Suppose I want to uniformly randomly shuffle a deck of cards in a single pass. I stick the deck on the table and call it the non-shuffled pile. My goal is to move the deck, one card at a time, into the shuffled pile. First I need to select a card, uniformly at random, to be the bottom card of the new pile, and I move it over. Then I select another card, uniformly at random from the still non-shuffled cards, and put it on top of the bottom shuffled card. I repeat this until I’ve moved all the cards, so that each card in the shuffled pile is a uniform random selection from all of the cards it could have been. And that’s it.
One can think of this as random selection, whereas the “forward” version is like random insertion of a not-random card into a shuffled pile. And for whatever reason I tend to think of the selection version first.
• loop counts downwards vs upwards
• the processed part of the array is a uniform sample of the whole array, or it is a segment that has been uniformly shuffled
Knuth described only the downwards sampling version, which is probably why it’s the most common.
The variants are compared quite well on wikipedia https://en.wikipedia.org/wiki/Fisher%E2%80%93Yates_shuffle
I suppose one of the benefits of having a poor memory is that one sometimes improves things in the course of rederiving them from an imperfect recollection.
Maybe it's because it's so easy to prove to yourself that Fisher-Yates generates every possible combination with the same probability[1], and so forwards or backwards just doesn't register as relevant.
[1]This of course makes the a hefty assumption about the source of random numbers which is not true in the vast majority of cases where the algorithm is put into practice as PRNGs are typically what's used. For example if you use a PRNG with a 64 bit seed then you cannot possibly reach the vast majority of states for a 52 card deck; you need 226 bits of state for that to even be possible. And of course even if you are shuffling with fewer combinations than the PRNG state can represent, you will always have some (extremely slight) bias if the state does not express an integer multiple of the number of permutations of your array size.
`mirror_shuffle` is the (GROUP) CONJUGATE of `fisher_yates_shuffle` by the cyclic permutation (n-1,n-2,...,1,0). In group theory, CONJUGATEs are like changes of coordinates. In the present application, they reverse the index labels.
The article said it, but it's worth distilling it.
PS: Oh, here's another link. Denote by "S!", or less formally, the factorial of a set S, to mean the set of permutations of S. Fisher-Yates is equivalent to a bijection between (S+1)! and S! × (S+1).
You could also call the last version the online version, as it will ensure the partial list is random at any point in time (and can be used for inputs with indeterminate length, or to extend a random list with new elements, sample k elements etc.)
Not too sure if the enumerate is necessary. I usually dislike using it just to have an index to play around with. A similar way of doing the same thing is:
for x in source:
a.append(x)
i = random.randint(0, len(a))
a[i], a[-1] = a[-1], a[i]
Not quite sure what you have in mind here, but you need reservoir sampling for this in order to make the selection uniformly random (which I assume is what's desired)
For me, the reason for reaching for the “backwards” version first is that it wasn’t as clear to me that the “forward” version makes a uniform distribution.
Even after reading the article.
I appreciated the comment here about inserting a card at a random location, but that also isn’t quite right, because you swap cards not insert. Nevertheless, that did it for me.
both seem intuitive from individual perspectives
(that was a joke)