sub to_key {
   my ($x, $y, $s) = @_;
   return pack 'L', ($x-MIN_X) * (MAX_Y-MIN_Y) * (MAX_S-MIN_S)
                  + ($y-MIN_Y)                 * (MAX_S-MIN_S)
                  + ($s-MIN_S)
}

sub fr_key {
   my $key = unpack 'L', $_[0];
   return (
      int( $key / (MAX_Y-MIN_Y) / (MAX_S-MIN_S) )                 + MIN_X,
      int( $key                 / (MAX_S-MIN_S) ) % (MAX_Y-MIN_Y) + MIN_Y,
      int( $key                                 ) % (MAX_S-MIN_S) + MIN_S,
   );
}

$paths->{to_key(0,0,0)} = join '',
   to_key(3,0,0),
);
  
$paths->{to_key(0,1,0)} = join '',
   to_key(0,2,0),
);

...