In BGL terminology the module Graph defines the graph concept (see www.boost.org/libs/graph/doc/graph_concepts.html). We however do not distinguish between the IncidenceGraph, EdgeListGraph and VertexListGraph concepts, which would complicate the interface too much. These concepts are defined in BGL to differentiate between efficient access to edges and vertices.
The RGL Graph concept contains only a few requirements that are common to all the graph concepts. These include, especially, the iterators defining the sets of vertices and edges (see each_vertex and each_adjacent). Most other functions are derived from these fundamental iterators, i.e. num_vertices or num_edges.
Each graph is an enumerable of vertices.
Returns true if the graph contains no cycles. This is only meaningful for directed graphs. Returns false for undirected graphs.
# File lib/rgl/topsort.rb, line 67 def acyclic? topsort_iterator.length == num_vertices end
Returns an array of vertices adjacent to vertex v.
# File lib/rgl/base.rb, line 168 def adjacent_vertices (v) r = [] each_adjacent(v) {|u| r << u} r end
Returns a BFSIterator, starting at vertex v.
# File lib/rgl/traversal.rb, line 230 def bfs_iterator (v = self.detect { |x| true}) BFSIterator.new(self, v) end
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.
# 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
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.
# 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
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.
# 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
Start a depth first search at vertex u. The block b is called on each finish_vertex event.
# 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
Returns a DFSIterator staring at vertex v.
# File lib/rgl/traversal.rb, line 292 def dfs_iterator (v = self.detect { |x| true }) DFSIterator.new(self, v) end
Is the graph directed? The default returns false.
# File lib/rgl/base.rb, line 140 def directed?; false; end
Call dotty for the graph which is written to the file 'graph.dot' in the # current directory.
# 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
Vertices get enumerated. A graph is thus an enumerable of vertices.
# File lib/rgl/base.rb, line 137 def each(&block); each_vertex(&block); end
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.
# File lib/rgl/base.rb, line 113 def each_adjacent (v) # :yields: v raise NotImplementedError end
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.
# 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
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).
# 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
The each_vertex iterator defines the set of vertices. This method must be defined by concrete graph classes. It defines the BGL VertexListGraph concept.
# File lib/rgl/base.rb, line 106 def each_vertex () # :yields: v raise NotImplementedError end
Returns the class for edges: DirectedEdge or UnDirectedEdge.
# File lib/rgl/base.rb, line 156 def edge_class; directed? ? DirectedEdge : UnDirectedEdge; end
Return the array of edges (DirectedEdge or UnDirectedEdge) of the graph using each_edge, depending whether the graph is directed or not.
# File lib/rgl/base.rb, line 160 def edges result = [] c = edge_class each_edge { |u,v| result << c.new(u,v) } result end
Return a new ImplicitGraph which has as edges all edges of the receiver which satisfy the predicate filter (a block with two parameters).
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]
# 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
Returns true if the graph has no vertices, i.e. num_vertices == 0.
# File lib/rgl/base.rb, line 150 def empty?; num_vertices.zero?; end
Equality is defined to be same set of edges and directed?
# 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
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).
# File lib/rgl/base.rb, line 145 def has_vertex?(v); include?(v); end
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.
# 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
Returns the number of edges.
# File lib/rgl/base.rb, line 189 def num_edges; r = 0; each_edge {|u,v| r +=1}; r; end
Returns the number of out-edges (for directed graphs) or the number of incident edges (for undirected graphs) of vertex v.
# File lib/rgl/base.rb, line 176 def out_degree (v) r = 0 each_adjacent(v) { |u| r += 1} r end
Output the DOT-graph to stream s.
# File lib/rgl/dot.rb, line 40 def print_dotted_on (params = {}, s = $stdout) s << to_dot_graph(params).to_s << "\n" end
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.
# 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
Returns the number of vertices.
# File lib/rgl/base.rb, line 183 def size() # Why not in Enumerable? inject(0) { |n, v| n + 1 } end
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.
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{Tarjan:1972:DFS,
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.
# 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
Convert a general graph to an AdjacencyGraph. If the graph is directed, returns a DirectedAdjacencyGraph; otherwise, returns an AdjacencyGraph.
# 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
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.
# 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
Utility method to show a string representation of the edges of the graph.
# File lib/rgl/base.rb, line 192 def to_s edges.sort.to_s end
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.
# File lib/rgl/adjacency.rb, line 200 def to_undirected return self unless directed? AdjacencyGraph.new(Set, self) end
Returns a TopsortIterator.
# File lib/rgl/topsort.rb, line 60 def topsort_iterator TopsortIterator.new(self) end
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.
# 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
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.
# 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
Return the array of vertices. Synonym for to_a inherited by Enumerable.
# File lib/rgl/base.rb, line 153 def vertices; to_a; end
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).
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)"
# 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
Use dot to create a graphical representation of the graph. Returns the filename of the graphics file.
# 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
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