Knuth's algorithm S

This is a method of randomly sampling n items from a set of M items, with equal probability; where M >= n and M, the number of items is unknown until the end. This means that the equal probability sampling should be maintained for all successive items > n as they become available (although the content of successive samples can change).

The algorithm

Select the first n items as the sample as they become available;
For the i-th item where i > n, have a random chance of n/i of keeping it. If failing this chance, the sample remains the same. If not, have it randomly (1/n) replace one of the previously selected n items of the sample.
Repeat #2 for any subsequent items.

The Task

Create a function s_of_n_creator that given $n$ the maximum sample size, returns a function s_of_n that takes one parameter, item.
Function s_of_n when called with successive items returns an equi-weighted random sample of up to n of its items so far, each time it is called, calculated using Knuths Algorithm S.
Test your functions by printing and showing the frequency of occurrences of the selected digits from 100,000 repetitions of:

Use the s_of_n_creator with n == 3 to generate an s_of_n.
call s_of_n with each of the digits 0 to 9 in order, keeping the returned three digits of its random sampling from its last call with argument item=9.

Note: A class taking n and generating a callable instance/function might also be used.

Reference

The Art of Computer Programming, Vol 2, 3.4.2 p.142

Cf.

Ada

Instead of defining a function S_of_N_Creator, we define a generic packgage with that name. The generic parameters are N (=Sample_Size) and the type of the items to be sampled:

<lang Ada>generic

  Sample_Size: Positive;
  type Item_Type is private;

package S_Of_N_Creator is

  subtype Index_Type is Positive range 1 .. Sample_Size;
  type Item_Array is array (Index_Type) of Item_Type;

  procedure Update(New_Item: Item_Type);
  function Result return Item_Array;

end S_Of_N_Creator;</lang>

Here is the implementation of that package:

<lang Ada>with Ada.Numerics.Float_Random, Ada.Numerics.Discrete_Random;

package body S_Of_N_Creator is

  package F_Rnd renames Ada.Numerics.Float_Random;
  F_Gen: F_Rnd.Generator;

  package D_Rnd is new Ada.Numerics.Discrete_Random(Index_Type);
  D_Gen: D_Rnd.Generator;

  Item_Count: Natural := 0; -- this is a global counter
  Sample: Item_Array; -- also used globally

  procedure Update(New_Item: Item_Type) is
  begin
     Item_Count := Item_Count + 1;
     if Item_Count <= Sample_Size then
        -- select the first Sample_Size items as the sample
        Sample(Item_Count) := New_Item;
     else
        -- for I-th item, I > Sample_Size: Sample_Size/I chance of keeping it
        if (Float(Sample_Size)/Float(Item_Count)) > F_Rnd.Random(F_Gen)  then
           -- randomly (1/Sample_Size) replace one of the items of the sample
           Sample(D_Rnd.Random(D_Gen)) := New_Item;
        end if;
     end if;
  end Update;

  function Result return Item_Array is
  begin
     Item_Count := 0; -- ready to start another run
     return Sample;
  end Result;

begin

  D_Rnd.Reset(D_Gen); -- at package instantiation, initialize rnd-generators
  F_Rnd.Reset(F_Gen);

end S_Of_N_Creator;</lang>

The main program:

<lang Ada>with S_Of_N_Creator, Ada.Text_IO;

procedure Test_S_Of_N is

  Repetitions: constant Positive := 100_000;
  type D_10 is range 0 .. 9;

  -- the instantiation of the generic package S_Of_N_Creator generates 
  -- a package with the desired functionality
  package S_Of_3 is new S_Of_N_Creator(Sample_Size => 3, Item_Type => D_10);

  Sample: S_Of_3.Item_Array;
  Result: array(D_10) of Natural := (others => 0);

begin

  for J in 1 .. Repetitions loop
     -- get Sample
     for Dig in D_10 loop
        S_Of_3.Update(Dig);
     end loop;
     Sample := S_Of_3.Result;

     -- update current Result
     for Item in Sample'Range loop
        Result(Sample(Item)) := Result(Sample(Item)) + 1;
     end loop;
  end loop;

  -- finally: output Result
  for Dig in Result'Range loop
     Ada.Text_IO.Put(D_10'Image(Dig) & ":"
                       & Natural'Image(Result(Dig)) & ";   ");
  end loop;

end Test_S_Of_N;</lang>

A sample output:

 0: 30008;    1: 30056;    2: 30080;    3: 29633;    4: 29910;    5: 30293;    6: 30105;    7: 29924;    8: 29871;    9: 30120;

BBC BASIC

Works with: BBC BASIC for Windows

At each of the 100000 repetitions not only is a new function created but also new copies of its PRIVATE variables index% and samples%(). Creating such a large number of variables at run-time impacts adversely on execution speed and isn't to be recommended, other than to meet the artificial requirements of the task. <lang bbcbasic> HIMEM = PAGE + 20000000

     PRINT "Single run samples for n = 3:"
     SofN% = FNs_of_n_creator(3)
     FOR I% = 0 TO 9
       !^a%() = FN(SofN%)(I%)
       PRINT " For item " ; I% " sample(s) = " FNshowarray(a%(), I%+1)
     NEXT
     
     DIM cnt%(9)
     PRINT '"Digit counts after 100000 runs:"
     FOR rep% = 1 TO 100000
       IF (rep% MOD 1000) = 0 PRINT ; rep% ; CHR$(13) ;
       F% = FNs_of_n_creator(3)
       FOR I% = 0 TO 9
         !^a%() = FN(F%)(I%)
       NEXT
       cnt%(a%(1)) += 1 : cnt%(a%(2)) += 1 : cnt%(a%(3)) += 1
     NEXT
     FOR digit% = 0 TO 9
       PRINT " " ; digit% " : " ; cnt%(digit%)
     NEXT
     END
     
     REM Dynamically creates this function:
     REM DEF FNfunction(item%) : PRIVATE samples%(), index%
     REM DIM samples%(n%) : = FNs_of_n(item%, samples%(), index%)
     DEF FNs_of_n_creator(n%)
     LOCAL p%, f$
     f$ = "(item%) : " + CHR$&0E + " samples%(), index% : " + \
     \    CHR$&DE + " samples%(" + STR$(n%) + ") : = " + \
     \    CHR$&A4 + "s_of_n(item%, samples%(), index%)"
     DIM p% LEN(f$) + 4 : $(p%+4) = f$ : !p% = p%+4
     = p%
     
     DEF FNs_of_n(D%, s%(), RETURN I%)
     LOCAL N%
     N% = DIM(s%(),1)
     I% += 1
     IF I% <= N% THEN
       s%(I%) = D%
     ELSE
       IF RND(I%) <= N% s%(RND(N%)) = D%
     ENDIF
     = !^s%()
     
     DEF FNshowarray(a%(), n%)
     LOCAL i%, a$
     a$ = "["
     IF n% > DIM(a%(),1) n% = DIM(a%(),1)
     FOR i% = 1 TO n%
       a$ += STR$(a%(i%)) + ", "
     NEXT
     = LEFT$(LEFT$(a$)) + "]"</lang>

Output:

Single run samples for n = 3:
 For item 0 sample(s) = [0]
 For item 1 sample(s) = [0, 1]
 For item 2 sample(s) = [0, 1, 2]
 For item 3 sample(s) = [0, 1, 2]
 For item 4 sample(s) = [0, 1, 4]
 For item 5 sample(s) = [0, 1, 4]
 For item 6 sample(s) = [0, 1, 6]
 For item 7 sample(s) = [0, 1, 6]
 For item 8 sample(s) = [8, 1, 6]
 For item 9 sample(s) = [8, 1, 9]

Digit counts after 100000 runs:
 0 : 30068
 1 : 30017
 2 : 30378
 3 : 29640
 4 : 30153
 5 : 29994
 6 : 29941
 7 : 29781
 8 : 29918
 9 : 30110

C

Instead of returning a closure we set the environment in a structure:

<lang c>#include <stdlib.h>

include <stdio.h>
include <string.h>
include <time.h>

struct s_env {

   unsigned int n, i;
   size_t size;
   void *sample;

};

void s_of_n_init(struct s_env *s_env, size_t size, unsigned int n) {

   s_env->i = 0;
   s_env->n = n;
   s_env->size = size;
   s_env->sample = malloc(n * size);

}

void sample_set_i(struct s_env *s_env, unsigned int i, void *item) {

   memcpy(s_env->sample + i * s_env->size, item, s_env->size);

}

void *s_of_n(struct s_env *s_env, void *item) {

   s_env->i++;
   if (s_env->i <= s_env->n)
       sample_set_i(s_env, s_env->i - 1, item);
   else if ((rand() % s_env->i) < s_env->n)
       sample_set_i(s_env, rand() % s_env->n, item);
   return s_env->sample;

}

int *test(unsigned int n, int *items_set, unsigned int num_items) {

   int i;
   struct s_env s_env;
   s_of_n_init(&s_env, sizeof(items_set[0]), n);
   for (i = 0; i < num_items; i++) {
       s_of_n(&s_env, (void *) &items_set[i]);
   }
   return (int *)s_env.sample;

}

int main() {

   unsigned int i, j;
   unsigned int n = 3;
   unsigned int num_items = 10;
   unsigned int *frequencies;
   int *items_set;
   srand(time(NULL));
   items_set = malloc(num_items * sizeof(int));
   frequencies = malloc(num_items * sizeof(int));
   for (i = 0; i < num_items; i++) {
       items_set[i] = i;
       frequencies[i] = 0;
   }
   for (i = 0; i < 100000; i++) {
       int *res = test(n, items_set, num_items);
       for (j = 0; j < n; j++) {
           frequencies[res[j]]++;
       }

free(res);

   }
   for (i = 0; i < num_items; i++) {
       printf(" %d", frequencies[i]);
   }
   puts("");
   return 0;

}</lang>

C++

Works with: C++11

<lang cpp>#include <iostream>

include <functional>
include <vector>
include <cstdlib>
include <ctime>

template <typename T> std::function<std::vector<T>(T)> s_of_n_creator(int n) {

 std::vector<T> sample;
 int i = 0;
 return [=](T item) mutable {
   i++;
   if (i <= n) {
     sample.push_back(item);
   } else if (std::rand() % i < n) {
     sample[std::rand() % n] = item;
   }
   return sample;
 };

}

int main() {

 std::srand(std::time(NULL));
 int bin[10] = {0};
 for (int trial = 0; trial < 100000; trial++) {
   auto s_of_n = s_of_n_creator<int>(3);
   std::vector<int> sample;
   for (int i = 0; i < 10; i++)
     sample = s_of_n(i);
   for (int s : sample)
     bin[s]++;
 }
 for (int x : bin)
   std::cout << x << std::endl;
 return 0;

}</lang>

Output:

Class-based version: <lang cpp>#include <iostream>

include <vector>
include <cstdlib>
include <ctime>

template <typename T> class SOfN {

 std::vector<T> sample;
 int i;
 const int n;
public:
 SOfN(int _n) : i(0), n(_n) { }
 std::vector<T> operator()(T item) {
   i++;
   if (i <= n) {
     sample.push_back(item);
   } else if (std::rand() % i < n) {
     sample[std::rand() % n] = item;
   }
   return sample;
 }

};

int main() {

 std::srand(std::time(NULL));
 int bin[10] = {0};
 for (int trial = 0; trial < 100000; trial++) {
   SOfN<int> s_of_n(3);
   std::vector<int> sample;
   for (int i = 0; i < 10; i++)
     sample = s_of_n(i);
   for (std::vector<int>::const_iterator i = sample.begin(); i != sample.end(); i++)
     bin[*i]++;
 }
 for (int i = 0; i < 10; i++)
   std::cout << bin[i] << std::endl;
 return 0;

}</lang>

Clojure

The Clojure approach to problems like this is to define a function which takes an accumulator state and an input item and produces the updated state. Here the accumulator state is the current sample and the number of items processed. This function is then used in a reduce call with an initial state and a list of items. <lang clojure>(defn s-of-n-fn-creator [n]

 (fn [[sample iprev] item]
   (let [i (inc iprev)]
     (if (<= i n)
       [(conj sample item) i]
       (let [r (rand-int i)]
         (if (< r n)
           [(assoc sample r item) i]
           [sample i]))))))

(def s-of-3-fn (s-of-n-fn-creator 3))

(->> #(reduce s-of-3-fn [[] 0] (range 10))

   (repeatedly 100000)
   (map first)
   flatten
   frequencies
   sort
   println)

</lang> Sample output: <lang>([0 29924] [1 30053] [2 30018] [3 29765] [4 29974] [5 30225] [6 30082] [7 29996] [8 30128] [9 29835])</lang>

If we really need a stateful (thread safe!) function for some reason, we can get it like this: <lang clojure>(defn s-of-n-creator [n]

 (let [state (atom [[] 0])
       s-of-n-fn (s-of-n-fn-creator n)]
   (fn [item]
     (first (swap! state s-of-n-fn item)))))</lang>

CoffeeScript

<lang coffeescript> s_of_n_creator = (n) ->

 arr = []
 cnt = 0
 (elem) ->
   cnt += 1
   if cnt <= n
     arr.push elem
   else
     pos = Math.floor(Math.random() * cnt)
     if pos < n
       arr[pos] = elem
   arr.sort()

sample_size = 3 range = [0..9] num_trials = 100000

counts = {}

for digit in range

 counts[digit] = 0

for i in [1..num_trials]

 s_of_n = s_of_n_creator(sample_size)
 for digit in range
   sample = s_of_n(digit)
 for digit in sample
   counts[digit] += 1

for digit in range

 console.log digit, counts[digit]

</lang> output <lang> > coffee knuth_sample.coffee 0 29899 1 29841 2 29930 3 30058 4 29932 5 29948 6 30047 7 30114 8 29976 9 30255 </lang>

Common Lisp

<lang lisp>(defun s-n-creator (n)

 (let ((sample (make-array n :initial-element nil))
       (i 0))
   (lambda (item)
     (if (<= (incf i) n)
         (setf (aref sample (1- i)) item)
       (when (< (random i) n)
         (setf (aref sample (random n)) item)))
     sample)))

(defun algorithm-s ()

 (let ((*random-state* (make-random-state t))
       (frequency (make-array '(10) :initial-element 0)))
   (loop repeat 100000
         for s-of-n = (s-n-creator 3)
         do (flet ((s-of-n (item)
                     (funcall s-of-n item)))
              (map nil
                   (lambda (i)
                     (incf (aref frequency i)))
                   (loop for i from 0 below 9
                         do (s-of-n i)
                         finally (return (s-of-n 9))))))
   frequency))

(princ (algorithm-s)) </lang>output<lang>#(30026 30023 29754 30017 30267 29997 29932 29990 29965 30029)</lang>

D

<lang d>import std.stdio, std.random;

auto sofN_creator(in int n) {

   size_t i;
   int[] sample;

   return (in int item) {
       i++;
       if (i <= n)
           sample ~= item;
       else if (uniform01 < (double(n) / i))
           sample[uniform(0, n)] = item;
       return sample;
   };

}

void main() {

   enum nRuns = 100_000;
   size_t[10] bin;

   foreach (immutable trial; 0 .. nRuns) {
       immutable sofn = sofN_creator(3);
       int[] sample;
       foreach (immutable item; 0 .. bin.length)
           sample = sofn(item);
       foreach (immutable s; sample)
           bin[s]++;
   }
   writefln("Item counts for %d runs:\n%s", nRuns, bin);

}</lang>

Output:

Item counts for 100000 runs:
[30191, 29886, 29988, 30149, 30251, 29997, 29748, 29909, 30041, 29840]

Faster Version

<lang d>import std.stdio, std.random, std.algorithm;

struct SOfN(size_t n) {

   size_t i;
   int[n] sample = void;

   int[] next(in size_t item, ref Xorshift rng) {
       i++;
       if (i <= n)
           sample[i - 1] = item;
       else if (rng.uniform01 < (double(n) / i))
           sample[uniform(0, n, rng)] = item;
       return sample[0 .. min(i, $)];
   }

}

void main() {

   enum nRuns = 100_000;
   size_t[10] bin;
   auto rng = Xorshift(0);

   foreach (immutable trial; 0 .. nRuns) {
       SOfN!3 sofn;
       foreach (immutable item; 0 .. bin.length - 1)
           sofn.next(item, rng);
       foreach (immutable s; sofn.next(bin.length - 1, rng))
           bin[s]++;
   }
   writefln("Item counts for %d runs:\n%s", nRuns, bin);

}</lang>

Elena

<lang elena>#import system.

import system'dynamic.
import extensions.
import system'routines.
import system'collections.

class(extension)algorithmOp

{

   #method s_of_n
   [
       #var counter := Integer new.
       
       ^ ArrayList new mix &into:
           {
               eval : n
               [
                   counter += 1.
                   
                   (this length < self)
                       ? [ this += n. ]
                       ! [ 
                           (randomGenerator eval:counter < self)
                               ? [ this@(randomGenerator eval:self) := n. ].
                       ].
                       
                   ^ this array.                        
               ]
           }.
   ]

}

symbol program =

[

   #var bin := Array new:10 set &every:(&index:n) [ Integer new ].
   0 till:10000 &doEach: trial
   [
       #var s_of_n := 3 s_of_n.
       
       0 till:10 &doEach:n
       [
           #var sample := s_of_n eval:n.
           
           (n == 9)
               ? [ sample run &each: i [ bin@i += 1. ]. ].
       ].
   ].    
   
   console writeLine:bin.

].</lang>

Output:

3050,3029,3041,2931,3040,2952,2901,2984,3069,3003

Elixir

<lang elixir> defmodule Knuth do

 def s_of_n_creator(n), do: {n, 1, []}

 def s_of_n({n, i, ys}, x) do
   cond do
     i <= n -> {n, i+1, [x|ys]}

     :rand.uniform(i) <= n ->
       {n, i+1, List.replace_at(ys, :rand.uniform(n)-1, x)}

     true -> {n, i+1, ys}
   end
 end

end

results = Enum.reduce(1..100000, %{}, fn _, freq ->

 {_, _, xs} = Enum.reduce(1..10, Knuth.s_of_n_creator(3), fn x, s ->
   Knuth.s_of_n(s, x)
 end)
 Enum.reduce(xs, freq, fn x, freq ->
   Map.put(freq, x, (freq[x] || 0) + 1)
 end)

end)

IO.inspect results </lang> Output:

%{1 => 30138, 2 => 29980, 3 => 29992, 4 => 29975, 5 => 30110, 6 => 29825,
  7 => 29896, 8 => 30188, 9 => 29898, 10 => 29998}

Go

<lang go>package main

import (

   "fmt"
   "math/rand"
   "time"

)

func sOfNCreator(n int) func(byte) []byte {

   s := make([]byte, 0, n)
   m := n
   return func(item byte) []byte {
       if len(s) < n {
           s = append(s, item)
       } else {
           m++
           if rand.Intn(m) < n {
               s[rand.Intn(n)] = item
           }
       }
       return s
   }

}

func main() {

   rand.Seed(time.Now().UnixNano())
   var freq [10]int
   for r := 0; r < 1e5; r++ {
       sOfN := sOfNCreator(3)
       for d := byte('0'); d < '9'; d++ {
           sOfN(d)
       }
       for _, d := range sOfN('9') {
           freq[d-'0']++
       }
   }
   fmt.Println(freq)

}</lang> Output:

[30075 29955 30024 30095 30031 30018 29973 29642 30156 30031]

Icon and Unicon

The following solution makes use of the makeProc procedure defined in the UniLib library and so is Unicon specific. However, the solution can be modified to work in Icon as well.

Technically, s_of_n_creator returns a co-expression, not a function. In Unicon, the calling syntax for this co-expression is indistinguishable from that of a function. <lang unicon>import Utils

procedure main(A)

   freq := table(0)
   every 1 to (\A[2] | 100000)\1 do {
       s_of_n := s_of_n_creator(\A[1] | 3)
       every sample := s_of_n(0 to 9)
       every freq[!sample] +:= 1
       }
   every write(i := 0 to 9,": ",right(freq[i],6))

end

procedure s_of_n_creator(n)

   items := []
   itemCnt := 0.0
   return makeProc {
              repeat {
                  item := (items@&source)[1]
                  itemCnt +:= 1
                  if *items < n then put(items, item)
                  else if ?0 < (n/itemCnt) then ?items := item
                  }
              }

end</lang> and a sample run:

J

Note that this approach introduces heavy inefficiencies, to achieve information hiding.

<lang j>s_of_n_creator=: 1 :0

 ctx=: conew&'inefficient' m
 s_of_n__ctx

)

coclass'inefficient'

 create=:3 :0
   N=: y
   ITEMS=: 
   K=:0
 )

 s_of_n=:3 :0
   K=: K+1
   if. N>:#ITEMS do.
     ITEMS=: ITEMS,y
   else.
     if. (N%K)>?0 do.
       ITEMS=: ((<<<?N){ITEMS),y
     else.
       ITEMS
     end.
   end.
 )

</lang>

Explanation: create is the constructor for the class named inefficient and it initializes three properties: N (our initial value), ITEMS (an initially empty list) and K (a counter which is initially 0).

Also, we have s_of_n which is a method of that class. It increments K and appends to the list, respecting the random value replacement requirement, once the list has reached the required length.

Finally, we have s_of_n_creator which is not a method of that class, but which will create an object of that class and return the resulting s_of_n method.

Required example:

<lang j>run=:3 :0

 nl=. conl 1
 s3_of_n=. 3 s_of_n_creator
 r=. {: s3_of_n"0 i.10
 coerase (conl 1)-.nl
 r

)

  (~.,._1 + #/.~) (i.10),,D=:run"0 i.1e5

0 40119 1 40050 2 40163 3 57996 4 42546 5 40990 6 38680 7 36416 8 33172 9 29868</lang>

Here, we have each of our digits along with how many times each appeared in a result from run.

Explanation of run:

First, we get a snapshot of the existing objects in nl.

Then, we get our s3_of_n which is a method in a new object.

Then we run that method on each of the values 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, keeping only the values from the last run, this will be the result of the run.

Then we delete any objects which did not previously exist.

Finally return our result.

Java

A class-based solution: <lang java>import java.util.*;

class SOfN<T> {

   private static final Random rand = new Random();

   private List<T> sample;
   private int i = 0;
   private int n;
   public SOfN(int _n) {

n = _n; sample = new ArrayList<T>(n);

   }
   public List<T> process(T item) {

i++; if (i <= n) {

           sample.add(item);

} else if (rand.nextInt(i) < n) { sample.set(rand.nextInt(n), item); } return sample;

}

public class AlgorithmS {

   public static void main(String[] args) {

int[] bin = new int[10]; for (int trial = 0; trial < 100000; trial++) { SOfN<Integer> s_of_n = new SOfN<Integer>(3); List<Integer> sample = null; for (int i = 0; i < 10; i++) sample = s_of_n.process(i); for (int s : sample) bin[s]++; } System.out.println(Arrays.toString(bin));

}</lang>

Output:

[30115, 30141, 30050, 29887, 29765, 30132, 29767, 30114, 30079, 29950]

Alternative solution without using an explicitly named type; instead using an anonymous class implementing a generic "function" interface: <lang java>import java.util.*;

interface Function<S, T> {

   public T call(S x);

}

public class AlgorithmS {

   private static final Random rand = new Random();
   public static <T> Function<T, List<T>> s_of_n_creator(final int n) {

return new Function<T, List<T>>() { private List<T> sample = new ArrayList<T>(n); private int i = 0; public List<T> call(T item) { i++; if (i <= n) { sample.add(item); } else if (rand.nextInt(i) < n) { sample.set(rand.nextInt(n), item); } return sample; } };

   public static void main(String[] args) {

int[] bin = new int[10]; for (int trial = 0; trial < 100000; trial++) { Function<Integer, List<Integer>> s_of_n = s_of_n_creator(3); List<Integer> sample = null; for (int i = 0; i < 10; i++) sample = s_of_n.call(i); for (int s : sample) bin[s]++; } System.out.println(Arrays.toString(bin));

}</lang>

Julia

<lang Julia> function makesofn(n::Int)

   buf = Any[]
   i = 0
   function sofn(item)
       i += 1
       if i <= n
           push!(buf, item)
       else
           j = rand(1:i)
           if j <= n
               buf[j] = item
           end
       end
       return buf
   end
   return sofn

end

nhist = zeros(Int, 10)

for i in 1:10^5

   kas = makesofn(3)
   for j in 0:8
       kas(j)
   end
   for k in kas(9)
       nhist[k+1] += 1
   end

end

println("Simulating sof3(0:9) 100000 times:") for (i, c) in enumerate(nhist)

   println(@sprintf "   %2d => %5d" i-1 c)

end </lang>

Output:

Simulating sof3(0:9) 100000 times:
    0 => 29994
    1 => 30026
    2 => 30173
    3 => 29590
    4 => 29967
    5 => 30104
    6 => 30185
    7 => 29761
    8 => 30147
    9 => 30053

Objective-C

Works with: Mac OS X version 10.6+

Uses blocks <lang objc>#import <Foundation/Foundation.h>

typedef NSArray *(^SOfN)(id);

SOfN s_of_n_creator(int n) {

 NSMutableArray *sample = [[NSMutableArray alloc] initWithCapacity:n];
 __block int i = 0;
 return [^(id item) {
   i++;
   if (i <= n) {
     [sample addObject:item];
   } else if (rand() % i < n) {
     sample[rand() % n] = item;
   }
   return sample;
 } copy];

}

int main(int argc, const char *argv[]) {

 @autoreleasepool {

   NSCountedSet *bin = [[NSCountedSet alloc] init];
   for (int trial = 0; trial < 100000; trial++) {
     SOfN s_of_n = s_of_n_creator(3);
     NSArray *sample;
     for (int i = 0; i < 10; i++)
       sample = s_of_n(@(i));
     [bin addObjectsFromArray:sample];
   }
   NSLog(@"%@", bin);
 
 }
 return 0;

}</lang>

Log:

<NSCountedSet: 0x100114120> (0 [29934], 9 [30211], 5 [29926], 1 [30067], 6 [30001], 2 [29972], 7 [30126], 3 [29944], 8 [29910], 4 [29909])

OCaml

<lang ocaml>let s_of_n_creator n =

 let i = ref 0
 and sample = ref [| |] in
 fun item ->
   incr i;
   if !i <= n then sample := Array.append [| item |] !sample
   else if Random.int !i < n then !sample.(Random.int n) <- item;
   !sample

let test n items_set =

 let s_of_n = s_of_n_creator n in
 Array.fold_left (fun _ v -> s_of_n v) [| |] items_set

let () =

 Random.self_init();
 let n = 3 in
 let num_items = 10 in
 let items_set = Array.init num_items (fun i -> i) in
 let results = Array.create num_items 0 in
 for i = 1 to 100_000 do
   let res = test n items_set in
   Array.iter (fun j -> results.(j) <- succ results.(j)) res
 done;
 Array.iter (Printf.printf " %d") results;
 print_newline()</lang>

Output:

 30051 29899 30249 30058 30012 29836 29998 29882 30148 29867

PARI/GP

This example is in need of improvement:

Does not return a function.

<lang parigp>KnuthS(v,n)={

 my(u=vector(n,i,i));
 for(i=n+1,#v,
   if(random(i)<n,u[random(n)+1]=i)
 );
 vecextract(v,u)

}; test()={

 my(v=vector(10),t);
 for(i=1,1e5,
   t=KnuthS([0,1,2,3,4,5,6,7,8,9],3);
   v[t[1]+1]++;v[t[2]+1]++;v[t[3]+1]++
 );
 v

};</lang>

Output:

%1 = [30067, 30053, 29888, 30161, 30204, 29990, 30175, 29980, 29622, 29860]

Perl

<lang perl>use strict;

sub s_of_n_creator {

   my $n = shift;
   my @sample;
   my $i = 0;
   sub {
       my $item = shift;
       $i++;
       if ($i <= $n) {
           # Keep first n items
           push @sample, $item;
       } elsif (rand() < $n / $i) {
           # Keep item
           @sample[rand $n] = $item;
       }
       @sample
   }

}

my @items = (0..9); my @bin;

foreach my $trial (1 .. 100000) {

   my $s_of_n = s_of_n_creator(3);
   my @sample;
   foreach my $item (@items) {
       @sample = $s_of_n->($item);
   }
   foreach my $s (@sample) {
       $bin[$s]++;
   }

} print "@bin\n"; </lang>

Sample output

30003 29923 30192 30164 29994 29976 29935 29860 30040 29913

Perl 6

<lang perl6>sub s_of_n_creator($n) {

   my @sample;
   my $i = 0;
   -> $item {
       if ++$i <= $n {
           push @sample, $item;
       }
       elsif $i.rand < $n {
           @sample[$n.rand] = $item;
       }
       @sample;
   }

}

my @items = 0..9; my @bin;

for ^100000 {

   my &s_of_n = s_of_n_creator(3);
   my @sample;
   for @items -> $item {
       @sample = s_of_n($item);
   }
   for @sample -> $s {
       @bin[$s]++;
   }

} say @bin;</lang> Output:

29975 30028 30246 30056 30004 29983 29836 29967 29924 29981

PHP

Works with: PHP version 5.3+

<lang php><?php function s_of_n_creator($n) {

   $sample = array();
   $i = 0;
   return function($item) use (&$sample, &$i, $n) {
       $i++;
       if ($i <= $n) {
           // Keep first n items
           $sample[] = $item;
       } else if (rand(0, $i-1) < $n) {
           // Keep item
           $sample[rand(0, $n-1)] = $item;
       }
       return $sample;
   };

}

$items = range(0, 9);

for ($trial = 0; $trial < 100000; $trial++) {

   $s_of_n = s_of_n_creator(3);
   foreach ($items as $item)
       $sample = $s_of_n($item);
   foreach ($sample as $s)
       $bin[$s]++;

} print_r($bin); ?></lang>

Sample output

Array
(
    [3] => 30158
    [8] => 29859
    [9] => 29984
    [6] => 29937
    [7] => 30361
    [4] => 29994
    [5] => 29849
    [0] => 29724
    [1] => 29997
    [2] => 30137
)

PicoLisp

<lang PicoLisp>(de s_of_n_creator (@N)

  (curry (@N (I . 0) (Res)) (Item)
     (cond
        ((>= @N (inc 'I)) (push 'Res Item))
        ((>= @N (rand 1 I)) (set (nth Res (rand 1 @N)) Item)) )
     Res ) )

(let Freq (need 10 0)

  (do 100000
     (let S_of_n (s_of_n_creator 3)
        (for I (mapc S_of_n (0 1 2 3 4 5 6 7 8 9))
           (inc (nth Freq (inc I))) ) ) )
  Freq )</lang>

Output:

-> (30003 29941 29918 30255 29848 29875 30056 29839 30174 30091)

Python

Works with: Python version 3.x

<lang python>from random import randrange

def s_of_n_creator(n):

   sample, i = [], 0
   def s_of_n(item):
       nonlocal i

       i += 1
       if i <= n:
           # Keep first n items
           sample.append(item)
       elif randrange(i) < n:
           # Keep item
           sample[randrange(n)] = item
       return sample
   return s_of_n

if __name__ == '__main__':

   bin = [0]* 10
   items = range(10)
   print("Single run samples for n = 3:")
   s_of_n = s_of_n_creator(3)
   for item in items:
       sample = s_of_n(item)
       print("  Item: %i -> sample: %s" % (item, sample))
   #
   for trial in range(100000):
       s_of_n = s_of_n_creator(3)
       for item in items:
           sample = s_of_n(item)
       for s in sample:
           bin[s] += 1
   print("\nTest item frequencies for 100000 runs:\n ",
         '\n  '.join("%i:%i" % x for x in enumerate(bin)))</lang>

Sample output

Single run samples for n = 3:
  Item: 0 -> sample: [0]
  Item: 1 -> sample: [0, 1]
  Item: 2 -> sample: [0, 1, 2]
  Item: 3 -> sample: [0, 1, 3]
  Item: 4 -> sample: [0, 1, 3]
  Item: 5 -> sample: [0, 1, 3]
  Item: 6 -> sample: [0, 1, 3]
  Item: 7 -> sample: [0, 3, 7]
  Item: 8 -> sample: [0, 3, 7]
  Item: 9 -> sample: [0, 3, 7]

Test item frequencies for 100000 runs:
  0:29983
  1:30240
  2:29779
  3:29921
  4:30224
  5:29967
  6:30036
  7:30050
  8:29758
  9:30042

Python Class based version

Only a slight change creates the following class-based implementation: <lang python>class S_of_n_creator():

   def __init__(self, n):
       self.n = n
       self.i = 0
       self.sample = []
   
   def __call__(self, item):
       self.i += 1
       n, i, sample = self.n, self.i, self.sample
       if i <= n:
           # Keep first n items
           sample.append(item)
       elif randrange(i) < n:
           # Keep item
           sample[randrange(n)] = item
       return sample</lang>

The above can be instantiated as follows after which s_of_n can be called in the same way as it is in the first example where it is a function instead of an instance. <lang python>s_of_n = S_of_n_creator(3)</lang>

Racket

<lang racket>#lang racket/base

(define (s-of-n-creator n)

 (define i 0)
 (define sample (make-vector n)) ; the sample of n items
 (lambda (item)
   (set! i (add1 i))
   (cond [(<= i n)               ; we're not full, so kind of boring
          (vector-set! sample (sub1 i) item)]
         [(< (random i) n)       ; we've already seen n items; swap one?
          (vector-set! sample (random n) item)])
   sample))

(define counts (make-vector 10 0))

(for ([i 100000])

 (define s-of-n (s-of-n-creator 3))
 (define sample (for/last ([digit 10]) (s-of-n digit)))
 (for ([d sample]) (vector-set! counts d (add1 (vector-ref counts d)))))

(for ([d 10]) (printf "~a ~a\n" d (vector-ref counts d)))</lang> Output:

REXX

<lang rexx>/*REXX program using Knuth's algorithm S (a random sampling N of M items).*/ parse arg trials size . /*obtain optional arguments from the CL*/ if trials== then trials=100000 /*Not specified? Then use the default.*/ if size== then size= 3 /* " " " " " " */

.=0 /*initialize frequency counter array. */

     do trials                        /*OK,  now let's light this candle.    */
     call s_of_n_creator    size      /*create an initial list of  N  items. */

        do gener=0  for 10
        call s_of_n gener             /*call s_of_n with a single decimal dig*/
        end       /*gener*/

            do count=1  for size      /*let's examine what  SofN  generated. */
            _=!.count                 /*get a decimal digit from the   Nth   */
            #._=#._+1                 /*  ··· item,  and count it, of course.*/
            end   /*count*/
     end          /*trials*/
                                                     @='trials, and with size='

say "Using Knuth's algorithm S for" commas(trials) @ || commas(size)":" say

     do dig=0  to 9                   /* [↓]  display the frequency of a dig.*/
     say left(,20)   "frequency of the"    dig    'digit is:'   commas(#.dig)
     end   /*dig*/

exit /*stick a fork in it, we're all done. */ /*────────────────────────────────────────────────────────────────────────────*/ commas: procedure; parse arg _; n=_'.9'; #=123456789; b=verify(n,#,"M")

       e=verify(n, #'0', , verify(n, #"0.", 'M')) - 4
          do j=e  to b  by -3;   _=insert(',',_,j);    end  /*j*/;     return _

/*────────────────────────────────────────────────────────────────────────────*/ s_of_n: parse arg item; items=items+1 /*get "item", bump the items counter.*/

       c=random(1, items)             /* [↓]  should replace a previous item?*/
       if c>size  then return         /*probability isn't good,  so skip it. */
       _=random(1, size);  !._=item   /*now, figure out which previous ···   */
       return                         /*      ··· item to replace with  ITEM.*/

/*────────────────────────────────────────────────────────────────────────────*/ s_of_n_creator: parse arg item 1 items /*generate ITEM number of items. */

                    do k=1  for item  /*traipse through the first  N  items. */
                    !.k=random(0, 9)  /*set the  Kth  item with random digit.*/
                    end   /*k*/
       return                         /*the piddly stuff is done  (for now). */</lang>

output when using the default input of: 100000 2

Using Knuth's algorithm  S  for 100,000 trials, and with size=3:

                     frequency of the 0 digit is: 29,843
                     frequency of the 1 digit is: 30,083
                     frequency of the 2 digit is: 29,966
                     frequency of the 3 digit is: 30,006
                     frequency of the 4 digit is: 30,137
                     frequency of the 5 digit is: 29,833
                     frequency of the 6 digit is: 30,160
                     frequency of the 7 digit is: 30,182
                     frequency of the 8 digit is: 29,941
                     frequency of the 9 digit is: 29,849

Ruby

Using a closure <lang ruby>def s_of_n_creator(n)

 sample = []
 i = 0
 Proc.new do |item|
   i += 1
   if i <= n
     sample << item
   elsif rand(i) < n
     sample[rand(n)] = item
   end
   sample
 end

end

frequency = Array.new(10,0) 100_000.times do

 s_of_n = s_of_n_creator(3)
 sample = nil
 (0..9).each {|digit| sample = s_of_n[digit]}
 sample.each {|digit| frequency[digit] += 1}

end

(0..9).each {|digit| puts "#{digit}\t#{frequency[digit]}"}</lang>

Example

0       29850
1       30015
2       29970
3       29789
4       29841
5       30075
6       30281
7       30374
8       29953
9       29852

Sidef

Translation of: Perl 6

<lang ruby>func s_of_n_creator(n) {

   var i = 0
   var sample = []
   { |item|
       if (++i <= n) {
           sample << item;
       }
       elsif (i.rand < n) {
           sample[n.rand] = item;
       }
       sample;
   }

}

var items = 0..9; var bin = [];

100000.times {

   var s_of_n = s_of_n_creator(3);
   var sample = []
   for item in items {
       sample = s_of_n(item);
   }
   for s in sample {
       bin[s] := 0 ++;
   }

}

say bin;</lang>

Output:

[30056, 29906, 30058, 29986, 30062, 29748, 29989, 29985, 30126, 30084]

Swift

<lang swift>import Darwin

func s_of_n_creator<T>(n: Int) -> T -> [T] {

 var sample = [T]()
 var i = 0
 return {(item: T) in
   i++
   if (i <= n) {
     sample.append(item)
   } else if (Int(arc4random_uniform(UInt32(i))) < n) {
     sample[Int(arc4random_uniform(UInt32(n)))] = item
   }
   return sample
 }

}

var bin = [Int](count:10, repeatedValue:0) for trial in 0..<100000 {

 let s_of_n: Int -> [Int] = s_of_n_creator(3)
 var sample: [Int] = []
 for i in 0..<10 {
   sample = s_of_n(i)
 }
 for s in sample {
   bin[s]++
 }

} println(bin)</lang>

Output:

[30038, 29913, 30047, 30069, 30159, 30079, 29773, 29962, 30000, 29960]

Tcl

<lang tcl>package require Tcl 8.6

oo::class create SofN {

   variable items size count
   constructor {n} {

set size $n

   }
   method item {item} {

if {[incr count] <= $size} { lappend items $item } elseif {rand()*$count < $size} { lset items [expr {int($size * rand())}] $item } return $items

}

Test code

for {set i 0} {$i < 100000} {incr i} {

   set sOf3 [SofN new 3]
   foreach digit {0 1 2 3 4 5 6 7 8 9} {

set digs [$sOf3 item $digit]

   }
   $sOf3 destroy
   foreach digit $digs {

incr freq($digit)

} parray freq</lang>

Sample output:

freq(0) = 29812
freq(1) = 30099
freq(2) = 29927
freq(3) = 30106
freq(4) = 30048
freq(5) = 29993
freq(6) = 29912
freq(7) = 30219
freq(8) = 30060
freq(9) = 29824

zkl

<lang zkl>fcn s_of_n_creator(n){

  fcn(item,ri,N,samples){
     i:=ri.inc(); // 1,2,3,4,...
     if(i<=N) samples.append(item);
     else if ((0).random(i) < N) samples[(0).random(N)] = item;
     samples
  }.fp1(Ref(1),n,L())

}</lang> One run: <lang zkl>s3:=s_of_n_creator(3); [0..9].pump(List,s3,"copy").println();</lang>

Output:

L(L(0),L(0,1),L(0,1,2),L(0,1,2),L(0,4,2),L(5,4,2),L(5,6,2),L(5,6,2),L(5,6,2),L(9,6,2))

100,000 runs: <lang zkl>dist:=L(0,0,0,0,0,0,0,0,0,0); do(0d100_000){

  (0).pump(10,Void,s_of_n_creator(3)).apply2('wrap(n){dist[n]=dist[n]+1})

} N:=dist.sum(); dist.apply('wrap(n){"%.2f%%".fmt(n.toFloat()/N*100)}).println();</lang>

Output:

L("10.00%","9.98%","10.00%","9.99%","10.00%","9.98%","10.01%","10.04%","9.98%","10.02%")