in reply to
A bad shuffle
The naive_shuffle() you object to is FischerYates, which is pretty well studied. I think you misapprehend the effect of the algorithm.
While it samples a permutation of the elements, it does not do so by selecting from a list of all permutations, and execution paths have nothing to do with it.
Re^2: A bad shuffle by tlm (Prior) on Mar 20, 2005 at 23:37 UTC 
Thank you for the information. Perhaps I should have made clearer in that my encounter with this algorithm, which motivated the whole write up, was in researchoriented/scientific code in which it was being used to sample permutations of an array uniformly at random. My contention is that this is a misuse of this algorithm, because it does not sample the space of permutations uniformly.
But in light of what you write about the algorithm's standing and pedigree, the title of my meditation is an awful one. Maybe the whole node should be retracted for the sake of not confusing others. Let me know what you think.
Update: I found a version of FisherYates online (linked from here):
#include <stdlib.h>
void shuffle(int *array, size_t n) {
if (n > 1) {
size_t i;
for (i = 0; i < n  1; i++) { /**/
size_t j = i + rand()/(RAND_MAX / (n  i) + 1); /**/
int t = array[j];
array[j] = array[i];
array[i] = t;
}
}
}
This is equivalent to the algorithm used by my random_perm, not the one used by naive_shuffle. To get the latter algorithm, the lines indicated with /**/ above would have to be changed to:
for (i = 0; i < n; i++) {
size_t j = rand()/(RAND_MAX / n);
the lowliest monk
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No, don't withdraw it. I think the thing to do is try and figure out the difference between a "fair shuffle" and a uniformly distributed selection over permutations.
Are the two the same if there are identical cards in the deck? What is the problem that motivated this?
Sometimes confusion, like greed, is good.
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Re^2: A bad shuffle by Anonymous Monk on Mar 21, 2005 at 01:43 UTC 
Actually, the original "naive_shuffle" is not a FisherYates shuffle, it is a "naive shuffle" implementation. The OP's analysis is correct.
His final algorithm, however, is a correct FisherYates implementation.  [reply] 

perlfaq4: How do I shuffle an array randomly? not only gives a "canonical" implementation of the FisherYates algorithm in Perl, but also refers to the List::Util module's shuffle function, which is an implementation of FisherYates in C. The algorithm is also implemented in Abigail's Algorithm::Numerical::Shuffle, the doco of which gives some citations into the literature (Knuth, Fisher&Yates, etc.).
The FisherYates has also been discussed many times on clpm.
(You can do a Google Groups search. My ego compels me to link to
this posting by yrs trly, which isn't about FY, but uses it.)
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sub shuffle (@) {
my @a=\(@_);
my $n;
my $i=@_;
map {
$n = rand($i);
(${$a[$n]}, $a[$n] = $a[$i])[0];
} @_;
}
Pretty gnarleous, IMO. Kind of like a hybrid between FY and Tanktalus's shuffler.
The C implementation of List::Util::shuffle is 1020x faster than the Perl implementation. For the practical programmer: end of story. Still, to satisfy my monkly preoccupation with how many angels can lambada on the head of a pin, and more importantly, in order to reduce this dead horse to a thin protein film, I benchmarked the three shufflers: ta = Tanktalus's shuffler; lu = Perl implementation of List::Util::shuffle; rp = random_perm:
N = 1000
Rate rp lu ta
rp 230/s  10% 22%
lu 257/s 12%  13%
ta 296/s 29% 15% 
N = 10_000
Rate ta lu rp
ta 14.3/s  27% 34%
lu 19.6/s 37%  10%
rp 21.7/s 52% 11% 
N = 100_000
Rate ta lu rp
ta 0.144/s  92% 93%
lu 1.75/s 1118%  13%
rp 2.01/s 1302% 15% 
the lowliest monk
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Re^2: A bad shuffle by bart (Canon) on Mar 21, 2005 at 09:16 UTC 
No, you're wrong. It looks like FisherYates, but there's a slight difference, in that all his array items can move again on every loop. His "correct algorithm" is actually FisherYates, where per loop, one item gets moved into its final position.  [reply] 

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Sorry Bart, I thouht you were replying to me...
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No, clearly his remarks were intended for me, not you.
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