October 20, 2009

The classic description of the array shuffle is given by Donald E. Knuth in The Art of Computer Programming, Volume 2, Section 3.4.2, Algorithm P:

(define (shuffle! v)
  (do ((n (vector-length v) (- n 1))) ((zero? n) v)
    (let* ((r (randint n)) (t (vector-ref v r)))
      (vector-set! v r (vector-ref v (- n 1)))
      (vector-set! v (- n 1) t))))

That algorithm has time complexity O(n). Using the same algorithm on a list, replacing vector-ref with list-ref, has time complexity O(n²), due to the cost of continually indexing through the list, starting each time at its head. We won’t show the list version of that algorithm.

Joe Marshall proposes shuffling a list by partitioning it into two pieces deterministically, shuffling them recursively, then merging them randomly:

(define (shuffle xs)
  (if (or (null? xs) (null? (cdr xs))) xs
      (let split ((xs xs) (odd '()) (even '()))
        (if (pair? xs)
            (split (cdr xs) (cons (car xs) even) odd)
            (let merge ((odd (shuffle odd)) (even (shuffle even)))
              (cond ((null? odd) even)
                    ((null? even) odd)
                    ((zero? (randint 2)) (cons (car odd) (merge (cdr odd) even)))
                    (else (cons (car even) (merge odd (cdr even))))))))))

Al Petrofsky proposes a somewhat faster method that first partitions the list randomly, then randomly merges the two pieces:

(define (shuffle xs)
  (let shuffle ((xs xs) (acc '()))
    (if (null? xs) acc
        (if (null? (cdr xs)) (cons (car xs) acc)
            (let split ((xs xs) (x1 '()) (x2 '()))
              (if (null? xs)
                  (if (null? x1)
                      (split x2 '() '())
                      (shuffle x1 (shuffle x2 acc)))
                  (if (zero? (randint 2))
                      (split (cdr xs) (cons (car xs) x1) x2)
                      (split (cdr xs) x1 (cons (car xs) x2)))))))))

If you want, you can always do Perl’s omigod Schwartzian transform:

(define (shuffle xs)
  (map cdr
    (sort (lambda (x y) (< (car x) (car y)))
      (map (lambda (x) (cons (rand) x)) xs))))

All the list shuffles have time complexity O(n log n); that’s the price of the linked list. But since Knuth’s method has time complexity O(n), the fastest algorithm for sorting a list is to convert it to an array, shuffle the array using Knuth’s method, then convert it back to a list, which is not only fast in theory but also faster in practice than any of the other methods, as for small lists the two type conversions get lost in the overhead of the partition/merge process, and for large lists the asymptotic behavior dwarfs everything else:

(define (shuffle x)
  (do ((v (list->vector x)) (n (length x) (- n 1)))
      ((zero? n) (vector->list v))
    (let* ((r (randint n)) (t (vector-ref v r)))
      (vector-set! v r (vector-ref v (- n 1)))
      (vector-set! v (- n 1) t))))

We used sort, rand and randint from the Standard Prelude. You can run the program at

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6 Responses to “Shuffle”

  1. Sam TH said

    Here’s a PLT version, which works like the Perl version:

    (define (shuffle x) (sort x #:key (lambda _ (random)) #:cache-keys? #t))

  2. Ed Siok said

    Here’s an example of the array shuffle in perl (without using the pre-existing library). Perl doesn’t generally support linked lists as all arrays have the push/pop/etc functionality included.

    Perl array shuffle example

  3. Michel S. said

    Here is an implementation in Clojure. Its native datatypes are immutable, so for the array shuffle I’m using Java’s ArrayList. The collection shuffler works on any ordered Clojure collection (immutable lists as well as vectors); because values are immutable, we are modifying what the variables point to, which has to be done within a software transaction with dosync.

    (import 'java.util.ArrayList)
    (defmulti shuffle type)
    (defmethod shuffle ArrayList [#^ArrayList arr]
      (letfn [(swap!
    	   [i j]
    	   (let [t (.get arr i)]
    	     (doto arr
    	       (.set i (.get arr j))
    	       (.set j t))))]
        (doseq [x (range (dec (.size arr)) -1 -1)]
          (swap! x (rand-int x)))
    ;; alternate shuffler:
    ;; #(sort (fn [x y] (dec (rand-int 3))) %)
    (defmethod shuffle clojure.lang.Ref [coll]
      (letfn [(shuffler
    	   (if (or (empty? coll) (empty? (next coll))) coll
    	       (let [c (count coll)
    		     pick (rand-int c)]
    		 (cons (nth coll pick)
    		       (concat (shuffler (take pick coll))
    			       (shuffler (drop (inc pick) coll)))))))]
        (dosync (alter coll shuffler))))
  4. Ben Goldberg said

    To shuffle a linked list without converting it into an array or treating it as if it were an array, create two temporary lists, left and right, and then for each element in the original list, move that element randomly into either left or right. Then use recursion shuffle each of the left and right lists, combine them, and return the combined list.

    Of course, doing two recursive calls can result in lots of memory usage, especially since the random selection could potentially put all elements of the original into left, or all into right… which adds to the stack depth, while accomplishing little work.

    Instead, after partitioning into left and right, join the lists back together before shuffling, so that one of the two recursive calls can be changed into a loop.

    Here’s some lisp code to do it:

    (defun partition (before exclude)
      "Destructively randomly partition part of a list (starting with
    (rest BEFORE), up to, but not including, EXCLUDE) into two lists,
    which (concatenated together) are then inserted into the original
    list in their original place."
      ;(format t "Partition ~A ~A~%" before exclude)
      (do (lft rgt lend rend llen rlen)
          ((and lend rend)
           (setf (rest before) lft (rest lend) rgt (rest rend) exclude)
           (values rgt (< llen rlen)))
        ;(format t "partition, looping~%")
        (setf lft nil lend nil rgt nil rend nil llen 0 rlen 0)
        (do ((y (rest before) (rest y))) ((eq y exclude))
          ; move each cons onto the end of either the left or
          ; right list, keeping track of where the list ends are.
          (if (zerop (random 2))
              (and (incf llen) (if (not lft)
                                   (setf lft y lend y)
                                   (setf (rest lend) y lend y)))
              (and (incf rlen) (if (not rgt)
                                   (setf rgt y rend y)
                                   (setf (rest rend) y rend y)))))))
    (defun shuf-internal (before exclude)
      "Destructively shuffles the list which starts with (rest BEFORE), up to and excluding
    the cons EXCLUDE.  Returns nothing."
      ;(format t "shuf-internal ~A ~A~%" before exclude)
      ; create a "working" list from mystart to myend, inclusive.
      ; Loop as long as the working list's length is > 1
      (do ()
          ((or (eq (rest before) exclude) (eq (cddr before) exclude)) (values 'foo!))
        ;(format t "m-v-b~%")
        (multiple-value-bind (rgt rec-left) (partition before exclude)
          ;(format t "After part, before is ~A, rgt is ~A, doleft is ~A~%" before rgt rec-left)
          (if rec-left
              (progn (shuf-internal before rgt) (setf before rgt))
              (progn (shuf-internal rgt exclude) (setf exclude rgt))))))
    (defun nshuffle (thelist)
      (let ((dummyhead (cons 'dummy-head-value thelist)))
        (shuf-internal dummyhead nil)
        (rest dummyhead)))

    Alternatively, here’s some C code:

    typedef struct node {
    	int data;
    	struct node * next;
    } node, *list;
    static void shl(list *before, node *after);
    void shufflelist(list * head) {
    	if(head != NULL) shl(head, NULL);
    #define RANDBOOL() (rand() > (RAND_MAX/2))
    static void shl(list *before, node *after) {
    	while( *before != after && (*before)->next != after ) {
    		list lft = NULL, rgt = NULL;
    		node *lend = NULL, *rend = NULL;
    		int llen = 0, rlen = 0;
    		for( node *i = *before; i != after; i = i->next )
    			if( RANDBOOL() ) {
    				if( lft ) lend->next = i; else lft = i;
    				lend = i;
    			} else {
    				if( rgt ) rend->next = i; else rgt = i;
    				rend = i;
    		if( !lft || !rgt ) goto redo;
    		*before = lft;
    		lend->next = rgt;
    		rend->next = after;
    		if( llen < rlen ) {
    			shl(before, rgt);
    			before = &(rgt->next);
    		} else {
    			shl(rgt, after);
    			after = rgt;

    This would be used as:

    list mylist = NULL;
    for( int i = 0; i < 10; ++i )
       insert_front(&mylist, i);

    PS: I’ve only tested the lisp code, since I’m still working on installing gcc on my computer.
    But since the algorithm is the same, they should work the same.

  5. programmingpraxis said

    I haven’t looked closely, but isn’t your algorithm O(n log n), like any other divide-and-conquer algorithm? The advantage of the list->vector->knuth-shuffle->list algorithm is that it works in O(n) time.

  6. Vikas Tandi said


    #include <stdlib.h>
    #include <time.h>
    static void swap(int arr[], int i, int j);
    void shuffle_array(int arr[], int n)
    	int i, j;
    	srand((unsigned int)time(NULL));
    	for(j = n; j > 1; j--)
    		i = (rand() % j) + 1;
    		swap(arr, i-1, j-1);
    void shuffle_linklist(List **p)
    	List *ptr, *currentp, *newp;
    	int size, randnum;
    	if(*p == NULL)
    	/* calculate Linklist size */
    	for(size = 0, currentp = *p; currentp != NULL; size++, currentp = currentp->next);
    	/* shuffle */
    	/* Shuffle single node */
    	srand((unsigned int)time(NULL));
    	randnum = (rand() % size) + 1;
    	ptr = *p;
    	currentp = *p;
    	while(--randnum > 1)
    		currentp = currentp->next;
    	/* first node */
    	if(currentp == ptr)
    		ptr = ptr->next;
    		*p = currentp;
    		*p = currentp->next;
    		currentp->next = currentp->next->next;
    	if(size == 1)
    	/* shuffle remaining node */
    	for(size = size-1, currentp = ptr, newp = *p; size > 1; currentp = ptr, size--, newp = newp->next)
    		randnum = (rand() % size) + 1;
    		while(--randnum > 1)
    			currentp = currentp->next;
    		/* first node */
    		if(currentp == ptr)
    			ptr = ptr->next;
    			newp->next = currentp;
    			newp->next = currentp->next;
    			currentp->next = currentp->next->next;
    		newp->next->next = NULL;
    	/* add last node */
    	newp->next = ptr;
    	newp->next->next = NULL;
    static void swap(int arr[], int i, int j)
    	int tmp;
    	tmp = arr[i];
    	arr[i] = arr[j];
    	arr[j] = tmp;

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