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1; # End of Algorithm::Graphs::TransitiveClosure::Tiny




__END__


=head1 NAME

Algorithm::Graphs::TransitiveClosure::Tiny - Calculate the transitive closure.


=head1 VERSION

Version 1.03


=head1 SYNOPSIS

    use Algorithm::Graphs::TransitiveClosure::Tiny qw(floyd_warshall);

    # The hash values here need not to be undef, but floyd_warshall()
    # only adds undef.
    my $graph = {
                 0 => {0 => undef},
                 1 => {1 => undef, 2 => undef, 3 => undef},
                 2 => {1 => undef, 2 => undef},
                 3 => {0 => undef, 2 => undef, 3 => undef},
                };

    floyd_warshall $graph;

    print "There is a path from 2 to 0.\n" if
        exists($graph->{2}) && exists($graph->{2}->{0});

The latter can also be written shorter provided you accept autovivification:

    print "There is a path from 2 to 0.\n" if exists($graph->{2}->{0});



=head1 DESCRIPTION

This module provides a single function, C<floyd_warshall>, which is exported
on demand. It is an implementation of the well known I<Floyd-Warshall>
algorithm for computing the transitive closure of a graph.

The code is taken from L<Algorithm::Graphs::TransitiveClosure> but has been
modified. The difference is that this implementation of C<floyd_warshall()>:

=over

=item *

works on hashes only,

=item *

uses C<undef> for hash values, so an incidence must be checked with
C<exists()> (but for the input hash you are not forced to use C<undef>),

=item *

fixes following problem of L<Algorithm::Graphs::TransitiveClosure>:

Example:

   my $g = {
            0 => { 2 => 1},
            1 => { 0 => 1},
           };

There is an edge from 0 to 2 and an edge from 1 to 0. So the transitive
closure would contain an edge from 1 to 2. But calling C<floyd_warshall($g)>
from L<Algorithm::Graphs::TransitiveClosure> results in:

           {
            0 => { 2 => 1},
            1 => { 0 => 1},
           }

No change. The edge from 1 to 2 is missing (you would need to add C<2=E<gt>{}>
to C<$g> to get it right). But if you call C<floyd_warshall($g)> from
C<Algorithm::Graphs::TransitiveClosure::Tiny>, then the result is correct:

           {
            0 => { 2 => 1},
            1 => { 0 => 1,
                   2 => undef},
           }

Edge from 1 to 2 has been added! (Also note that it was possible to use 1
instead of C<undef> as hash value. This value is kept, but the value added by
the function is still C<undef>!)


=item *

By default, C<floyd_warshall($graph)> removes empty subhashes from C<$graph>,
e.g.

    my $graph = {
                 this => {that => undef},
                 that => {}
                };
    floyd_warshall($graph);

will result in

   {
    this => {that => undef}
   }

This behavior can be changed by setting optional second argument of
C<floyd_warshall> to a true value, i.e., calling C<floyd_warshall($graph, 1)>
with the above example hash will not remove C<that =E<gt> {}>.


=back