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Author image Ed J
and 1 contributors


Graph::TransitiveClosure::Matrix - create and query transitive closure of graph


    use Graph::TransitiveClosure::Matrix;
    use Graph::Directed; # or Undirected

    my $g  = Graph::Directed->new;
    $g->add_...(); # build $g

    # Compute the transitive closure matrix.
    my $tcm = Graph::TransitiveClosure::Matrix->new($g);

    # Being reflexive is the default,
    # meaning that null transitions are included.
    my $tcm = Graph::TransitiveClosure::Matrix->new($g, reflexive => 1);
    $tcm->is_reachable($u, $v)

    # is_reachable(u, v) is always reflexive.
    $tcm->is_reachable($u, $v)

    # The reflexivity of is_transitive(u, v) depends on the reflexivity
    # of the transitive closure.
    $tcg->is_transitive($u, $v)

    my $tcm = Graph::TransitiveClosure::Matrix->new($g, path_length => 1);
    my $n = $tcm->path_length($u, $v)

    my $tcm = Graph::TransitiveClosure::Matrix->new($g, path_vertices => 1);
    my @v = $tcm->path_vertices($u, $v)

    my $tcm =
                                              attribute_name => 'length');
    my $n = $tcm->path_length($u, $v)

    my @v = $tcm->vertices


You can use Graph::TransitiveClosure::Matrix to compute the transitive closure matrix of a graph and optionally also the minimum paths (lengths and vertices) between vertices, and after that query the transitiveness between vertices by using the is_reachable() and is_transitive() methods, and the paths by using the path_length() and path_vertices() methods.

If you modify the graph after computing its transitive closure, the transitive closure and minimum paths may become invalid.


Class Methods


Construct the transitive closure matrix of the graph $g.

new($g, options)

Construct the transitive closure matrix of the graph $g with options as a hash. The known options are

attribute_name => attribute_name

By default the edge attribute used for distance is weight. You can change that by giving another attribute name with the attribute_name attribute to the new() constructor.

reflexive => boolean

By default the transitive closure matrix is not reflexive: that is, the adjacency matrix has zeroes on the diagonal. To have ones on the diagonal, use true for the reflexive option.

path => boolean

If set true, sets path_length and path_vertices. If either of those are true (and path_vertices is by default), then both are calculated.

path_length => boolean

By default "false", but see above as overridden by default path_vertices being true. If calculated, they can be retrieved using the path_length() method.

path_vertices => boolean

By default the paths are computed, with the boolean transitivity, they can be retrieved using the path_vertices() method.

path_count => boolean

As an alternative to setting path_length, if this is true then the matrix will store the quantity of paths between the two vertices. This is still retrieved using the path_length() method. The path vertices will not be available. You should probably only use this on a DAG, and not with reflexive.

Object Methods

is_reachable($u, $v)

Return true if the vertex $v is reachable from the vertex $u, or false if not.

path_length($u, $v)

Return the minimum path length from the vertex $u to the vertex $v, or undef if there is no such path.

path_vertices($u, $v)

Return the minimum path (as a list of vertices) from the vertex $u to the vertex $v, or an empty list if there is no such path, OR also return an empty list if $u equals $v.

has_vertices($u, $v, ...)

Return true if the transitive closure matrix has all the listed vertices, false if not.

is_transitive($u, $v)

Return true if the vertex $v is transitively reachable from the vertex $u, false if not.


Return the list of vertices in the transitive closure matrix.

path_successor($u, $v)

Return the successor of vertex $u in the transitive closure path towards vertex $v.

all_paths($u, $v)

Return list of array-refs with all the paths from $u to $v. Will ignore self-loops.


For path_length() the return value will be the sum of the appropriate attributes on the edges of the path, weight by default. If no attribute has been set, one (1) will be assumed.

If you try to ask about vertices not in the graph, undefs and empty lists will be returned.


The transitive closure algorithm used is Warshall and Floyd-Warshall for the minimum paths, which is O(V**3) in time, and the returned matrices are O(V**2) in space.




Jarkko Hietaniemi jhi@iki.fi


This module is licensed under the same terms as Perl itself.