I'd seen
Abigail do nasty things with regexes before, but never really understood them at all. However, in the recent
N-Queens solver, it was a pure regex (backrefs only), there were no
(?{}) constructs, and I was finally able to understand how
Abigail does it..
Then, on a regex kick, I discovered Abigail's pure regex 3-SAT reduction. If you haven't already seen it, it's the coolest couple of lines on the planet. With all this regex excitement, I decided to try my hand at solving some NP-complete problem(s) with pure regexes.
I tried a handful of different NP-complete problems, but after a few emails with MJD, I understood why these attempts went wrong. I kept getting hung up on the string and regex size being exponential on the size of input (not meaning my solution was incorrect, but invalidating its use as an NP-completeness reduction proof). Finally, I think I may have gotten it right with Hamiltonian Circuits. Abigail has done this before using extended regexes, but claimed to be stuck on a pure regex solution that wasn't exponential.
So without further ado, here's my solution. Given a (simple, undirected) graph with E edges and V vertices, it finds a Hamiltonian Circuit (a cycle that visits every vertex exactly once, starting and ending in the same place). The size of the string it creates is bounded by O(V^4), and the size of the regex it creates is bounded by O(V^2).
my @E = ([1,3],[1,5],[2,3],[2,4],[2,5],[4,5]);
my $V = 5;
my $verbose = 1;
my @all_edges = map { my $x = $_; map { [$x, $_] } $x+1 .. $V } 1 .. $
+V-1;
my $string = (join(' ', 1 .. $V) . "\n") x $V
. "\n"
. (join(' ', map { join "-", @$_ } @all_edges ) . "\n")
x @all_edges
. "\n"
. (join(' ', map { join "-", @$_ } @E ) . "\n") x $V;
my $regex = "^ "
. ".* \\b (\\d+) \\b .* \\n\n" x $V
. "\\n\n"
. join("", map { my ($x, $y) = @$_;
".* \\b (?: \\$x-\\$y | \\$y-\\$x ) \\b .*
+\\n\n"
} @all_edges)
. "\\n\n"
. join("", map { my ($x, $y) = ($_, $_+1);
".* \\b (?: \\$x-\\$y | \\$y-\\$x ) \\b .*
+\\n\n"
} 1 .. ($V-1))
. ".* \\b (?: \\$V-\\1 | \\1-\\$V ) \\b .* \\n \$\n";
print "'$string' =~ /\n$regex\n/x\n" if $verbose;
if (my @c = $string =~ /$regex/x) {
local $" = " -> ";
print "Hamiltonian circuit: [ @c -> $1 ]\n";
} else {
print "No Hamiltonian circuit\n";
}
Now the dirty details.. Here's the value of
$string and
$regex for the example graph included:
$string = q[
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-2 1-3 1-4 1-5 2-3 2-4 2-5 3-4 3-5 4-5
1-3 1-5 2-3 2-4 2-5 4-5
1-3 1-5 2-3 2-4 2-5 4-5
1-3 1-5 2-3 2-4 2-5 4-5
1-3 1-5 2-3 2-4 2-5 4-5
1-3 1-5 2-3 2-4 2-5 4-5
];
$regex = q[^
.* \b (\d+) \b .* \n
.* \b (\d+) \b .* \n
.* \b (\d+) \b .* \n
.* \b (\d+) \b .* \n
.* \b (\d+) \b .* \n
\n
.* \b (?: \1-\2 | \2-\1 ) \b .* \n
.* \b (?: \1-\3 | \3-\1 ) \b .* \n
.* \b (?: \1-\4 | \4-\1 ) \b .* \n
.* \b (?: \1-\5 | \5-\1 ) \b .* \n
.* \b (?: \2-\3 | \3-\2 ) \b .* \n
.* \b (?: \2-\4 | \4-\2 ) \b .* \n
.* \b (?: \2-\5 | \5-\2 ) \b .* \n
.* \b (?: \3-\4 | \4-\3 ) \b .* \n
.* \b (?: \3-\5 | \5-\3 ) \b .* \n
.* \b (?: \4-\5 | \5-\4 ) \b .* \n
\n
.* \b (?: \1-\2 | \2-\1 ) \b .* \n
.* \b (?: \2-\3 | \3-\2 ) \b .* \n
.* \b (?: \3-\4 | \4-\3 ) \b .* \n
.* \b (?: \4-\5 | \5-\4 ) \b .* \n
.* \b (?: \5-\1 | \1-\5 ) \b .* \n $
];
__OUTPUT__
Hamiltonian circuit: [ 5 -> 4 -> 2 -> 3 -> 1 -> 5 ]
Here's how it works:
- In the first "paragraph", $1 through $5 each get a vertex assigned to them. This paragraph of $string is bounded by O(V^2), and $regex by O(V).
- The second "paragraph" is the ugly bit. We have to make sure all the vertices chosen are different. This could have been done by originally choosing $1 through $5 to be any permutation from an exhaustive list of permutations, but such a list would have been exponential in size.
The way I checked is by picking any 5 vertices, then ensuring that they are all pairwise distinct. In $string, we have a repeated list of all "valid" (distinct) vertex pairs. In $regex, we make sure that every pair of two vertices in {$1,..,$5} shows up in that list.
This isn't exponential, because there are V(V-1)/2 "valid" (distinct) pairs, and V(V-1)/2 pairs to verify. So this paragraph of $string is bounded by O(V^4), and $regex by O(V^2).
- The third "paragraph" is where we verify that our distinct vertex set has edges from $1 to $2 ... to $5 and then back to $1. This paragraph of $string is bounded by O(E*V) = O(V^3), and $regex by O(V).
If all these conditions matched, then the regex matches, and $1 through $5 must be the vertices of a Hamiltonian Circuit.
There you have it. Your feedback is welcome. I really hope I haven't made any mistakes in my analysis. This code isn't one-tenth as beautiful and elegant as Abigail's 3-SAT reduction, and perhaps it could be improved upon. But hey, we all have to start somewhere ;)
Update: modified code so that $string is a little clearer to read. Instead of comma-separating the edge and vertex lists, they are separated by spaces. $regex is modified accordingly, matching on \b where appropriate, instead of a comma.
