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.

HISTORY

    $Date: 1999/03/01 19:51:11 $

    $Log: TransitiveClosure.pm,v $
    Revision 1.3  1999/03/01 19:51:11  abigail
    Renamed primary namespace to Algorithm::

    Revision 1.2  1998/03/23 04:00:37  abigail
    Fixed order of loop variables in the ARRAYREF case.
    It's k, i, j, not i, j, k.

    Revision 1.1  1998/03/22 06:54:42  abigail
    Initial revision

AUTHOR

This package was written by Abigail.

COPYRIGHT

Copyright 1998 by Abigail.

LICENSE

This package is free and open software.

You may use, copy, modify, distribute and sell this package or any modifications there of in any form you wish, provided you do not do any of the following:

    - claim that any of the original code was written by someone
      else than the original author(s).
    - restrict someone in using, copying, modifying, distributing or
      selling this program or module or any modifications of it.

THIS PACKAGE IS PROVIDED "AS IS" AND WITHOUT ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.