sub lower_bound1 {
  my @sorted;
  for my $idx ($start..$stop) {
    my $x=0;
    my @order = sort { $a->[1] <=> $b->[1] } map { [$x++,$_] } @{ $data[$idx] };
    push @sorted,\@order;
  }
  #pp \@sorted;
  my $min=0;
  for my $idx ($start..$stop) {
    my $shortest1=$sorted[$idx][1][1];	     # 1. shortest
    my $shortest2=$sorted[$idx][2][1];	     # 2. shortest
    $min+=$shortest1;
    unless ($idx==$start or $idx == $stop) {
      $min+=$shortest2;
    }
    #pp $min,$shortest1,$shortest2, $sorted[$idx];
  }
  return $min/2;
}

##</code><code>##

     0    1    2    3    4      min1          min2
0    0    1   (2)   3    1      1                
1    1    0  (10)  (7)   1      1+1           10+7 
2   (2) (10)   0   15    3      2+3          10+15
3    3   (7)  15    0   (2)     2+3           7+15
4    1    1    3   (2)   0      1                
                             
shortest0-4:     length:21
0--2--1--3--4
  2 10  7  2   

lower_bound1: sum(min1)/2  =  14/2  =  7

lower_bound2: 19 = sum(min2)/2 - 15 + 1 + 1
   (excluding col 0 and 4 for double rows, striking the maximum)