# module TSort

`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¶ ↑

1. 1. Tarjan, “Depth First Search and Linear Graph Algorithms”,

SIAM Journal on Computing, Vol. 1, No. 2, pp. 146-160, June 1972.

### Public Class Methods

each_strongly_connected_component(each_node, each_child) { |nodes| ... } click to toggle source

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]

    # File lib/tsort.rb, line 342
    def TSort.each_strongly_connected_component(each_node, each_child) # :yields: nodes
      return to_enum(__method__, each_node, each_child) unless block_given?

      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

each_strongly_connected_component_from(node, each_child, id_map={}, stack=[]) { |nodes| ... } click to toggle source

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]

    # File lib/tsort.rb, line 408
    def TSort.each_strongly_connected_component_from(node, each_child, id_map={}, stack=[]) # :yields: nodes
      return to_enum(__method__, node, each_child, id_map, stack) unless block_given?

      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

strongly_connected_components(each_node, each_child) click to toggle source

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]]

    # File lib/tsort.rb, line 280
    def TSort.strongly_connected_components(each_node, each_child)
      TSort.each_strongly_connected_component(each_node, each_child).to_a
    end

tsort(each_node, each_child) click to toggle source

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

    # File lib/tsort.rb, line 175
    def TSort.tsort(each_node, each_child)
      TSort.tsort_each(each_node, each_child).to_a
    end

tsort_each(each_node, each_child) { |node| ... } click to toggle source

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

    # File lib/tsort.rb, line 223
    def TSort.tsort_each(each_node, each_child) # :yields: node
      return to_enum(__method__, each_node, each_child) unless block_given?

      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

### Public Instance Methods

each_strongly_connected_component() { |nodes| ... } click to toggle source

The iterator version of the `strongly_connected_components` method. `obj.each_strongly_connected_component` is similar to `obj.strongly_connected_components.each`, 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]

    # File lib/tsort.rb, line 313
    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

each_strongly_connected_component_from(node, id_map={}, stack=[]) { |nodes| ... } click to toggle source

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]

    # File lib/tsort.rb, line 383
    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

strongly_connected_components() click to toggle source

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]]

    # File lib/tsort.rb, line 254
    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

tsort() click to toggle source

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

    # File lib/tsort.rb, line 149
    def tsort
      each_node = method(:tsort_each_node)
      each_child = method(:tsort_each_child)
      TSort.tsort(each_node, each_child)
    end

tsort_each() { |node| ... } click to toggle source

The iterator version of the `tsort` method. `obj.tsort_each` is similar to `obj.tsort.each`, 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

    # File lib/tsort.rb, line 202
    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

tsort_each_child(node) { |child| ... } click to toggle source

Should be implemented by a extended class.

`tsort_each_child` is used to iterate for child nodes of node.

    # File lib/tsort.rb, line 449
    def tsort_each_child(node) # :yields: child
      raise NotImplementedError.new
    end

tsort_each_node() { |node| ... } click to toggle source

Should be implemented by a extended class.

`tsort_each_node` is used to iterate for all nodes over a graph.

    # File lib/tsort.rb, line 441
    def tsort_each_node # :yields: node
      raise NotImplementedError.new
    end