Random Shuffles

This is more difficult. We have a list (x₁,..., x_n) and we wantto shuffle it randomly.

The following, surprisingly simple, algorithm does the trick:

---

algorithm shuffle(x,n)   // shuffle x₁,..., x_n
begin
  for i = n downto 2 begin
    k = irand(1,i)
    swap over x_i and x_k
  end
end

 

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Why does this work? It seems that it is not doing enough `mixing up'to produce a genuinely random shuffle. We can only prove that it worksif we have some notion of what we mean by a `random shuffle'. Let mekeep things simple and say that a shuffling process is random if eachelement of the list can end up in any position in the list with equalprobability. That is to say, in a list of n items there are npossible places that any given element can eventually occupy. I wanteach of these places to be equally likely.

So we now have to calculate the probability that our algorithm putselement x_i into position j, where 1i, jn. If you lookcarefully at the algorithm you will see that element x_i end up inposition j if it is not chosen on the first n - j steps ofthe algorithm and is chosen on the step n - j + 1. Since we knowthat irand(n, m) is genuine we can use elementary methods tofind that the probability of x_i ending up in position j is

  ^{. . ... .}   = ^. = ,