Here is a recent article describing a simpler algorithm for sampling derangements. I also found slides for the presentation. Since the paper is so recent, I guess this means that a small modification of FisherYates is unlikely to generate derangements, since someone would have already come up with it by now. Still, their algorithm is inplace and has better expected running time than retrying FisherYates until you get a derangement.
Here is a Perl implementation I whipped up. It is slightly odd because I followed their lead and used array indexing from 1.
sub rand_derangement {
my $n = shift;
return if $n == 1; ## no derangements of size 1
## precompute $D[n] == number of derangements of size n
my @D = (1,0);
push @D, $#D * ($D[1] + $D[2]) while $#D < $n;
my @A = (undef, 1 .. $n);
my @mark = (1, (0) x $n);
my ($i, $u) = ($n, $n);
while ($u > 1) {
if (! $mark[$i]) {
my $j = 0;
$j = 1 + int rand($i2) while $mark[$j];
@A[$i,$j] = @A[$j,$i];
if ( rand(1) < ($u1) * $D[$u2] / $D[$u] ) {
$mark[$j] = 1;
$u;
}
$u;
}
$i;
}
return @A[1..$n];
}
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