Implementing DigraphEdgeConnectivity

doc/oper.xml

@@ -1954,6 +1954,46 @@ rec( idom := [ 2, fail, 2, 2, 2 ], preorder := [ 2, 1, 3, 4, 5 ] )
 </ManSection>
 <#/GAPDoc>
 
+<#GAPDoc Label="DigraphDominatingSet">
+<ManSection>
+  <Oper Name="DigraphDominatingSet" Arg="digraph"/>
+  <Returns>A List</Returns>
+  <Description>
+      This creates a dominating set of <A>digraph</A>, which is a subset of the vertices in 
+      <A>digraph</A> such that the neighbourhood of this subset of vertices is equal to all 
+      the vertices in <A>digraph</A>. </P>
+
+      The domating set returned will be one of potentially multiple possible dominating sets for the digraph.
+  <Example><![CDATA[
+D := Digraph([[2,3,4], [],[],[]]);;
+gap> DigraphDominatingSet(D);
+[ 4, 1 ]
+gap> DigraphDominatingSet(D);
+[ 1 ]]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
+<#GAPDoc Label="DigraphGetNeighbourhood">
+<ManSection>
+  <Oper Name="DigraphGetNeighbourhood" Arg="digraph, vertices"/>
+  <Returns>A List</Returns>
+  <Description>
+    This returns the set of vertices that are adjacent to a vertex in <A>vertices</A>,
+    not including any vertices that exist in <A>vertices</A>. 
+  <Example><![CDATA[
+gap> D := Digraph([[2, 3, 4], [1], [], []]);;
+gap> DigraphGetNeighbourhood(D, [1]);
+[ 2, 3, 4 ]
+gap> DigraphGetNeighbourhood(D, [1, 2]);
+[ 3, 4 ]
+gap> D := Digraph([[2, 3, 4], [], [], []]);;
+gap> DigraphGetNeighbourhood(D, [2]);
+[  ]]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
 <#GAPDoc Label="PartialOrderDigraphMeetOfVertices">
 <ManSection>
   <Oper Name="PartialOrderDigraphMeetOfVertices"

doc/weights.xml

@@ -272,6 +272,31 @@ gap> Sum(flow[1]);
 </ManSection>
 <#/GAPDoc>
 
+<#GAPDoc Label="DigraphEdgeConnectivity">
+<ManSection>
+  <Attr Name="DigraphEdgeConnectivity" Arg="digraph"/>
+  <Returns>An integer</Returns>
+  <Description>
+    This returns a record representing the edge connectivity of <A>digraph</A>.<P/>
+
+    It makes use of  <C>DigraphMaximumFlow(<A>digraph</A>)</C>, by constructing an edge-weighted Digraph
+    with edge weights of 1, then using the Max-flow min-cut theorem to determine the size of the minimum cut.
+    For a digraph, the minimum cut is the set of edges that need to be removed to ensure the digraph is 
+    no longer Strongly Connected. </P>
+    <Example><![CDATA[
+gap> D := Digraph([[2, 3, 4], [1, 3, 4], [1, 2], [2, 3]]);;
+gap> DigraphEdgeConnectivity(D);
+2
+gap> D := Digraph([[], [1, 2], [2]]);;
+gap> DigraphEdgeConnectivity(D);
+0
+gap> D := Digraph([[1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5]]);;
+gap> DigraphEdgeConnectivity(D);
+4]]></Example>
+  </Description>
+</ManSection>
+<#/GAPDoc>
+
 <#GAPDoc Label="RandomUniqueEdgeWeightedDigraph">
 <ManSection>
   <Oper Name="RandomUniqueEdgeWeightedDigraph" Arg="[filt, ]n[, p]"/>

gap/oper.gd

@@ -154,6 +154,8 @@ DeclareOperation("IsOrderIdeal", [IsDigraph, IsList]);
 DeclareOperation("IsOrderFilter", [IsDigraph, IsList]);
 DeclareOperation("Dominators", [IsDigraph, IsPosInt]);
 DeclareOperation("DominatorTree", [IsDigraph, IsPosInt]);
+DeclareOperation("DigraphDominatingSet", [IsDigraph]);
+DeclareOperation("DigraphGetNeighbourhood", [IsDigraph, IsList]);
 DeclareOperation("DigraphCycleBasis", [IsDigraph]);
 
 # 10. Operations for vertices . . .

gap/oper.gi

@@ -2522,6 +2522,45 @@ function(D, root)
   return result;
 end);
 
+# For calculating a dominating set for a digraph
+# Algorithm 7 in :
+# https://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
+InstallMethod(DigraphDominatingSet, "for a digraph",
+[IsDigraph],
+function(digraph)
+  local D, neighbourhood, VerticesLeft, Vertices;
+
+  Vertices := [1 .. DigraphNrVertices(digraph)];
+  Shuffle(Vertices);
+
+  # Shuffling not technically necessary - may be better to just do D := [1]?
+  D := [Vertices[1]];
+  neighbourhood := DigraphGetNeighbourhood(digraph, D);
+  VerticesLeft := Difference(Vertices, Union(neighbourhood, D));
+
+  while not IsEmpty(VerticesLeft) do;
+    Append(D, [VerticesLeft[1]]);
+    neighbourhood := DigraphGetNeighbourhood(digraph, D);
+    VerticesLeft := Difference(Vertices, Union(neighbourhood, D));
+  od;
+
+  return D;
+end);
+
+# For getting the neighbourhood for a given List of Vertices in a Digraph
+InstallMethod(DigraphGetNeighbourhood, "for a digraph and a List of vertices",
+[IsDigraph, IsList],
+function(digraph, vertices)
+  local v, neighbourhood;
+  neighbourhood := [];
+  for v in vertices do
+    neighbourhood := Difference(Union(neighbourhood,
+      OutNeighbours(digraph)[v]), vertices);
+  od;
+
+  return neighbourhood;
+end);
+
 # Computes the fundamental cycle basis of a symmetric digraph
 # First, notice that the cycle space is composed of orthogonal subspaces
 # corresponding to the cycle spaces of the connected components.

gap/weights.gd

@@ -39,6 +39,10 @@ DeclareGlobalFunction("DIGRAPHS_Edge_Weighted_Dijkstra");
 DeclareOperation("DigraphMaximumFlow",
                  [IsDigraph and HasEdgeWeights, IsPosInt, IsPosInt]);
 
+# Digraph Edge Connectivity
+DeclareOperation("DigraphEdgeConnectivity", [IsDigraph]);
+DeclareOperation("DigraphEdgeConnectivityDS", [IsDigraph]);
+
 # 6. Random edge weighted digraphs
 DeclareOperation("RandomUniqueEdgeWeightedDigraph", [IsPosInt]);
 DeclareOperation("RandomUniqueEdgeWeightedDigraph", [IsPosInt, IsFloat]);

gap/weights.gi

@@ -772,6 +772,211 @@ function(D, start, destination)
   return flows;
 end);
 
+#############################################################################
+# Digraph Edge Connectivity
+#############################################################################
+
+# Algorithms constructed off the algorithms detailed in:
+# https://www.cse.msu.edu/~cse835/Papers/Graph_connectivity_revised.pdf
+# Each Algorithm uses a different method to decrease the time complexity,
+# of calculating Edge Connectivity, though all make use of DigraphMaximumFlow()
+# due to the Max-Flow, Min-Cut Theorem
+
+# Algorithm 1: Calculating the Maximum Flow of every possible source and sink
+# Algorithm 2: Calculating the Maximum Flow to all sinks of an arbitrary source
+# Algorithm 3: Finding Maximum Flow within the non-leaves of a Spanning Tree
+# Algorithm 4: Constructing a spanning tree with a high number of leaves
+# Algorithm 5: Using the spanning tree^ to find Maximum Flow within non-leaves
+# Algorithm 6: Finding Maximum Flow within a dominating set of the digraph
+# Algorithm 7: Constructing a dominating set for use in Algorithm 6
+
+# Algorithms 4-7 are used below:
+
+# DigraphEdgeConnectivity calculated using Spanning Trees (Algorithm 4 & 5)
+InstallMethod(DigraphEdgeConnectivity, "for a digraph",
+[IsDigraph],
+function(digraph)
+  local w, weights, EdgeD, VerticesED, min, u, v,
+    sum, a, b, st, added, NeighboursV, Edges, notadded,
+    max, NextVertex, notAddedNeighbours, non_leaf;
+
+  if DigraphNrVertices(digraph) = 1 then
+    return 0;
+  fi;
+
+  if DigraphNrStronglyConnectedComponents(digraph) > 1 then
+    return 0;
+  fi;
+
+  weights := List([1 .. DigraphNrVertices(digraph)],
+  x -> List([1 .. Length(OutNeighbours(digraph)[x])],
+  y -> 1));
+  EdgeD := EdgeWeightedDigraph(digraph, weights);
+  VerticesED := [1 .. DigraphNrVertices(EdgeD)];
+
+  min := -1;
+
+  # Algorithm 4: Constructing a Spanning Tree with a large numbers of leaves
+
+  st := EmptyDigraph(DigraphNrVertices(EdgeD));
+  v := 1;
+  added := [v];
+  notadded := Difference(VerticesED, added);
+
+  while (DigraphNrEdges(st) < DigraphNrVertices(EdgeD) - 1) and
+              Length(notadded) > 0 do
+
+    # Add all edges incident from v
+    NeighboursV := Difference(OutNeighbors(EdgeD)[v], added);
+
+    Edges := List([1 .. Length(NeighboursV)],
+    x -> [v, NeighboursV[x]]);
+
+    # Edges := Difference(Edges, [[v, v]]);
+
+    st := DigraphAddEdges(st, Edges);
+    Append(added, Difference(OutNeighbours(EdgeD)[v], added));
+
+    # Select the neighbour to v with the highest number of not-added neighbours:
+    notadded := Difference(VerticesED, added);
+    max := 0;
+    NextVertex := v;
+
+    # Preventing infinite iteration if v has no non-added neighbours
+    if (Length(NeighboursV) = 0) then
+      # Pick from a vertex that hasn't been added yet
+      NextVertex := notadded[1];
+    fi;
+
+    for w in NeighboursV do;
+      notAddedNeighbours := Intersection(notadded, OutNeighbours(EdgeD)[w]);
+      if (Length(notAddedNeighbours) > max) then
+        max := Length(notAddedNeighbours);
+        NextVertex := w;
+      fi;
+    od;
+
+    v := NextVertex;
+
+  od;
+
+  # Algorithm 5: Iterating through the non-leaves of the
+  # Spanning Tree created in Algorithm 4 to find the Edge Connectivity
+  non_leaf := [];
+  for b in VerticesED do
+    if not IsEmpty(OutNeighbours(st)[b]) then
+      Append(non_leaf, [b]);
+    fi;
+  od;
+
+  if (Length(non_leaf) > 1) then
+    u := non_leaf[1];
+
+    for v in [2 .. Length(non_leaf)] do
+      a := DigraphMaximumFlow(EdgeD, u, non_leaf[v])[u];
+      b := DigraphMaximumFlow(EdgeD, non_leaf[v], u)[non_leaf[v]];
+
+      sum := Minimum(Sum(a), Sum(b));
+      if (sum < min or min = -1) then
+        min := sum;
+      fi;
+    od;
+
+    min := Minimum(min, Minimum(Minimum(OutDegrees(EdgeD)),
+      Minimum(InDegrees(EdgeD))));
+  else
+    # In the case of spanning trees with only one non-leaf node,
+    # the above algorithm does not work
+    # Revert to iterating through all vertices of the original digraph
+
+    u := 1;
+    for v in [2 .. DigraphNrVertices(EdgeD)] do
+      a := DigraphMaximumFlow(EdgeD, u, v)[u];
+      b := DigraphMaximumFlow(EdgeD, v, u)[v];
+
+      sum := Minimum(Sum(a), Sum(b));
+      if (sum < min or min = -1) then
+        min := sum;
+      fi;
+    od;
+  fi;
+
+  if (min = -1) then
+    return 0;
+  fi;
+
+  return min;
+end);
+
+# Digraph EdgeConnectivity calculated with Dominating Sets (Algorithm 6-7)
+InstallMethod(DigraphEdgeConnectivityDS, "for a digraph",
+[IsDigraph],
+function(digraph)
+  # Form an identical but edge weighted digraph with all edge weights as 1:
+  local weights, i, u, v, w, neighbourhood, EdgeD,
+    maxFlow, min, sum, a, b, V, added, st, non_leaf, max,
+    notAddedNeighbours, notadded, NextVertex, NeighboursV,
+    neighbour, Edges, D, VerticesLeft, VerticesED;
+
+  if DigraphNrVertices(digraph) = 1 then
+    return 0;
+  fi;
+
+  if DigraphNrStronglyConnectedComponents(digraph) > 1 then
+    return 0;
+  fi;
+
+  weights := List([1 .. DigraphNrVertices(digraph)],
+  x -> List([1 .. Length(OutNeighbours(digraph)[x])],
+  y -> 1));
+  EdgeD := EdgeWeightedDigraph(digraph, weights);
+
+  min := -1;
+
+  # Algorithm 7: Creating a dominating set of the digraph
+  D := DigraphDominatingSet(digraph);
+
+  # Algorithm 6: Using the dominating set created to determine the Maximum Flow
+
+  if Length(D) > 1 then
+
+    v := D[1];
+    for i in [2 .. Length(D)] do
+      w := D[i];
+      a := DigraphMaximumFlow(EdgeD, v, w)[v];
+      b := DigraphMaximumFlow(EdgeD, w, v)[w];
+
+      sum := Minimum(Sum(a), Sum(b));
+      if (sum < min or min = -1) then
+        min := sum;
+      fi;
+    od;
+
+  else
+    # If the dominating set of EdgeD is of Length 1,
+    # the above algorithm will not work
+    # Revert to iterating through all vertices of the original digraph
+
+    u := 1;
+
+    for v in [2 .. DigraphNrVertices(EdgeD)] do
+      a := DigraphMaximumFlow(EdgeD, u, v)[u];
+      b := DigraphMaximumFlow(EdgeD, v, u)[v];
+
+      sum := Minimum(Sum(a), Sum(b));
+      if (sum < min or min = -1) then
+        min := sum;
+      fi;
+
+    od;
+  fi;
+
+  return Minimum(min,
+    Minimum(Minimum(OutDegrees(EdgeD)),
+    Minimum(InDegrees(EdgeD))));
+
+end);
+
 #############################################################################
 # 6. Random edge weighted digraphs
 #############################################################################