I fully agree with your arguments and admit that I got carried away in my strive to make the algorithm as short as possible. In particular your comment on a small change could break it, is very valid.

For something more maintainable, it should be written like this which does exactly the same thing (with the exception of the test in the while loop which is redundant). 

hdb => sub {
  my $input = shift;
  my $length = length $$input;
  my $i = 0;
  while( $i < $length ) {                    # redundant, but to be on
+ the safe side
      $i++;                                  # length of next candidat
+e
      my $possible = substr $$input, 0, $i;  # next candidate
      my $j = index $$input, $possible, $i;  # try to find it to the r
+ight
      if( $j > 0 ) {                         # if found, skip ahead
          $possible = substr $$input, 0, $j; # the candidate has to be
+ this long at least
          $i = $j;                           # store its length
      }
      # check if we have a solution already
      if( substr( $$input, $i ) eq substr($$input, 0, $length - $i) ) 
+{
          return $possible;                  # success !
      }
  }
  # will never reach here
  return '';
},

[download]

And an explanation why the skip ahead is allowed should probably also be added...

In reply to Re^5: Finding repeat sequences. by hdb
in thread Finding repeat sequences. by BrowserUk

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