#!/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 );
	}
}