// Builds a neighborhood for a set X from the ground on up, without relying on what has been saved previously in O(n^2) time
public dNeighborhood(BitSet X, BitSet vector, Graph graph)
{
// O(n) time copy
Vector = vector.Copy();
Occurrences = new Dictionary<int, int>();
// Loops in O(|Vector|) time over all vertices in the vector
foreach (int v in Vector)
{
// Bitset operation of O(n) time
BitSet nv = graph.OpenNeighborhood(v) * X;
Occurrences[v] = Math.Min(nv.Count, dValue);
}
}
// Returns all connected components in a certain subgraph, where we define a subgraph by the vertices that are contained in it
// We apply multiple dfs searches to find all connected parts of the graph
public static List<BitSet> ConnectedComponents(Graph graph, BitSet subgraph)
{
// Each connected component is defined as a bitset, thus the result is a list of these bitsets
List<BitSet> result = new List<BitSet>();
// Working stack for the dfs algorithm
Stack<int> stack = new Stack<int>();
// Each vertex of the subgraph will either be explored, or it will be the starting point of a new dfs search
BitSet todo = subgraph.Copy();
while (!todo.IsEmpty)
{
int s = todo.First();
stack.Push(s);
// Start tracking the new component
BitSet component = new BitSet(0, graph.Size);
// Default dfs exploring part of the graph
while (stack.Count > 0)
{
int v = stack.Pop();
// If we have not encountered this vertex before (meaning it isn't in this component), then we check for all neighbors if they are part of the subgraph
// If a neighbor is part of the subgraph it means that we have to push it on the stack to explore it at a later stage for this component
if (!component.Contains(v))
{
component.Add(v);
// Remove this vertex from the 'todo' list, since it can never be the starting point of a new component
todo.Remove(v);
foreach (int w in graph.OpenNeighborhood(v))
if (subgraph.Contains(w))
stack.Push(w);
}
}
// The whole connected component has been found, so we can add it to the list of results
result.Add(component);
}
return result;
}
// Counts the number of independent sets in a graph, such that:
// - All vertices in R are included in the independent set, initially empty
// - Some of the vertices in P are included in the independent set, initially all vertices of the graph
// - None of the vertices in X are included in the independent set, initially empty
private static int Compute(Graph graph, BitSet R, BitSet P, BitSet X)
{
// Base case, when P and X are both empty we cannot expand the IS
if (P.IsEmpty && X.IsEmpty)
return 1;
int count = 0;
BitSet copy = P.Copy();
// Foreach vertex v in P we include it in the IS and compute how many maximal IS will include v by going into recursion.
foreach (int v in copy)
{
count += Compute(graph, R + v, P - graph.ClosedNeighborhood(v), X - graph.OpenNeighborhood(v));
P.Remove(v);
X.Add(v);
}
return count;
}
/*************************/
// Candidate strategy
/*************************/
public static BitSet Candidates(Graph graph, BitSet left, BitSet right, CandidateStrategy candidateStrategy)
{
BitSet nl = graph.Neighborhood(left) * right;
return candidateStrategy == CandidateStrategy.All ?
right.Copy() : (nl + graph.Neighborhood(nl)) * right;
}
private void FillTable(Graph graph, List<int> sequence)
{
int n = graph.Size;
// Processed vertices
BitSet left = new BitSet(0, n);
// Unprocessed vertices
BitSet right = graph.Vertices;
// Lists of representatives that we keep track of on each level of the algorithm
LinearRepresentativeList reps = new LinearRepresentativeList();
// Initial insertions, the empty set always has the empty neighborhood set as a representative initially
reps.Update(new BitSet(0, n), new BitSet(0, n));
for (int i = 0; i < sequence.Count; i++)
{
/// We give v the possibility to be a representative instead of being contained in neighborhoods
int v = sequence[i];
// Actually move v from one set to the other set
left.Add(v);
right.Remove(v);
// We don't want to disturb any pointers so we create new empty datastructures to save everything for the next iteration
LinearRepresentativeList nextReps = new LinearRepresentativeList();
// We are iterating over all representatives saved inside the list of the previous step. For each entry there are only two possibilities to create a new neighborhood
foreach (BitSet representative in reps)
{
// Case 1: The neighborhood possibly contained v (thus v has to be removed), but is still valid
BitSet neighborhood = reps.GetNeighborhood(representative) - v;
nextReps.Update(representative, neighborhood);
// Case 2: The union of the old neighborhood, together with v's neighborhood, creates a new entry. The representative is uniond together with v and saved.
BitSet representative_ = representative + v;
BitSet neighborhood_ = neighborhood + (graph.OpenNeighborhood(v) * right);
nextReps.Update(representative_, neighborhood_);
}
// Update the values for the next iteration
reps = nextReps;
// Save the maximum size that we encounter during all iterations; this will be the boolean dimension of the graph.
MaxDimension = Math.Max(MaxDimension, reps.Count);
// Save the representatives at the current cut in the table
Table[left.Copy()] = reps;
}
}
public int this[BitSet representative, int k]
{
get
{
if (!Data[k].ContainsKey(representative))
return 0;
return Data[k][representative];
}
set
{
Data[k][representative.Copy()] = value;
}
}
/*************************/
// Basic operations
/*************************/
// When adding a new entry we check if the neighborhood is already contained in the set
// If true, then we might replace the previous representative that is not connected to this neighborhood if the new one is lexicographically smaller
public void Update(BitSet representative, dNeighborhood neighborhood)
{
Map.Add(representative.Copy(), neighborhood);
}
/*************************/
// Basic operations
/*************************/
// When adding a new entry we check if the neighborhood is already contained in the set
// If true, then we might replace the previous representative that is not connected to this neighborhood if the new one is lexicographically smaller
public void Update(BitSet representative, BitSet neighborhood)
{
if (Map.ContainsValue(neighborhood))
{
BitSet previousRep = Map.Reverse[neighborhood];
if (representative.IsLexicographicallySmaller(previousRep))
{
Map.Remove(previousRep, neighborhood);
Map.Add(representative.Copy(), neighborhood.Copy());
}
}
else
{
Map.Add(representative.Copy(), neighborhood.Copy());
}
}
// Basic constructor
private BipartiteGraph(BitSet _left, BitSet _right, int size)
: base(size)
{
left = _left.Copy();
right = _right.Copy();
}
public int this[BitSet inner, BitSet outer, int k]
{
get
{
Tuple<BitSet, BitSet> t = new Tuple<BitSet, BitSet>(inner, outer);
if (!Data[k].ContainsKey(t))
return 0;
return Data[k][t];
}
set
{
Tuple<BitSet, BitSet> t = new Tuple<BitSet, BitSet>(inner.Copy(), outer.Copy());
Data[k][t] = value;
}
}
// Implementation of Algorithm 1 of 'Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems' (Bui-Xuan et al.)
// Used to compute the representatives and their corresponding dNeighborhoods for a given node of the decomposition tree
private void FillTable(Graph graph, BitSet cut)
{
int n = graph.Size;
BitSet _cut = graph.Vertices - cut;
// Lists of representatives that we keep track of on each level of the algorithm
RepresentativeList representatives = new RepresentativeList();
RepresentativeList lastLevel = new RepresentativeList();
// Initial insertions, the empty set always has the empty neighborhood set as a representative initially
dNeighborhood dInitial = new dNeighborhood(_cut);
representatives.Update(new BitSet(0, n), dInitial);
lastLevel.Update(new BitSet(0, n), dInitial);
while (lastLevel.Count > 0)
{
RepresentativeList nextLevel = new RepresentativeList();
foreach (BitSet r in lastLevel)
{
foreach (int v in cut)
{
// avoid that r_ = r, since we already saved all sets of that size
if (r.Contains(v))
continue;
BitSet r_ = r + v;
dNeighborhood dn = representatives.GetNeighborhood(r).CopyAndUpdate(graph, v);
if (!representatives.ContainsNeighborhood(dn) && !dn.Equals(representatives.GetNeighborhood(r)))
{
nextLevel.Update(r_, dn);
representatives.Update(r_, dn);
}
}
}
// Update the values for the next iteration
lastLevel = nextLevel;
}
// Save the representatives at the current cut in the table
Table[cut.Copy()] = representatives;
// Save the maximum size that we encounter during all iterations; this will be the boolean dimension of the graph is d = 1.
MaxDimension = Math.Max(MaxDimension, representatives.Count);
}
