainsKey(edgeDestination))
{
int cmp = pathWeight.CompareTo(weightAndDistance[edgeDestination].Key);
if (cmp > 0)
{
continue;
}
else if (cmp == 0)// if path is equal to an already computed path
{
// we continue with the next edge only if the distance is bigger or equal
if (curDistance + 1 >= weightAndDistance[edgeDestination].Value)
{
continue;
}
// if the distance is smaller we update the path with this one
}
}
if (i == lastStep)// if we need to relax on the last iteration we have a negative cycle
{
return(false);
}
// Else we save the path
previousVertices[edgeDestination] = new KeyValuePair <TVertex, TWeight>(edgeSource, edgeWeight);
weightAndDistance[edgeDestination] = new KeyValuePair <TWeight, int>(pathWeight, curDistance + 1);
}
}
}
// Remove source vertex from weight and distance dictionary
weightAndDistance.Remove(source);
shortestPaths = new WeightedGraphShortestPaths <TVertex, TWeight>(source, previousVertices, weightAndDistance);
return(true);
}
public DepthFirstSearch(IWeightedGraph graph)
{
this.graph = graph;
}
public L[] Search(L start, L goal, IWeightedGraph <L, C> graph)
{
return(Search(start, goal, graph, defaultHeuristic));
}
/// <summary>
/// Uses Kruskal's algorithm to find the minimum spanning tree of the undirected weighted graph. Returns a <see cref="WeightedALGraph{TVertex, TWeight}"/> representing the MST.
/// </summary>
/// <typeparam name="TVertex">The data type of the vertices. TVertex implements <see cref="IComparable{T}"/>.</typeparam>
/// <typeparam name="TWeight">The data type of weight of the edges. TWeight implements <see cref="IComparable{T}"/>.</typeparam>
/// <param name="graph">The graph structure that implements <see cref="IWeightedGraph{TVertex, TWeight}"/>.</param>
/// <returns>Returns a <see cref="WeightedALGraph{TVertex, TWeight}"/> representing the MST.</returns>
public static WeightedALGraph <TVertex, TWeight> KruskalMST <TVertex, TWeight>(this IWeightedGraph <TVertex, TWeight> graph)
where TVertex : IComparable <TVertex>
where TWeight : IComparable <TWeight>
{
if (graph.IsDirected)
{
throw new ArgumentException("Graph is directed!");
}
var mst = new WeightedALGraph <TVertex, TWeight>();
// A dictionary of the added vertices to the MST and their tree identifiers (first vertex from the tree)
var addedVertices = new Dictionary <TVertex, TVertex>();
// Edge comparer for the edge sorting
var edgeComparer = Comparer <IWeightedEdge <TVertex, TWeight> > .Create((x, y) =>
{
int cmp = 0;
// null checking because TWeight can be null
if (x.Weight == null)
{
if (y.Weight == null)
{
cmp = -1; // change cmp to skip next comparisons
}
else
{
return(int.MinValue);
}
}
if (cmp == 0)// if Weights are not both null
{
if (y.Weight == null)
{
return(int.MaxValue);
}
// normal check if both weights are not null
cmp = x.Weight.CompareTo(y.Weight);
}
else
{
cmp = 0; // if both Weights were null, compare the kvp value
}
if (cmp == 0)
{
cmp = x.Source.CompareTo(y.Source);
}
if (cmp == 0)
{
cmp = x.Destination.CompareTo(y.Destination);
}
return(cmp);
});
var vertices = graph.Vertices.ToList();
if (vertices.Count == 0)
{
return(mst);
}
mst.AddVertices(vertices);
// Get edges
var edges = graph.Edges.ToList();
if (edges.Count == 0)
{
return(mst);
}
// Sort them
edges.QuickSort(edgeComparer);
for (int i = 0; i < edges.Count; i++)
{
var curEdge = edges[i];
if (addedVertices.ContainsKey(curEdge.Source))
{
if (addedVertices.ContainsKey(curEdge.Destination))// if both vertices are added to the MST
{
// Find tree roots
var srcRoot = FindTreeRoot(addedVertices, curEdge.Source);
var dstRoot = FindTreeRoot(addedVertices, curEdge.Destination);
// If vertices belong to the same tree we continue. Edge will create a cycle.
if (object.Equals(srcRoot, dstRoot))
{
continue;
}
else // if not add edge and connect the trees
{
// Add edge to MST
mst.AddEdge(curEdge.Source, curEdge.Destination, curEdge.Weight);
// Connect destination vertex tree with the source
addedVertices[curEdge.Destination] = srcRoot;
addedVertices[dstRoot] = srcRoot;
}
}
else // if the source is added to the MST but not the destination
{
// Add edge to MST
mst.AddEdge(curEdge.Source, curEdge.Destination, curEdge.Weight);
// Find tree root
var srcRoot = FindTreeRoot(addedVertices, curEdge.Source);
// Add to the tree
addedVertices.Add(curEdge.Destination, srcRoot);
}
}
else // if the source is not added to the MST
{
if (addedVertices.ContainsKey(curEdge.Destination))// if the destination is added
{
// Add edge to MST
mst.AddEdge(curEdge.Source, curEdge.Destination, curEdge.Weight);
// Find tree root
var dstRoot = FindTreeRoot(addedVertices, curEdge.Destination);
// Add to the tree
addedVertices.Add(curEdge.Source, dstRoot);
}
else// if both vertices are not added to the MST
{
// Add edge to MST
mst.AddEdge(curEdge.Source, curEdge.Destination, curEdge.Weight);
// Connect the vertices in a tree
addedVertices.Add(curEdge.Source, curEdge.Source);
addedVertices.Add(curEdge.Destination, curEdge.Source);
}
}
}
// Return MST
return(mst);
}
public DijkstraSP(IWeightedGraph <V> g)
{
_g = g;
}
public static List <TNode> ShortestPath <TNode>(
IWeightedGraph <TNode> graph,
TNode sourceNode,
TNode targetNode,
int maxiterations = 1000
)
{
// Initialize values
Dictionary <TNode, double> distance = new Dictionary <TNode, double>();
Dictionary <TNode, TNode> previous = new Dictionary <TNode, TNode>();
// Instead of PriorityQueue
List <TNode> localNodes = new List <TNode>();
localNodes.Add(sourceNode);
distance.Add(sourceNode, 0);
// For iterations limit:
int i = 0;
while (localNodes.Count > 0 && i < maxiterations)
{
i++;
// Return and remove best vertex (that is, connection with minimum distance
TNode minNode = localNodes.OrderBy(n => distance[n]).First();
localNodes.Remove(minNode);
// Loop all connected nodes
// foreach (TNode neighbor in connections[minNode])
foreach (TNode neighbor in graph.Neighbors(minNode))
{
if (!distance.ContainsKey(neighbor))
{
localNodes.Add(neighbor);
distance.Add(neighbor, double.PositiveInfinity);
}
// The positive distance between node and it's neighbor, added to the distance of the current node
double dist = distance[minNode] + graph.Weight(minNode, neighbor);
if (dist < distance[neighbor])
{
distance[neighbor] = dist;
previous[neighbor] = minNode;
}
}
// If we're at the target node, break
if (graph.Equals(minNode, targetNode))
{
break;
}
}
// Construct a list containing the complete path. We'll start by looking at the previous node of the target and then making our way to the beginning.
// We'll reverse it to get a source->target list instead of the other way around. The source node is manually added.
List <TNode> result = new List <TNode>();
TNode target = targetNode;
while (previous.ContainsKey(target))
{
result.Add(target);
target = previous[target];
}
result.Add(sourceNode);
result.Reverse();
return(result);
}
public List <string> search(IWeightedGraph graph, object startVertex, object endVertex)
{
return(depthFirstSearch(graph, startVertex, endVertex));
}
/// <summary>
/// Initializes a new instance of the <see cref="BellmanFord"/> class.
/// </summary>
/// <param name="graph">The graph.</param>
/// <param name="startNode">The start node.</param>
public BellmanFordSearch([NotNull] IWeightedGraph graph, uint startNode)
: this(graph, startNode, startNode)
{
}
public ShortestPathSearch(IWeightedGraph graph)
{
this.graph = graph;
}
/**
* Finds a minimum spanning tree in the graph.
*
* @note The MST is computed using the Prim's (also knows as DJP)
* algorithm.
* TODO: describe the algorithm
*
* @param mst Unweighted tree instance that will become
* an MST in this method. The MST instance must
* already contain all the original's graph
* vertices.
*
* @return The weighted graph instance representing the MST. Note
* that NULL is returned if the graph is disconnected (in other
* words, there is not path from any given vertex to all other
* vertices in the graph).
*
* @throws InvalidOperationException if graph is empty.
*/
protected IWeightedGraph <VertexT> FindMinimumSpanningTree(IWeightedGraph <VertexT> mst)
{
if (Size == 0)
{
// It is illegal to call this method on an empty graph
throw new InvalidOperationException();
}
// Priority queue that orders the edges in ascending order
// by their weight. The next edge to place into MST is
// taken from the priority queue.
ArrayHeap <Edge> priority_queue =
new ArrayHeap <Edge>(
Size,
(Edge edge_1, Edge edge_2) =>
{
// TODO: floats shouldn't be compared like this.
if (edge_1.Weight < edge_2.Weight)
{
return(-1);
}
else if (edge_1.Weight > edge_2.Weight)
{
return(1);
}
else
{
return(0);
}
});
// A vertex is 'marked' once an edge ending in that vertex
// has been added to MST, that is once vertex becomes part
// of the MST. This helps ensure that we never add two edges
// that end in the same vertex, which in turn makes sure
// that generated graph is in fact a MST.
BitArray marked_vertices = new BitArray(Size, false);
marked_vertices[0] = true;
// The vertex whose outward edges will be processed next
int vertex_index = 0;
// The following loop will run Size - 1 times, as we need
// to select Size - 1 edges
for (int i = 1; i < Size; ++i)
{
for (int column = 0; column < Size; ++column)
{
float?edge_value = Edges.GetEdge(vertex_index, column);
if (edge_value != null && !marked_vertices[column])
{
// Add this edge to the priority queue
priority_queue.Insert(new Edge(vertex_index, column, edge_value.Value));
}
}
// Find the next edge to insert to the MST. We cannot simply pick
// an edge at the head of the priority queue because the end
// vertex of that edge might already be 'marked', hence edges
// towards it must not be considered
Edge min_weight_edge = null;
do
{
if (priority_queue.IsEmpty)
{
// This means that we're unable to reach all vertices
// of the graph (disconnected graph). Return NULL.
return(null);
}
min_weight_edge = priority_queue.Remove();
} while (marked_vertices[min_weight_edge.EndVertex]);
// Add the edge to the MST
mst.AddEdge(
min_weight_edge.StartVertex,
min_weight_edge.EndVertex,
min_weight_edge.Weight);
// 'Mark' the end vertex of the newly added edge
marked_vertices[min_weight_edge.EndVertex] = true;
// The next vertex whose outward edges will be processed is
// the end vertex of the edge we just added to MST
vertex_index = min_weight_edge.EndVertex;
}
return(mst);
}
