Interesting problem, interesting enough for me to implement a sub-optimal solution. Optimal solutions be damned, full steam ahead.

The Page object maintains a list of openings. When a Rect is put in an opening, (up to) two new openings are made. Putting in Rect 'a' generates openings 'b' and 'c':

aaaabbbb
aaaabbbb
cccccccc
cccccccc

Really putting in Rect 'a' should generate two openings 'bd' and 'cd' like this:

aaaabbbb
aaaabbbb
ccccdddd
ccccdddd

With this improvement it would be a 'first fit' type solution (still sub-optimal). Nevertheless, this solution is not half bad and maybe worth a look.

YuckFoo

#!/usr/bin/perl

   use strict;

   my ($WIDE, $HIGH) = qw(48 24);

   my ($rects, $rect, $page);

   # Make list of random rectangles
   for my $char ('a'..'z', 'A'..'Z') {

      my $wide = int(rand($WIDE/2)+1);
      my $high = int(rand($HIGH/2)+1);

      push (@{$rects}, Rect->new($wide, $high, $char));
   }

   # Sort list of rectangles
   $rects = Rect::sortrecs($rects);
   for $rect (@{$rects}) { $rect->show(); }

   # While there are rectangles, create a page and pack it
   while (@{$rects}) {
      $page = Page->new($WIDE, $HIGH);
      $rects = $page->packit($rects);
      $page->show();
   }

#-----------------------------------------------------------
package Page;
#-----------------------------------------------------------
sub new {

   my ($pkg, $wide, $high) = @_;

   my $r = {}; bless $r, $pkg;

   $r->{wide} = $wide;
   $r->{high} = $high;
   $r->{page} = [];
   $r->{open} = [];

   push (@{$r->{open}}, [0, 0, $wide-1, $high-1, $wide, $high]);

   $r->fill(0, 0, $wide, $high, '.');

   return $r;
}

#-----------------------------------------------------------
sub fill {

   my ($p, $begx, $begy, $endx, $endy, $char) = @_;

   for (my $y = $begy; $y < $begy+$endy; $y++) {
      for (my $x = $begx; $x < $begx+$endx; $x++) {
         $p->{page}[$x][$y] = $char;
      }
   }
}

#-----------------------------------------------------------
sub packit {
   
   my ($p, $rs) = @_;
   my (@notdone);
   
   for my $r (@{$rs}) {
      
      # Try packing rectangle into page
      my $done = packtry($p, $r);
      
      if ($done == -1) {
         
         # Flip rectangle and try again
         ($r->{wide}, $r->{high}) = ($r->{high}, $r->{wide});
         $done = packtry($p, $r);
      }
      if ($done >= 0) {

         my (@add, $nwx, $nwy, $sex, $sey, $wide, $high);

         # The opening that was used
         my $o = $p->{open}[$done];

         # Make new opening
         $nwx = $o->[0] + $r->{wide};
         $nwy = $o->[1];
         $sex = $o->[2];
         $sey = $o->[1] + $r->{high} - 1;
         $wide = $sex - $nwx + 1;
         $high = $sey - $nwy + 1;
         
         @add = ();
         if ($wide * $high > 0) {
            push (@add, [$nwx, $nwy, $sex, $sey, $wide, $high]);
         }
         
         # Make new opening
         $nwx = $o->[0];
         $nwy = $o->[1] + $r->{high};
         $sex = $o->[2];
         $sey = $o->[3];
         $wide = $sex - $nwx + 1;
         $high = $sey - $nwy + 1;
         
         if ($wide * $high > 0) {
            push (@add, [$nwx, $nwy, $sex, $sey, $wide, $high]);
         }
         
         # Replace old opening with new openings
         splice(@{$p->{open}}, $done, 1, @add);
         
         # to show each step
         # $p->show();
      }
      else {
         push(@notdone, $r);
      }
   }
   
   return \@notdone;
}

#-----------------------------------------------------------
sub packtry {

   my ($p, $r) = @_;

   for (my $i = 0; $i < @{$p->{open}}; $i++) {

      my $o = $p->{open}[$i];

      if ($r->{wide} <= $o->[4] && $r->{high} <= $o->[5]) { 
         $p->fill($o->[0], $o->[1], $r->{wide}, $r->{high}, $r->{char}
+);
         return($i);
      }
   }
   
   return (-1);
}

#-----------------------------------------------------------
sub show {
   
   my ($p) = @_;
   
   for (my $y = 0; $y < $p->{high}; $y++) {
      for (my $x = 0; $x < $p->{wide}; $x++) {
         print "$p->{page}[$x][$y]";
      }
      print "\n";
   }
   print "\n";
   
   for my $o (@{$p->{open}}) {
      print "open: $o->[0],$o->[1] ".
                  "$o->[2],$o->[3] ".
                  "$o->[4],$o->[5]\n";
   }
   print "\n";
}

#-----------------------------------------------------------
package Rect;
#-----------------------------------------------------------
sub new {

   my ($pkg, $wide, $high, $char) = @_;

   my $r = {}; bless $r, $pkg;

   $r->{wide} = $wide;
   $r->{high} = $high;
   $r->{char} = $char;

   return $r;
}

#-----------------------------------------------------------
sub sortrecs {
   
   my ($rs) = @_;
   
   my @rs = sort {
               $b->{wide} * $b->{high} <=> $a->{wide} * $a->{high}
            } @{$rs};
   
   return \@rs;
}

#-----------------------------------------------------------
sub show {
   
   my ($r) = @_;
   
   print "$r->{wide} x $r->{high}\n";
   
   my $str = $r->{char} x $r->{wide};
   
   for (1..$r->{high}) { print "$str\n"; }
   
   print "\n";
}
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