NAME
    Algorithms::Graphs::TransitiveClosure - Calculate the transitive
    closure.

SYNOPSIS
        use Algorithms::Graphs::TransitiveClosure qw /floyd_warshall/;

        my $graph = [[1, 0, 0, 0], [0, 1, 1, 1], [0, 1, 1, 0], [1, 0, 1, 1]];
        floyd_warshall $graph;
        print "There is a path from 2 to 0.\n" if $graph -> [2] -> [0];

        my $graph2 = {one   => {one => 1},
                      two   => {two => 1, three => 1, four => 1},
                      three => {two => 1, three => 1},
                      four  => {one => 1, four  => 1}};
        floyd_warshall $graph2;
        print "There is a path from three to one.\n" if
            $graph2 -> {three} -> {one};

DESCRIPTION
    This is an implementation of the well known *Floyd-Warshall* algorithm.
    [1,2]

    The subroutine "floyd_warshall" takes a directed graph, and calculates
    its transitive closure, which will be returned. The given graph is
    actually modified, so be sure to pass a copy of the graph to the routine
    if you need to keep the original graph.

    The subroutine takes graphs in one of the two following formats:

    floyd_warshall ARRAYREF
        The graph *G = (V, E)* is described with a list of lists, $graph,
        representing *V x V*. If there is an edge between vertices $i and $j
        (or if "$i == $j"), then "$graph -> [$i] -> [$j] == 1". For all
        other pairs "($k, $l)" from *V x V*, "$graph -> [$k] -> [$l] == 0".

        The resulting $graph will have "$graph -> [$i] -> [$j] == 1" iff "$i
        == $j" or there is a path in *G* from $i to $j, and "$graph -> [$i]
        -> [$j] == 0" otherwise.

    floyd_warshall HASHREF
        The graph *G = (V, E)*, with labeled vertices, is described with a
        hash of hashes, $graph, representing *V x V*. If there is an edge
        between vertices $label1 and $label2 (or if "$label1 eq $label2"),
        then "$graph -> {$label1} -> {$label2} == 1". For all other pairs
        "($label3, $label4)" from *V x V*, "$graph -> {$label3} ->
        {$label4}" does not exist.

        The resulting $graph will have "$graph -> {$label1} -> {$label2} ==
        1" iff "$label1 eq $label2" or there is a path in *G* from $label1
        to $label2, and "$graph -> {$label1} -> {$label2}" does not exist
        otherwise.

EXAMPLES
        my $graph = [[1, 0, 0, 0],
                     [0, 1, 1, 1],
                     [0, 1, 1, 0],
                     [1, 0, 1, 1]];
        floyd_warshall $graph;
        foreach my $row (@$graph) {print "@$row\n"}

        1 0 0 0
        1 1 1 1
        1 1 1 1
        1 1 1 1

        my $graph = {one   => {one => 1},
                     two   => {two => 1, three => 1, four => 1},
                     three => {two => 1, three => 1},
                     four  => {one => 1, three => 1, four => 1}};
        floyd_warshall $graph;
        foreach my $l1 (qw /one two three four/) {
            print "$l1: ";
            foreach my $l2 (qw /one two three four/) {
                next if $l1 eq $l2;
                print "$l2 " if $graph -> {$l1} -> {$l2};
            }
            print "\n";
        }

        one: 
        two: one three four 
        three: one two four 
        four: one two three

COMPLEXITY
    The running time of the algorithm is cubed in the number of vertices of
    the graph. The author of this package is not aware of any faster
    algorithms, nor of a proof if this is optimal. Note than in specific
    cases, when the graph can be embedded on surfaces of bounded genus, or
    in the case of sparse connection matrices, faster algorithms than cubed
    in the number of vertices exist.

    The space used by this algorithm is at most quadratic in the number of
    vertices, which is optimal as the resulting transitive closure can have
    a quadratic number of edges. In case when the graph is represented as a
    list of lists, the quadratic bound will always be achieved, as the list
    of lists already has that size. The hash of hashes however will use
    space linear to the size of the resulting transitive closure.

LITERATURE
    The Floyd-Warshall algorithm is due to Floyd [2], and based on a theorem
    of Warshall [3]. The implemation of this package is based on an
    implementation of Floyd-Warshall found in Cormen, Leiserson and Rivest
    [1].

REFERENCES
    [1] Thomas H. Cormen, Charles E. Leiserson and Ronald L. Rivest:
        *Introduction to Algorithms*. Cambridge: MIT Press, 1990. ISBN
        0-262-03141-8.

    [2] Robert W. Floyd: "Algorithm 97 (SHORTEST PATH)". *Communications of
        the ACM*, 5(6):345, 1962.

    [3] Stephan Warshall: "A theorem on boolean matrices." *Journal of the
        ACM*, 9(1):11-12, 1962.

DEVELOPMENT
    The current sources of this module are found on github,
    <git://github.com/Abigail/test--regexp.git>.

AUTHOR
    Abigail <mailto:algorithm-graphs-transitiveclosure@abigail.be>.

COPYRIGHT and LICENSE
    Copyright (C) 1998, 2009 by Abigail

    Permission is hereby granted, free of charge, to any person obtaining a
    copy of this software and associated documentation files (the
    "Software"), to deal in the Software without restriction, including
    without limitation the rights to use, copy, modify, merge, publish,
    distribute, sublicense, and/or sell copies of the Software, and to
    permit persons to whom the Software is furnished to do so, subject to
    the following conditions:

    The above copyright notice and this permission notice shall be included
    in all copies or substantial portions of the Software.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
    OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
    MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
    IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
    CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
    TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
    SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.