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Re: random derangement

  • To: mathgroup at
  • Subject: [mg37435] Re: [mg37416] random derangement
  • From: Daniel Lichtblau <danl at>
  • Date: Tue, 29 Oct 2002 00:09:36 -0500 (EST)
  • References: <>
  • Sender: owner-wri-mathgroup at

"DIAMOND Mark R." wrote:
> This is not quite aposite to either NG, but there appear to be none better
> ... My apologies.
> I am searching for an algorithm for producing a random derangement of, for
> instance, the integers 1 to approx 10000.
> I thought Skiena's site might have such an algorithm, but I could not locate
> one. ... Producing all derangements and choosing one at random is marginally
> beyond the capacity of my machine :-)
> Cheers,
> Mark R. Diamond

One thing that will work efficiently is a modification of a basic random
shuffle. The basic shuffle can be found at

The modification is that at step j we insist on moving something between
position j+1 (rather than j) and the end into position j.

derangement = Compile[{{n,_Integer}}, Module[
  {deck=Range[n], newj},
    newj = Random[Integer, {j+1,n}];
    deck[[{j,newj}]] = deck[[{newj,j}]],

In[4]:= Timing[dd = derangement[10^6];]
Out[4]= {5.25 Second, Null}

Check that this is indeed a derangement:

In[5]:= Select[Transpose[{dd,Range[10^6]}], #[[1]]==#[[2]]&]
Out[5]= {}

(Or you can use MapIndexed for this test):
In[12]:= Apply[Or, MapIndexed[#1==#2[[1]]&, dd]]
Out[12]= False

I think this will give random derangements with equal probabilities
though I don't have a proof of that off the top of my head.

Daniel Lichtblau
Wolfram Research

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