Do you have two arrays, before and after sorting (yeah, I read that you don't want to compare the arrays...)? If yes, then for big arrays, you could write a simple test that compares n indices:

for my $r (@rand) {
    return 0 if $a[$r] ne $b[$r];
}
return 1;
[download]

Not 100% safe of course, but if the input lines are not related, it should work. To be sure, you could apply a second test if the first test returned 1: Array::Compare uses join to compare two arrays. (This is probably fast enough even without the first test.)

If you don't have the two arrays, you won't get better than O(n) or you use a similar (unsafe) test like above - check if n blocks of size 2 are sorted.

UPDATE: md5 seems to be a little bit faster than join:

#!/usr/bin/perl
use strict;
use warnings;
use Benchmark qw(:all) ;

use List::Util qw(shuffle);
use Digest::MD5 qw(md5);

open my $JOHN, '<', 'dictionary' or die $!;
my @array =  <$JOHN>;
my @sorted = sort @array;

$a = \@array; $b = \@sorted;

timethese(5000, {
     'join' =>
             sub {
                 return join('|',@$a) eq join('|', @$b);
             },  

     'md5' =>
             sub {
                 return Digest::MD5::md5(@$a) eq Digest::MD5::md5(@$b)
+;
             },  
});
[download]

perl cmparray.pl
Benchmark: timing 5000 iterations of join, md5...
      join: 29 wallclock secs (27.91 usr +  0.18 sys = 28.09 CPU) @ 17
+8.00/s (n=5000)
       md5: 27 wallclock secs (25.20 usr +  0.19 sys = 25.39 CPU) @ 19
+6.93/s (n=5000)
[download]