You shouldn't use grep in this context, because grep will search through an entire array no matter what--in other words, it'll do a lot of extra work. If you write up your own routine, you can break out of the compare loop as soon as you find two elements that aren't equal. This is good.

Use the code from the FAQ!

$are_equal = compare_arrays(\@frogs, \@toads);

sub compare_arrays {
    my ($first, $second) = @_;
    local $^W = 0;  # silence spurious -w undef complaints
    return 0 unless @$first == @$second;
    for (my $i = 0; $i < @$first; $i++) {
        return 0 if $first->[$i] ne $second->[$i];
    }
    return 1;
}
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In particular, note this line:

return 0 unless @$first == @$second;
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So the routine returns automatically if the arrays aren't of equal size. In your algorithm, you compare the elements in the arrays even if they're of different sizes. In fact, you explicitly wrote this in using your max function! Why? If the arrays aren't the same size, how could they possibly be equal?

Also, look at the FreezeThaw examples in the FAQ.