Module: RGL::Graph

Includes:
Enumerable, Edge
Included in:
BidirectionalGraph, ImplicitGraph, MutableGraph
Defined in:
lib/rgl/base.rb,
lib/rgl/dot.rb,
lib/rgl/topsort.rb,
lib/rgl/implicit.rb,
lib/rgl/adjacency.rb,
lib/rgl/traversal.rb,
lib/rgl/traversal.rb,
lib/rgl/condensation.rb,
lib/rgl/transitivity.rb,
lib/rgl/connected_components.rb

Overview

class AdjacencyGraph

Defined Under Namespace

Classes: TarjanSccVisitor

Instance Method Summary collapse

Methods included from Enumerable

#length

Instance Method Details

#acyclic? ⇒ Boolean

Returns true if the graph contains no cycles. This is only meaningful for directed graphs. Returns false for undirected graphs.

Returns:

  • (Boolean) 


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# File 'lib/rgl/topsort.rb', line 67

def acyclic?
  topsort_iterator.length == num_vertices
end

#adjacent_vertices(v) ⇒ Object

Returns an array of vertices adjacent to vertex v.



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# File 'lib/rgl/base.rb', line 168

def adjacent_vertices (v)
  r = []
  each_adjacent(v) {|u| r << u}
  r
end

#bfs_iterator(v = self.detect { |x| true}) ⇒ Object

Returns a BFSIterator, starting at vertex v.



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# File 'lib/rgl/traversal.rb', line 230

def bfs_iterator (v = self.detect { |x| true})
  BFSIterator.new(self, v)
end

#bfs_search_tree_from(v) ⇒ Object

Returns a DirectedAdjacencyGraph, which represents a BFS search tree starting at v. This method uses the tree_edge_event of BFSIterator to record all tree edges of the search tree in the result.



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# File 'lib/rgl/traversal.rb', line 238

def bfs_search_tree_from (v)
  require 'rgl/adjacency'
  bfs  = bfs_iterator(v)
  tree = DirectedAdjacencyGraph.new
  bfs.set_tree_edge_event_handler { |from, to|
    tree.add_edge(from, to)
  }
  bfs.set_to_end        # does the search
  tree
end

#condensation_graph ⇒ Object

Returns an RGL::ImplicitGraph where the strongly connected components of this graph are condensed into single nodes represented by Set instances containing the members of each strongly connected component. Edges between the different strongly connected components are preserved while edges within strongly connected components are omitted.

Raises RGL::NotDirectedError if run on an undirected graph.

Raises:



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# File 'lib/rgl/condensation.rb', line 13

def condensation_graph
  raise NotDirectedError,
    "condensation_graph only supported for directed graphs" unless directed?

  # Get the component map for the strongly connected components.
  comp_map = strongly_connected_components.comp_map
  # Invert the map such that for any number, n, in the component map a Set
  # instance is created containing all of the nodes which map to n.  The Set
  # instances will be used to map to the number, n, with which the elements
  # of the set are associated.
  inv_comp_map = {}
  comp_map.each { |v, n| (inv_comp_map[n] ||= Set.new) << v }

  # Create an ImplicitGraph where the nodes are the strongly connected
  # components of this graph and the edges are the edges of this graph which
  # cross between the strongly connected components.
  ImplicitGraph.new do |g|
    g.vertex_iterator do |b|
      inv_comp_map.each_value(&b)
    end
    g.adjacent_iterator do |scc, b|
      scc.each do |v|
        each_adjacent(v) do |w|
          # Do not make the cluster reference itself in the graph.
          if comp_map[v] != comp_map[w] then
            b.call(inv_comp_map[comp_map[w]])
          end
        end
      end
    end
    g.directed = true
  end
end

#depth_first_search(vis = DFSVisitor.new(self), &b) ⇒ Object

Do a recursive DFS search on the whole graph. If a block is passed, it is called on each finish_vertex event. See strongly_connected_components for an example usage.



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# File 'lib/rgl/traversal.rb', line 300

def depth_first_search (vis = DFSVisitor.new(self), &b)
  each_vertex do |u|
    unless vis.finished_vertex?(u)
      vis.handle_start_vertex(u)
      depth_first_visit(u, vis, &b)
    end
  end
end

#depth_first_visit(u, vis = DFSVisitor.new(self), &b) ⇒ Object

Start a depth first search at vertex u. The block b is called on each finish_vertex event.



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# File 'lib/rgl/traversal.rb', line 312

def depth_first_visit (u, vis = DFSVisitor.new(self), &b)
  vis.color_map[u] = :GRAY
  vis.handle_examine_vertex(u)
  each_adjacent(u) { |v|
    vis.handle_examine_edge(u, v)
    if vis.follow_edge?(u, v)            # (u,v) is a tree edge
      vis.handle_tree_edge(u, v)         # also discovers v
      vis.color_map[v] = :GRAY           # color of v was :WHITE
      depth_first_visit(v, vis, &b)
    else                                 # (u,v) is a non tree edge
      if vis.color_map[v] == :GRAY
        vis.handle_back_edge(u, v)       # (u,v) has gray target
      else
        vis.handle_forward_edge(u, v)    # (u,v) is a cross or forward edge 
      end
    end
  }
  vis.color_map[u] = :BLACK
  vis.handle_finish_vertex(u)           # finish vertex
  b.call(u)
end

#dfs_iterator(v = self.detect { |x| true }) ⇒ Object

Returns a DFSIterator staring at vertex v.



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# File 'lib/rgl/traversal.rb', line 292

def dfs_iterator (v = self.detect { |x| true })
  DFSIterator.new(self, v)
end

#directed? ⇒ Boolean

Is the graph directed? The default returns false.

Returns:

  • (Boolean) 


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# File 'lib/rgl/base.rb', line 140

def directed?; false; end

#dotty(params = {}) ⇒ Object

Call dotty for the graph which is written to the file ‘graph.dot’ in the # current directory.



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# File 'lib/rgl/dot.rb', line 47

def dotty (params = {})
  dotfile = "graph.dot"
  File.open(dotfile, "w") {|f|
    print_dotted_on(params, f)
  }
  system("dotty", dotfile)
end

#each(&block) ⇒ Object

Vertices get enumerated. A graph is thus an enumerable of vertices.

Testing



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# File 'lib/rgl/base.rb', line 137

def each(&block); each_vertex(&block); end

#each_adjacent(v) ⇒ Object

The each_adjacent iterator defines the out edges of vertex v. This method must be defined by concrete graph classes. Its defines the BGL IncidenceGraph concept.

Raises:

  • (NotImplementedError) 


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# File 'lib/rgl/base.rb', line 113

def each_adjacent (v) # :yields: v
  raise NotImplementedError
end

#each_connected_component {|comp| ... } ⇒ Object

Compute the connected components of an undirected graph, using a DFS (Depth-first search)-based approach. A _connected component_ of an undirected graph is a set of vertices that are all reachable from each other.

The function is implemented as an iterator which calls the client with an array of vertices for each component.

It raises an exception if the graph is directed.

Yields:

  • (comp)

Raises:



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# File 'lib/rgl/connected_components.rb', line 23

def each_connected_component
  raise NotUndirectedError,
    "each_connected_component only works " +
    "for undirected graphs." if directed?
  comp = []
  vis  = DFSVisitor.new(self)
  vis.set_finish_vertex_event_handler { |v| comp << v }
  vis.set_start_vertex_event_handler { |v|
    yield comp unless comp.empty?
    comp = []
  }
  depth_first_search(vis) { |v| }
  yield comp unless comp.empty?
end

#each_edge(&block) ⇒ Object

The each_edge iterator should provide efficient access to all edges of the graph. Its defines the EdgeListGraph concept.

This method must not be defined by concrete graph classes, because it can be implemented using each_vertex and each_adjacent. However for undirected graph the function is inefficient because we must not yield (v,u) if we already visited edge (u,v).



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# File 'lib/rgl/base.rb', line 124

def each_edge (&block)
  if directed?
    each_vertex { |u|
      each_adjacent(u) { |v| yield u,v }
    }
  else
    each_edge_aux(&block)       # concrete graphs should to this better
  end
end

#each_vertex ⇒ Object

The each_vertex iterator defines the set of vertices. This method must be defined by concrete graph classes. It defines the BGL VertexListGraph concept.

Raises:

  • (NotImplementedError) 


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# File 'lib/rgl/base.rb', line 106

def each_vertex () # :yields: v
  raise NotImplementedError
end

#edge_class ⇒ Object

Returns the class for edges: DirectedEdge or UnDirectedEdge.



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# File 'lib/rgl/base.rb', line 156

def edge_class; directed? ? DirectedEdge : UnDirectedEdge; end

#edges ⇒ Object

Return the array of edges (DirectedEdge or UnDirectedEdge) of the graph using each_edge, depending whether the graph is directed or not.



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# File 'lib/rgl/base.rb', line 160

def edges
  result = []
  c = edge_class
  each_edge { |u,v| result << c.new(u,v) }
  result
end

#edges_filtered_by(&filter) ⇒ Object

Return a new ImplicitGraph which has as edges all edges of the receiver which satisfy the predicate filter (a block with two parameters).

Example

g = complete(7).edges_filtered_by {|u,v| u+v == 7}
g.to_s     => "(1=6)(2=5)(3=4)"
g.vertices => [1, 2, 3, 4, 5, 6, 7]



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# File 'lib/rgl/implicit.rb', line 146

def edges_filtered_by (&filter)
  implicit_graph { |g|
    g.adjacent_iterator { |v, b|
      self.each_adjacent(v) { |u|
        b.call(u) if filter.call(v, u)
      }
    }
    g.edge_iterator { |b|
      self.each_edge { |u,v| b.call(u, v) if filter.call(u, v) }
    }
  }
end

#empty? ⇒ Boolean

Returns true if the graph has no vertices, i.e. num_vertices == 0.

accessing vertices and edges

Returns:

  • (Boolean) 


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# File 'lib/rgl/base.rb', line 150

def empty?; num_vertices.zero?; end

#eql?(g) ⇒ Boolean Also known as: ==

Equality is defined to be same set of edges and directed?

Returns:

  • (Boolean) 


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# File 'lib/rgl/base.rb', line 197

def eql?(g)
  equal?(g) or
    begin
      g.is_a?(Graph) and directed? == g.directed? and
        g.inject(0) { |n, v| has_vertex?(v) or return false; n+1} ==
        num_vertices and begin
                           ng = 0
                           g.each_edge {|u,v| has_edge? u,v or return false; ng += 1}
                           ng == num_edges
                         end
    end
end

#has_vertex?(v) ⇒ Boolean

Returns true if v is a vertex of the graph. Same as #include? inherited from Enumerable. Complexity is O(num_vertices) by default. Concrete graph may be better here (see AdjacencyGraph).

Returns:

  • (Boolean) 


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# File 'lib/rgl/base.rb', line 145

def has_vertex?(v); include?(v); end

#implicit_graph {|result| ... } ⇒ Object

Return a new ImplicitGraph which is isomorphic (i.e. has same edges and vertices) to the receiver. It is a shortcut, also used by edges_filtered_by and vertices_filtered_by.

Yields:

  • (result) 


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# File 'lib/rgl/implicit.rb', line 163

def implicit_graph
  result = ImplicitGraph.new { |g|
    g.vertex_iterator { |b| self.each_vertex(&b) }
    g.adjacent_iterator { |v, b| self.each_adjacent(v, &b) }
    g.directed = self.directed?
  }
  yield result if block_given? # let client overwrite defaults
  result
end

#num_edges ⇒ Object

Returns the number of edges.



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# File 'lib/rgl/base.rb', line 189

def num_edges; r = 0; each_edge {|u,v| r +=1}; r; end

#out_degree(v) ⇒ Object

incident edges (for undirected graphs) of vertex v.



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# File 'lib/rgl/base.rb', line 176

def out_degree (v)
  r = 0
  each_adjacent(v) { |u| r += 1}
  r
end

#print_dotted_on(params = {}, s = $stdout) ⇒ Object

Output the DOT-graph to stream s.



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# File 'lib/rgl/dot.rb', line 40

def print_dotted_on (params = {}, s = $stdout)
  s << to_dot_graph(params).to_s << "\n"
end

#reverse ⇒ Object

Return a new DirectedAdjacencyGraph which has the same set of vertices. If (u,v) is an edge of the graph, then (v,u) is an edge of the result.

If the graph is undirected, the result is self.



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# File 'lib/rgl/adjacency.rb', line 186

def reverse
  return self unless directed?
  result = DirectedAdjacencyGraph.new
  each_vertex { |v| result.add_vertex v }
  each_edge { |u,v| result.add_edge(v, u) }
  result
end

#size ⇒ Object Also known as: num_vertices

Returns the number of vertices.



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# File 'lib/rgl/base.rb', line 183

def size()                  # Why not in Enumerable?
  inject(0) { |n, v| n + 1 }
end

#strongly_connected_components ⇒ Object

This is Tarjan’s algorithm for strongly connected components, from his paper “Depth first search and linear graph algorithms”. It calculates the components in a single application of DFS. We implement the algorithm with the help of the DFSVisitor TarjanSccVisitor.

Definition

A _strongly connected component_ of a directed graph G=(V,E) is a maximal set of vertices U which is in V, such that for every pair of vertices u and v in U, we have both a path from u to v and a path from v to u. That is to say, u and v are reachable from each other.

@Article 

author =       "R. E. Tarjan",
key =          "Tarjan",
title =        "Depth First Search and Linear Graph Algorithms",
journal =      "SIAM Journal on Computing",
volume =       "1",
number =       "2",
pages =        "146--160",
month =        jun,
year =         "1972",
CODEN =        "SMJCAT",
ISSN =         "0097-5397 (print), 1095-7111 (electronic)",
bibdate =      "Thu Jan 23 09:56:44 1997",
bibsource =    "Parallel/Multi.bib, Misc/Reverse.eng.bib",

The output of the algorithm is recorded in a TarjanSccVisitor vis. vis.comp_map will contain numbers giving the component ID assigned to each vertex. The number of components is vis.num_comp.

Raises:



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# File 'lib/rgl/connected_components.rb', line 129

def strongly_connected_components
  raise NotDirectedError,
    "strong_components only works for directed graphs." unless directed?
  vis = TarjanSccVisitor.new(self)
  depth_first_search(vis) { |v| }
  vis
end

#to_adjacency ⇒ Object

Convert a general graph to an AdjacencyGraph. If the graph is directed, returns a DirectedAdjacencyGraph; otherwise, returns an AdjacencyGraph.



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# File 'lib/rgl/adjacency.rb', line 174

def to_adjacency
  result = (directed? ? DirectedAdjacencyGraph : AdjacencyGraph).new
  each_vertex { |v| result.add_vertex(v) }
  each_edge { |u,v| result.add_edge(u, v) }
  result
end

#to_dot_graph(params = {}) ⇒ Object

Return a RGL::DOT::Digraph for directed graphs or a DOT::Subgraph for an undirected Graph. params can contain any graph property specified in rdot.rb.



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# File 'lib/rgl/dot.rb', line 19

def to_dot_graph (params = {})
params['name'] ||= self.class.name.gsub(/:/,'_')
fontsize   = params['fontsize'] ? params['fontsize'] : '8'
graph      = (directed? ? DOT::Digraph : DOT::Subgraph).new(params)
edge_class = directed? ? DOT::DirectedEdge : DOT::Edge
each_vertex do |v|
  name = v.to_s
  graph << DOT::Node.new('name'     => name,
                         'fontsize' => fontsize,
                         'label'    => name)
end
each_edge do |u,v|
  graph << edge_class.new('from'     => u.to_s,
                          'to'       => v.to_s,
                          'fontsize' => fontsize)
  end
  graph
end

#to_s ⇒ Object

Utility method to show a string representation of the edges of the graph.



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# File 'lib/rgl/base.rb', line 192

def to_s
  edges.sort.to_s
end

#to_undirected ⇒ Object

Return a new AdjacencyGraph which has the same set of vertices. If (u,v) is an edge of the graph, then (u,v) and (v,u) (which are the same edges) are edges of the result.

If the graph is undirected, the result is self.



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# File 'lib/rgl/adjacency.rb', line 200

def to_undirected
  return self unless directed?
  AdjacencyGraph.new(Set, self)
end

#topsort_iterator ⇒ Object

Returns a TopsortIterator.



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# File 'lib/rgl/topsort.rb', line 60

def topsort_iterator
  TopsortIterator.new(self)
end

#transitive_closure ⇒ Object

Returns an RGL::DirectedAdjacencyGraph which is the transitive closure of this graph. Meaning, for each path u -> … -> v in this graph, the path is copied and the edge u -> v is added. This method supports working with cyclic graphs by ensuring that edges are created between every pair of vertices in the cycle, including self-referencing edges.

This method should run in O(|V||E|) time, where |V| and |E| are the number of vertices and edges respectively.

Raises RGL::NotDirectedError if run on an undirected graph.

Raises:



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# File 'lib/rgl/transitivity.rb', line 19

def transitive_closure
  raise NotDirectedError,
    "transitive_closure only supported for directed graphs" unless directed?

  # Compute a condensation graph in order to hide cycles.
  cg = condensation_graph

  # Use a depth first search to calculate the transitive closure over the
  # condensation graph.  This ensures that as we traverse up the graph we
  # know the transitive closure of each subgraph rooted at each node
  # starting at the leaves.  Subsequent root nodes which consume these
  # subgraphs by way of the nodes' immediate successors can then immediately
  # add edges to the roots of the subgraphs and to every successor of those
  # roots.
  tc_cg = DirectedAdjacencyGraph.new
  cg.depth_first_search do |v|
    # For each vertex v, w, and x where the edges v -> w and w -> x exist in
    # the source graph, add edges v -> w and v -> x to the target graph.
    cg.each_adjacent(v) do |w|
      tc_cg.add_edge(v, w)
      tc_cg.each_adjacent(w) do |x|
        tc_cg.add_edge(v, x)
      end
    end
    # Ensure that a vertex with no in or out edges is added to the graph.
    tc_cg.add_vertex(v)
  end

  # Expand the condensed transitive closure.
  #
  # For each trivial strongly connected component in the condensed graph,
  # add the single node it contains to the new graph and add edges for each
  # edge the node begins in the original graph.
  # For each NON-trivial strongly connected component in the condensed
  # graph, add each node it contains to the new graph and add edges to
  # every node in the strongly connected component, including self
  # referential edges.  Then for each edge of the original graph from any
  # of the contained nodes, add edges from each of the contained nodes to
  # all the edge targets.
  g = DirectedAdjacencyGraph.new
  tc_cg.each_vertex do |scc|
    scc.each do |v|
      # Add edges between all members of non-trivial strongly connected
      # components (size > 1) and ensure that self referential edges are
      # added when necessary for trivial strongly connected components.
      if scc.size > 1 || has_edge?(v, v) then
        scc.each do |w|
          g.add_edge(v, w)
        end
      end
      # Ensure that a vertex with no in or out edges is added to the graph.
      g.add_vertex(v)
    end
    # Add an edge from every member of a strongly connected component to
    # every member of each strongly connected component to which the former
    # points.
    tc_cg.each_adjacent(scc) do |scc2|
      scc.each do |v|
        scc2.each do |w|
          g.add_edge(v, w)
        end
      end
    end
  end

  # Finally, the transitive closure...
  g
end

#transitive_reduction ⇒ Object

Returns an RGL::DirectedAdjacencyGraph which is the transitive reduction of this graph. Meaning, that each edge u -> v is omitted if path u -> … -> v exists. This method supports working with cyclic graphs; however, cycles are arbitrarily simplified which may lead to variant, although equally valid, results on equivalent graphs.

This method should run in O(|V||E|) time, where |V| and |E| are the number of vertices and edges respectively.

Raises RGL::NotDirectedError if run on an undirected graph.

Raises:



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# File 'lib/rgl/transitivity.rb', line 98

def transitive_reduction
  raise NotDirectedError,
    "transitive_reduction only supported for directed graphs" unless directed?

  # Compute a condensation graph in order to hide cycles.
  cg = condensation_graph

  # Use a depth first search to compute the transitive reduction over the
  # condensed graph.  This is similar to the computation of the transitive
  # closure over the graph in that for any node of the graph all nodes
  # reachable from the node are tracked.  Using a depth first search ensures
  # that all nodes reachable from a target node are known when considering
  # whether or not to add an edge pointing to that target.
  tr_cg = DirectedAdjacencyGraph.new
  paths_from = {}
  cg.depth_first_search do |v|
    paths_from[v] = Set.new
    cg.each_adjacent(v) do |w|
      # Only add the edge v -> w if there is no other edge v -> x such that
      # w is reachable from x.  Make sure to completely skip the case where
      # x == w.
      unless Enumerable::Enumerator.new(cg, :each_adjacent, v).any? do |x|
        x != w && paths_from[x].include?(w)
      end then
        tr_cg.add_edge(v, w)

        # For each vertex v, track all nodes reachable from v by adding node
        # w to the list as well as all the nodes readable from w.
        paths_from[v] << w
        paths_from[v].merge(paths_from[w])
      end
    end
    # Ensure that a vertex with no in or out edges is added to the graph.
    tr_cg.add_vertex(v)
  end

  # Expand the condensed transitive reduction.
  #
  # For each trivial strongly connected component in the condensed graph,
  # add the single node it contains to the new graph and add edges for each
  # edge the node begins in the original graph.
  # For each NON-trivial strongly connected component in the condensed
  # graph, add each node it contains to the new graph and add arbitrary
  # edges between the nodes to form a simple cycle.  Then for each strongly
  # connected component adjacent to the current one, find and add the first
  # edge which exists in the original graph, starts in the first strongly
  # connected component, and ends in the second strongly connected
  # component.
  g = DirectedAdjacencyGraph.new
  tr_cg.each_vertex do |scc|
    # Make a cycle of the contents of non-trivial strongly connected
    # components.
    scc_arr = scc.to_a
    if scc.size > 1 || has_edge?(scc_arr.first, scc_arr.first) then
      0.upto(scc_arr.size - 2) do |idx|
        g.add_edge(scc_arr[idx], scc_arr[idx + 1])
      end
      g.add_edge(scc_arr.last, scc_arr.first)
    end

    # Choose a single edge between the members of two different strongly
    # connected component to add to the graph.
    edges = Enumerable::Enumerator.new(self, :each_edge)
    tr_cg.each_adjacent(scc) do |scc2|
      g.add_edge(
        *edges.find do |v, w|
          scc.member?(v) && scc2.member?(w)
        end
      )
    end

    # Ensure that a vertex with no in or out edges is added to the graph.
    scc.each do |v|
      g.add_vertex(v)
    end
  end

  # Finally, the transitive reduction...
  g
end

#vertices ⇒ Object

Return the array of vertices. Synonym for #to_a inherited by Enumerable.



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# File 'lib/rgl/base.rb', line 153

def vertices; to_a; end

#vertices_filtered_by(&filter) ⇒ Object

Graph adaptors

Return a new ImplicitGraph which has as vertices all vertices of the receiver which satisfy the predicate filter.

The methods provides similar functionaty as the BGL graph adapter filtered_graph (see BOOST_DOC/filtered_graph.html).

Example

def complete (n)
  set = n.integer? ? (1..n) : n
  RGL::ImplicitGraph.new { |g|
        g.vertex_iterator { |b| set.each(&b) }
        g.adjacent_iterator { |x, b|
          set.each { |y| b.call(y) unless x == y }
        }
  }
end

complete(4).to_s =>     "(1=2)(1=3)(1=4)(2=3)(2=4)(3=4)"
complete(4).vertices_filtered_by {|v| v != 4}.to_s => "(1=2)(1=3)(2=3)"



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# File 'lib/rgl/implicit.rb', line 126

def vertices_filtered_by (&filter)
  implicit_graph { |g|
    g.vertex_iterator { |b|
      self.each_vertex { |v| b.call(v) if filter.call(v) }
    }
    g.adjacent_iterator { |v, b|
      self.each_adjacent(v) { |u| b.call(u) if filter.call(u) }
    }
  }
end

#write_to_graphic_file(fmt = 'png', dotfile = "graph") ⇒ Object

Use dot to create a graphical representation of the graph. Returns the filename of the graphics file.



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# File 'lib/rgl/dot.rb', line 58

def write_to_graphic_file (fmt='png', dotfile="graph")
  src = dotfile + ".dot"
  dot = dotfile + "." + fmt

  File.open(src, 'w') do |f|
    f << self.to_dot_graph.to_s << "\n"
  end

  system( "dot -T#{fmt} #{src} -o #{dot}" )
  dot
end