sub invert {
 my $x = shift;
 my $y = {};
 foreach my $xk1 (keys %$x) {
  foreach my $xk2 (keys %{$x->{$xk1}}) {
   $y->{$xk2}{$xk1} = $x->{$xk1}{$xk2};
  }
 }
 return $y;
}