Question

I have an array of pairs like this:

[["a", "b"], ["b", "d"], ["a", "c"], ["e", "d"], ["a", "d"], ..., ["s", "f"]]

1. What is an efficient way to check if the given array can express a partial ordering? That is, there is no “loop” in the given array like ["a", "b"], ["b", "c"], ["c", "a"].

2. If it is confirmed that the array expresses a partial order, I want to normalize this by removing all of the pairs that can be derived by reflexivity or transitivity. For example, in the above, since there is ["a", "b"] and ["b", "d"], the pair ["a", "d"] is redundant, and should be removed.

The order between 1 and 2 does not matter. If 2 should be done before or within the process of 1, then, that is fine.

Preferably I want it in Ruby 1.9.3, but just pseudo-code will suffice.

Answer 1

For the first part of the question, I came up with my own answer here with the help of an answer at a mathematics site.

For the second part of the question, after following the suggestions given in the other answers, I implemented in Ruby (i) Floyd-Warshall algorithm to calculate the transitive closure, (ii) composition, and (iii) transitive reduction using the formula R^- = R – R \cdot R^+.

module Digraph; module_function
    def vertices graph; graph.flatten(1).uniq end
    ## Floyd-Warshall algorithm
    def transitive_closure graph
        vs = vertices(graph)
        path = graph.inject({}){|path, e| path[e] = true; path}
        vs.each{|k| vs.each{|i| vs.each{|j| path[[i, j]] ||= true if path[[i, k]] && path[[k, j]]}}}
        path.keys
    end
    def compose graph1, graph2
        vs = (vertices(graph1) + vertices(graph2)).uniq
        path1 = graph1.inject({}){|path, e| path[e] = true; path}
        path2 = graph2.inject({}){|path, e| path[e] = true; path}
        path = {}
        vs.each{|k| vs.each{|i| vs.each{|j| path[[i, j]] ||= true if path1[[i, k]] && path2[[k, j]]}}}
        path.keys
    end
    def transitive_reduction graph
            graph - compose(graph, transitive_closure(graph))
    end
end

Usage examples:

Digraph.transitive_closure([[1, 2], [2, 3], [3, 4]])
#=> [[1, 2], [2, 3], [3, 4], [1, 3], [1, 4], [2, 4]]

Digraph.compose([[1, 2], [2, 3]], [[2, 4], [3, 5]])
#=> [[1, 4], [2, 5]]

Digraph.transitive_reduction([[1, 2], [2, 3], [3, 4], [1, 3], [1, 4], [2, 4]])
#=> [[1, 2], [2, 3], [3, 4]]