ruby/lib/tsort.rb

#--
# tsort.rb - provides a module for topological sorting and strongly connected components.
#++
#

#
# TSort implements topological sorting using Tarjan's algorithm for
# strongly connected components.
#
# TSort is designed to be able to be used with any object which can be
# interpreted as a directed graph.
#
# TSort requires two methods to interpret an object as a graph,
# tsort_each_node and tsort_each_child.
#
# * tsort_each_node is used to iterate for all nodes over a graph.
# * tsort_each_child is used to iterate for child nodes of a given node.
#
# The equality of nodes are defined by eql? and hash since
# TSort uses Hash internally.
#
# == A Simple Example
#
# The following example demonstrates how to mix the TSort module into an
# existing class (in this case, Hash). Here, we're treating each key in
# the hash as a node in the graph, and so we simply alias the required
# #tsort_each_node method to Hash's #each_key method. For each key in the
# hash, the associated value is an array of the node's child nodes. This
# choice in turn leads to our implementation of the required #tsort_each_child
# method, which fetches the array of child nodes and then iterates over that
# array using the user-supplied block.
#
#   require 'tsort'
#
#   class Hash
#     include TSort
#     alias tsort_each_node each_key
#     def tsort_each_child(node, &block)
#       fetch(node).each(&block)
#     end
#   end
#
#   {1=>[2, 3], 2=>[3], 3=>[], 4=>[]}.tsort
#   #=> [3, 2, 1, 4]
#
#   {1=>[2], 2=>[3, 4], 3=>[2], 4=>[]}.strongly_connected_components
#   #=> [[4], [2, 3], [1]]
#
# == A More Realistic Example
#
# A very simple `make' like tool can be implemented as follows:
#
#   require 'tsort'
#
#   class Make
#     def initialize
#       @dep = {}
#       @dep.default = []
#     end
#
#     def rule(outputs, inputs=[], &block)
#       triple = [outputs, inputs, block]
#       outputs.each {|f| @dep[f] = [triple]}
#       @dep[triple] = inputs
#     end
#
#     def build(target)
#       each_strongly_connected_component_from(target) {|ns|
#         if ns.length != 1
#           fs = ns.delete_if {|n| Array === n}
#           raise TSort::Cyclic.new("cyclic dependencies: #{fs.join ', '}")
#         end
#         n = ns.first
#         if Array === n
#           outputs, inputs, block = n
#           inputs_time = inputs.map {|f| File.mtime f}.max
#           begin
#             outputs_time = outputs.map {|f| File.mtime f}.min
#           rescue Errno::ENOENT
#             outputs_time = nil
#           end
#           if outputs_time == nil ||
#              inputs_time != nil && outputs_time <= inputs_time
#             sleep 1 if inputs_time != nil && inputs_time.to_i == Time.now.to_i
#             block.call
#           end
#         end
#       }
#     end
#
#     def tsort_each_child(node, &block)
#       @dep[node].each(&block)
#     end
#     include TSort
#   end
#
#   def command(arg)
#     print arg, "\n"
#     system arg
#   end
#
#   m = Make.new
#   m.rule(%w[t1]) { command 'date > t1' }
#   m.rule(%w[t2]) { command 'date > t2' }
#   m.rule(%w[t3]) { command 'date > t3' }
#   m.rule(%w[t4], %w[t1 t3]) { command 'cat t1 t3 > t4' }
#   m.rule(%w[t5], %w[t4 t2]) { command 'cat t4 t2 > t5' }
#   m.build('t5')
#
# == Bugs
#
# * 'tsort.rb' is wrong name because this library uses
#   Tarjan's algorithm for strongly connected components.
#   Although 'strongly_connected_components.rb' is correct but too long.
#
# == References
#
# R. E. Tarjan, "Depth First Search and Linear Graph Algorithms",
# <em>SIAM Journal on Computing</em>, Vol. 1, No. 2, pp. 146-160, June 1972.
#

module TSort
  class Cyclic < StandardError
  end

  # Returns a topologically sorted array of nodes.
  # The array is sorted from children to parents, i.e.
  # the first element has no child and the last node has no parent.
  #
  # If there is a cycle, TSort::Cyclic is raised.
  #
  #   class G
  #     include TSort
  #     def initialize(g)
  #       @g = g
  #     end
  #     def tsort_each_child(n, &b) @g[n].each(&b) end
  #     def tsort_each_node(&b) @g.each_key(&b) end
  #   end
  #
  #   graph = G.new({1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]})
  #   p graph.tsort #=> [4, 2, 3, 1]
  #
  #   graph = G.new({1=>[2], 2=>[3, 4], 3=>[2], 4=>[]})
  #   p graph.tsort # raises TSort::Cyclic
  #
  def tsort
    each_node = method(:tsort_each_node)
    each_child = method(:tsort_each_child)
    TSort.tsort(each_node, each_child)
  end

  # Returns a topologically sorted array of nodes.
  # The array is sorted from children to parents, i.e.
  # the first element has no child and the last node has no parent.
  #
  # The graph is represented by _each_node_ and _each_child_.
  # _each_node_ should have +call+ method which yields for each node in the graph.
  # _each_child_ should have +call+ method which takes a node argument and yields for each child node.
  #
  # If there is a cycle, TSort::Cyclic is raised.
  #
  #   g = {1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]}
  #   each_node = lambda {|&b| g.each_key(&b) }
  #   each_child = lambda {|n, &b| g[n].each(&b) }
  #   p TSort.tsort(each_node, each_child) #=> [4, 2, 3, 1]
  #
  #   g = {1=>[2], 2=>[3, 4], 3=>[2], 4=>[]}
  #   each_node = lambda {|&b| g.each_key(&b) }
  #   each_child = lambda {|n, &b| g[n].each(&b) }
  #   p TSort.tsort(each_node, each_child) # raises TSort::Cyclic
  #
  def TSort.tsort(each_node, each_child)
    result = []
    TSort.tsort_each(each_node, each_child) {|element| result << element}
    result
  end

  # The iterator version of the #tsort method.
  # <tt><em>obj</em>.tsort_each</tt> is similar to <tt><em>obj</em>.tsort.each</tt>, but
  # modification of _obj_ during the iteration may lead to unexpected results.
  #
  # #tsort_each returns +nil+.
  # If there is a cycle, TSort::Cyclic is raised.
  #
  #   class G
  #     include TSort
  #     def initialize(g)
  #       @g = g
  #     end
  #     def tsort_each_child(n, &b) @g[n].each(&b) end
  #     def tsort_each_node(&b) @g.each_key(&b) end
  #   end
  #
  #   graph = G.new({1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]})
  #   graph.tsort_each {|n| p n }
  #   #=> 4
  #   #   2
  #   #   3
  #   #   1
  #
  def tsort_each(&block) # :yields: node
    each_node = method(:tsort_each_node)
    each_child = method(:tsort_each_child)
    TSort.tsort_each(each_node, each_child, &block)
  end

  # The iterator version of the TSort.tsort method.
  #
  # The graph is represented by _each_node_ and _each_child_.
  # _each_node_ should have +call+ method which yields for each node in the graph.
  # _each_child_ should have +call+ method which takes a node argument and yields for each child node.
  #
  #   g = {1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]}
  #   each_node = lambda {|&b| g.each_key(&b) }
  #   each_child = lambda {|n, &b| g[n].each(&b) }
  #   TSort.tsort_each(each_node, each_child) {|n| p n }
  #   #=> 4
  #   #   2
  #   #   3
  #   #   1
  #
  def TSort.tsort_each(each_node, each_child) # :yields: node
    TSort.each_strongly_connected_component(each_node, each_child) {|component|
      if component.size == 1
        yield component.first
      else
        raise Cyclic.new("topological sort failed: #{component.inspect}")
      end
    }
  end

  # Returns strongly connected components as an array of arrays of nodes.
  # The array is sorted from children to parents.
  # Each elements of the array represents a strongly connected component.
  #
  #   class G
  #     include TSort
  #     def initialize(g)
  #       @g = g
  #     end
  #     def tsort_each_child(n, &b) @g[n].each(&b) end
  #     def tsort_each_node(&b) @g.each_key(&b) end
  #   end
  #
  #   graph = G.new({1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]})
  #   p graph.strongly_connected_components #=> [[4], [2], [3], [1]]
  #
  #   graph = G.new({1=>[2], 2=>[3, 4], 3=>[2], 4=>[]})
  #   p graph.strongly_connected_components #=> [[4], [2, 3], [1]]
  #
  def strongly_connected_components
    each_node = method(:tsort_each_node)
    each_child = method(:tsort_each_child)
    TSort.strongly_connected_components(each_node, each_child)
  end

  # Returns strongly connected components as an array of arrays of nodes.
  # The array is sorted from children to parents.
  # Each elements of the array represents a strongly connected component.
  #
  # The graph is represented by _each_node_ and _each_child_.
  # _each_node_ should have +call+ method which yields for each node in the graph.
  # _each_child_ should have +call+ method which takes a node argument and yields for each child node.
  #
  #   g = {1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]}
  #   each_node = lambda {|&b| g.each_key(&b) }
  #   each_child = lambda {|n, &b| g[n].each(&b) }
  #   p TSort.strongly_connected_components(each_node, each_child)
  #   #=> [[4], [2], [3], [1]]
  #
  #   g = {1=>[2], 2=>[3, 4], 3=>[2], 4=>[]}
  #   each_node = lambda {|&b| g.each_key(&b) }
  #   each_child = lambda {|n, &b| g[n].each(&b) }
  #   p TSort.strongly_connected_components(each_node, each_child)
  #   #=> [[4], [2, 3], [1]]
  #
  def TSort.strongly_connected_components(each_node, each_child)
    result = []
    TSort.each_strongly_connected_component(each_node, each_child) {|component| result << component}
    result
  end

  # The iterator version of the #strongly_connected_components method.
  # <tt><em>obj</em>.each_strongly_connected_component</tt> is similar to
  # <tt><em>obj</em>.strongly_connected_components.each</tt>, but
  # modification of _obj_ during the iteration may lead to unexpected results.
  #
  # #each_strongly_connected_component returns +nil+.
  #
  #   class G
  #     include TSort
  #     def initialize(g)
  #       @g = g
  #     end
  #     def tsort_each_child(n, &b) @g[n].each(&b) end
  #     def tsort_each_node(&b) @g.each_key(&b) end
  #   end
  #
  #   graph = G.new({1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]})
  #   graph.each_strongly_connected_component {|scc| p scc }
  #   #=> [4]
  #   #   [2]
  #   #   [3]
  #   #   [1]
  #
  #   graph = G.new({1=>[2], 2=>[3, 4], 3=>[2], 4=>[]})
  #   graph.each_strongly_connected_component {|scc| p scc }
  #   #=> [4]
  #   #   [2, 3]
  #   #   [1]
  #
  def each_strongly_connected_component(&block) # :yields: nodes
    each_node = method(:tsort_each_node)
    each_child = method(:tsort_each_child)
    TSort.each_strongly_connected_component(each_node, each_child, &block)
  end

  # The iterator version of the TSort.strongly_connected_components method.
  #
  # The graph is represented by _each_node_ and _each_child_.
  # _each_node_ should have +call+ method which yields for each node in the graph.
  # _each_child_ should have +call+ method which takes a node argument and yields for each child node.
  #
  #   g = {1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]}
  #   each_node = lambda {|&b| g.each_key(&b) }
  #   each_child = lambda {|n, &b| g[n].each(&b) }
  #   TSort.each_strongly_connected_component(each_node, each_child) {|scc| p scc }
  #   #=> [4]
  #   #   [2]
  #   #   [3]
  #   #   [1]
  #
  #   g = {1=>[2], 2=>[3, 4], 3=>[2], 4=>[]}
  #   each_node = lambda {|&b| g.each_key(&b) }
  #   each_child = lambda {|n, &b| g[n].each(&b) }
  #   TSort.each_strongly_connected_component(each_node, each_child) {|scc| p scc }
  #   #=> [4]
  #   #   [2, 3]
  #   #   [1]
  #
  def TSort.each_strongly_connected_component(each_node, each_child) # :yields: nodes
    id_map = {}
    stack = []
    each_node.call {|node|
      unless id_map.include? node
        TSort.each_strongly_connected_component_from(node, each_child, id_map, stack) {|c|
          yield c
        }
      end
    }
    nil
  end

  # Iterates over strongly connected component in the subgraph reachable from
  # _node_.
  #
  # Return value is unspecified.
  #
  # #each_strongly_connected_component_from doesn't call #tsort_each_node.
  #
  #   class G
  #     include TSort
  #     def initialize(g)
  #       @g = g
  #     end
  #     def tsort_each_child(n, &b) @g[n].each(&b) end
  #     def tsort_each_node(&b) @g.each_key(&b) end
  #   end
  #
  #   graph = G.new({1=>[2, 3], 2=>[4], 3=>[2, 4], 4=>[]})
  #   graph.each_strongly_connected_component_from(2) {|scc| p scc }
  #   #=> [4]
  #   #   [2]
  #
  #   graph = G.new({1=>[2], 2=>[3, 4], 3=>[2], 4=>[]})
  #   graph.each_strongly_connected_component_from(2) {|scc| p scc }
  #   #=> [4]
  #   #   [2, 3]
  #
  def each_strongly_connected_component_from(node, id_map={}, stack=[], &block) # :yields: nodes
    TSort.each_strongly_connected_component_from(node, method(:tsort_each_child), id_map, stack, &block)
  end

  # Iterates over strongly connected components in a graph.
  # The graph is represented by _node_ and _each_child_.
  #
  # _node_ is the first node.
  # _each_child_ should have +call+ method which takes a node argument
  # and yields for each child node.
  #
  # Return value is unspecified.
  #
  # #TSort.each_strongly_connected_component_from is a class method and
  # it doesn't need a class to represent a graph which includes TSort.
  #
  #   graph = {1=>[2], 2=>[3, 4], 3=>[2], 4=>[]}
  #   each_child = lambda {|n, &b| graph[n].each(&b) }
  #   TSort.each_strongly_connected_component_from(1, each_child) {|scc|
  #     p scc
  #   }
  #   #=> [4]
  #   #   [2, 3]
  #   #   [1]
  #
  def TSort.each_strongly_connected_component_from(node, each_child, id_map={}, stack=[]) # :yields: nodes
    minimum_id = node_id = id_map[node] = id_map.size
    stack_length = stack.length
    stack << node

    each_child.call(node) {|child|
      if id_map.include? child
        child_id = id_map[child]
        minimum_id = child_id if child_id && child_id < minimum_id
      else
        sub_minimum_id =
          TSort.each_strongly_connected_component_from(child, each_child, id_map, stack) {|c|
            yield c
          }
        minimum_id = sub_minimum_id if sub_minimum_id < minimum_id
      end
    }

    if node_id == minimum_id
      component = stack.slice!(stack_length .. -1)
      component.each {|n| id_map[n] = nil}
      yield component
    end

    minimum_id
  end

  # Should be implemented by a extended class.
  #
  # #tsort_each_node is used to iterate for all nodes over a graph.
  #
  def tsort_each_node # :yields: node
    raise NotImplementedError.new
  end

  # Should be implemented by a extended class.
  #
  # #tsort_each_child is used to iterate for child nodes of _node_.
  #
  def tsort_each_child(node) # :yields: child
    raise NotImplementedError.new
  end
end