Transitive closure

[larsen]

Conjuring on an adjacency matrix (thanks to Math::MatrixReal), this snippet finds every couple of nodes (i, j) such that there's a path from i to j. The matrix $sum is called transitive closure of the graph $graph.

#!/usr/bin/perl

# Transitive closure of a directed graph

use strict;

use Math::MatrixReal;

my $n = 4; # Matrix' size
my $graph = Math::MatrixReal->new_from_string( <<'MATRIX' );
[ 0 1 0 1 ]
[ 0 0 1 0 ]
[ 0 0 0 0 ]
[ 1 1 0 0 ]
MATRIX

my $sum = $graph->shadow();
my $p = $graph->shadow();
$p->one();

# "One Ring to rule them all, One Ring to find them..."

# Sum of A^i, for i = 0..n-1 (A is the adjacency matrix of our graph)

foreach (0 .. ($n - 1)) {
    $p = $p * $graph;
    $sum = $sum + $p;
}

# Finished.
# Now we print every couple (i, j) such that
# there's a path from i to j.

foreach my $i (1..$n) {
    foreach my $j (1..$n) {
        print "There's a path from $i to $j.\n" if $sum->element( $i, 
+$j );
    }
}

[Zaxo]

For efficient use on larger networks, Graph is my new favorite toy:

use Graph::Directed;
my $net = new Graph::Directed;
# populate with $net->add_vertex(), $net->add_edge(),
# $net->add_edges(), $net->add_path()
my $tc = $net->TransitiveClosure_Floyd_Warshall;

After Compline,
Zaxo

[larsen]

Yes, sure. I was interested in is the application of algebric methods to problems concerning directed graphs.

Interpreting a graph like a relation allows to compose such relation more and more, as in my snippet (the interesting part is that multiplying two matrices is like composing two relations).

So, x A y means there's an arc from x to y. And x AA y (i.e. x A² y) means: there is some z such that x A z and z A y, so there's a path from x to y through z. When I learned that, I said "hey so those tricky rules for multiplying matrices make sense!".

Another interesting point is that working on algebric structures one could use different operators than sum and product, on condition that the underlying structure is preserved (this is the meaning of my Tolkien quote). Try to reimplement + and * in the matrix product algorithm with min and +.