Graph::TransitiveClosure::Matrix(3) create and query transitive closure of graph

## SYNOPSIS

use Graph::TransitiveClosure::Matrix;
use Graph::Directed; # or Undirected
my \$g = Graph::Directed->new;
# 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 of 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 =
Graph::TransitiveClosure::Matrix->new(\$g,
attribute_name => 'length');
my \$n = \$tcm->path_length(\$u, \$v)
my @v = \$tcm->vertices

## DESCRIPTION

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

new(\$g)
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 "w". 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.

NOTE: this behaviour has changed from Graph 0.2xxx: transitive closure graphs were by default reflexive.

path_length => boolean
By default the path lengths are not computed, only the boolean transitivity. By using true for "path_length" also the path lengths will be computed, they can be retrieved using the path_length() method.
path_vertices => boolean
By default the paths are not computed, only the boolean transitivity. By using true for "path_vertices" also the paths will be computed, they can be retrieved using the path_vertices() method.

## 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.
vertices
Return the list of vertices in the transitive closure matrix.
path_predecessor
Return the predecessor of vertex \$v in the transitive closure path going back to vertex \$u.

## RETURN VALUES

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.

## ALGORITHM

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.