Imagine I have two lists of elements like the following:

listOne = RandomReal[{0,2}, {100,2}];
listTwo = RandomReal[{0,2}, {100,2}];

Let $f: X \to Y$ be a bijection or one-to-one correspondence between the elements in listOne and the elements in listTwo.

I can shuffle the elements in listOne as follows:

listOne = RandomSample[listOne, Length[listOne]];

However, is there a way for me to perform the above shuffling operation while preserving the bijection $f$? In other words, can I shuffle the elements in listOne and listTwo in precisely the same manner?

  • 1
    $\begingroup$ p.s. There's no need for Length[listone]. $\endgroup$
    – Kuba
    Oct 29, 2013 at 21:07
  • $\begingroup$ @MikeHoneychurch I just saw your comment. I was a little slow on the pickup for rule replacement, but I think I get it now. $\endgroup$
    – RTaylor
    Oct 29, 2013 at 21:42
  • $\begingroup$ Personally I don't see this as a duplicate. Could one of those who closed the question show how one of the answers to the other question can be applied to this question? $\endgroup$ Oct 29, 2013 at 21:59
  • $\begingroup$ @MikeHoneychurch To my mind, both of the highest-voted answers there apply to this question quite directly, so much so that I don't know what else to add. $\endgroup$ Oct 29, 2013 at 22:15
  • 1
    $\begingroup$ @LeonidShifrin I agree with Mike Honeychurch; those seem to require that one list start off life ordered. [If I'm wrong, please accept Mike's humble apology..] $\endgroup$ Oct 29, 2013 at 23:41

3 Answers 3


Just create an ordering list and use Part:

listOne = {1, 2, 3};
listTwo = {4, 5, 6};

order = RandomSample[Range[Length[listOne]]]
(* {2, 3, 1} *)

(* {2, 3, 1} *)

(* {5, 6, 4} *)
  • $\begingroup$ +1 Geez I normally always choose Part over other methods but overlooked that. My morning coffee hasn't kicked in yet -- brain still frozen $\endgroup$ Oct 29, 2013 at 21:11
  • 1
    $\begingroup$ @MikeHoneychurch, I have a time-zone advantage, my brain has been steeping in coffee all day :-) $\endgroup$ Oct 29, 2013 at 21:17

If it's a bijection you can for example do this:

s1 = RandomSample[listOne]
s2 = s1 /. MapThread[Rule, {listOne, listTwo}, 1]
  • 3
    $\begingroup$ I would have gone with Dispatch@Thread[Rule[listOne, listTwo]]. Not sure but might be faster for large lists. $\endgroup$ Oct 29, 2013 at 21:08
  • $\begingroup$ @Kuba Thanks it's good to know that you can do this. $\endgroup$
    – RTaylor
    Oct 29, 2013 at 21:20
  • $\begingroup$ This doesn't work if listOne contains duplicates. (Since the question is about Mathematica lists/arrays, other site users should be aware of this limitation.) $\endgroup$
    – Michael E2
    Oct 30, 2013 at 9:58
  • $\begingroup$ @MichaelE2 OP gives the assumption that it is bijection. So there are no duplicates, or if there are they are reffering to duplicate positions in the second list. Or I missed something? $\endgroup$
    – Kuba
    Oct 30, 2013 at 11:10
  • $\begingroup$ A one-to-one correspondence between two Lists, which are ordered and permit duplicates, is different than a one-to-one correspondence between sets. The site being about Mathematica, it seems fair to presume that an array is a List, not a mathematical set. Because the Q&A format is for everyone, I think the distinction ought to be made clear for those who would adapt a solution to their own use-case. If sets are meant - I agree the possibility is suggested by "bijection" - then the question should make that clear. I had a similar disagreement about another of the OP's questions. :) $\endgroup$
    – Michael E2
    Oct 30, 2013 at 12:49

Although I have used Simon's method many times I think it is somewhat cleaner in this case to transpose the data, shuffle, and transpose again:

l1 = {a, b, c, d, e};
l2 = {v, w, x, y, z};

RandomSample[{l1, l2}\[Transpose]]\[Transpose]
{{e, c, b, a, d}, {z, x, w, v, y}}

The \[Transpose] symbol appears as T in the Front End:

enter image description here

  • $\begingroup$ I agree that this is cleaner. In fact I'd normally only use the method I posted if there was some need later on to undo the shuffle. $\endgroup$ Oct 31, 2013 at 22:18

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