The runtime of the subroutine isn't that big.

Update:

Change the testing part at the bottom to read:

my $a = "abbacbacccabbacbacccccccccccgcccc";
my $b = "aabaabcbavcvcbvbvnbmhgfdgf";
print "LCS: ", lcs($a, $b), "\n\n";
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real    0m0.043s
user    0m0.039s
sys     0m0.002s
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# lcs.pl
use strict;
use Memoize;

sub longerOf {
    my ($x, $y) = @_;
    return (length $x > length $y) ? $x : $y;
}

memoize('lcs');
sub lcs {
    my ($a, $b) = @_;

    if ($a eq "" || $b eq ""){
        return "";
    }

    my ($az, $bz) = (chop $a, chop $b);
    if ($az eq $bz){
        return lcs($a, $b) . $az;
    } else {
        return longerOf(
                  lcs($a . $az, $b),
                  lcs($b . $bz, $a));
    }
}

while (1){
    print "1: "; my $a = <>; chomp $a;
    print "2: "; my $b = <>; chomp $b;
$start = time();
    print "LCS: ", lcs($a, $b), "\n\n";
  $end = time();
   print "<br>Time taken was ", ($end - $start), " seconds";

$start = time();
    print "Brute Force: ", lcsbruteforce($a, $b), "\n\n";
  $end = time();
   print "<br>Time taken was ", ($end - $start), " seconds";
}


sub lcsbruteforce {
  my($x, $y) = @_;
  my(@v, $cx, $cy, $left, $above);
  for my $xi (0 .. length($x) - 1) {
    $cx = substr $x, $xi, 1;
    for my $yi (0 .. length($y) - 1) {
      $cy = substr $y, $yi, 1;
      if ($cx eq $cy) {
        $v[$xi][$yi] = 1 + (($xi && $yi) ? $v[$xi - 1][$yi - 1] : 0);
      } else {
        $left = ($xi && $v[$xi - 1][$yi]) || 0;
        $above = ($xi && $v[$xi][$yi - 1]) || 0;
        $v[$xi][$yi] = ($left > $above) ? $left : $above;
      }
    }
  }
  return $v[length($x) - 1][length($y) - 1];
}
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Don't let the name of the subroutine fool you. The "brute force" algorithm is not really "brute forcing" the problem. A brute force approach would be to consider every possible subsequence of the strings, taking O(2^min(x,y)) time.

In fact, the "brute force" algorithm is doing the same thing as the recursive algorithm (i.e., doing the dynamic programming solution), but iteratively. It uses a standard trick for making a recursive memoizing dynamic programming algorithm iterative. Since the two algorithms solve the problem in essentially the same way, but the iterative one doesn't have the overhead of subroutine calls (which are slow in Perl), it is no surprise that the iterative one is faster.

Usually it's easier and more intuitive to write a dynamic programming problem in terms of recursive calls. However, it's necessary to memoize the result of each recursive call, because several other subproblems might use that result of this subproblem in their computation.

Now imagine a table that holds all of these memoized results. What happens to this table while the recursive algorithm is running? The table is gradually filling up. How does it fill up? Well, in this case, to compute the value of the subproblem ($a,$b), I need to get the solutiosn for at most these three subproblems:

($a,substr($b,0,-1)), (substr($a,0,-1),b), (substr($a,0,-1),substr($b,0,-1))

In other words, I need to have those 3 cells in the table filled in before I can fill in this cell.

So suppose I now do things iteratively instead of recursively, and just concentrate on filling up the table. I'll visit the table's cells in such a way so that I visit the cell ($a,$b) after I visit the three above cells. That way, to fill up the cell ($a,$b), I just check those 3 other cells, do some local comparisons, and I'm done. Finally, the last cell in the table is generally the answer to the "main" subproblem, and I can return that. That's exactly what this "brute force" algorithm is doing.

blokhead

Thanks for the explanation. How would I modify the above brute force code to be truely brute force? Also, the code only print out the length of the sequence but does not print out the characters of the sequence. I tried putting in some print statement in between but it does not seem to work correctly. Any helps?

Thank you Blokhead. I still have lots to learn about Perl. I have another question for you. The sub lcs takes 2 parameters ($a, $b) but inside the loop where the recursive call is made, the program did a call with

return lcs($a, $b) . $az;

what does the . $az do? I tried printing out the $a, $b but did not see any differences. The new string is one character shorter. When I remove $az from the return, the program produce the wrong result. Thanks again for your help.

I haven't look too close at your code, but it seems the "brute force" approach takes O(length($x) * length($y)) time.

The recursive method is a bit harder to estimate (at least for me), I'll try it anyway. With Memoize it will run in quadratic time as well in the worst case, since lcs will be called with all possible pairs of position shifts in the arguments. Memoize will only reduce this runtime if at least one of the strings is of the from $s x $n with $n >= 2.

But it uses many more method calls, which tend to be slow in Perl.

To test if this explanation really works you could benchmark both subs with increasingly long strings, and test if their runtime actually evolves similarly.