Knuth's algorithm S

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Task
Knuth's algorithm S
You are encouraged to solve this task according to the task description, using any language you may know.

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
  1. Select the first n items as the sample as they become available;
  2. 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.
  3. Repeat #2 for any subsequent items.
The Task
  1. 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.
  2. 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.
  3. Test your functions by printing and showing the frequency of occurrences of the selected digits from 100,000 repetitions of:
  1. Use the s_of_n_creator with n == 3 to generate an s_of_n.
  2. 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.

Contents

[edit] 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:

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;

Here is the implementation of that package:

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;

The main program:

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;

A sample output:

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

[edit] BBC BASIC

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.

      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$(a$)) + "]"

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

[edit] C

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

#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;
}

[edit] C++

Works with: C++11
#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;
}
Output:
30052
29740
30197
30223
29857
29688
30095
29803
30098
30247

Class-based version:

#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;
}

[edit] 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.

(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)
 

Sample output:

([0 29924] [1 30053] [2 30018] [3 29765] [4 29974] [5 30225] [6 30082] [7 29996] [8 30128] [9 29835])

If we really need a stateful (thread safe!) function for some reason, we can get it like this:

(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)))))

[edit] 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]
 

output

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


[edit] Common 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))
 
output
#(30026 30023 29754 30017 30267 29997 29932 29990 29965 30029)

[edit] 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);
}
Output:
Item counts for 100000 runs:
[30191, 29886, 29988, 30149, 30251, 29997, 29748, 29909, 30041, 29840]

[edit] Faster Version

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);
}

[edit] 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)
}

Output:

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

[edit] 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.

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

and a sample run:

->kas    
0:  29941
1:  29963
2:  29941
3:  30005
4:  30087
5:  29895
6:  30075
7:  30059
8:  29962
9:  30072
->

[edit] J

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

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=: (((i.#ITEMS)-.?N){ITEMS),y
else.
ITEMS
end.
end.
)
 
 
s_of_n_creator_base_=: 1 :0
ctx=: conew&'inefficient' m
s_of_n__ctx
)

Required example:

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 30099
1 29973
2 29795
3 29995
4 29996
5 30289
6 29903
7 29993
8 30215
9 29742

[edit] Java

A class-based solution:

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));
}
}

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:

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));
}
}

[edit] Objective-C

Works with: Mac OS X version 10.6+

Uses blocks

#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;
}

Log:

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

[edit] 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()

Output:

 30051 29899 30249 30058 30012 29836 29998 29882 30148 29867

[edit] PARI/GP

This example is in need of improvement:
Does not return a function.
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
};

Output:

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

[edit] 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";
 
Sample output
30003 29923 30192 30164 29994 29976 29935 29860 30040 29913

[edit] Perl 6

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;

Output:

29975 30028 30246 30056 30004 29983 29836 29967 29924 29981

[edit] PHP

Works with: PHP version 5.3+
<?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);
?>
Sample output
Array
(
    [3] => 30158
    [8] => 29859
    [9] => 29984
    [6] => 29937
    [7] => 30361
    [4] => 29994
    [5] => 29849
    [0] => 29724
    [1] => 29997
    [2] => 30137
)

[edit] 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 )

Output:

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

[edit] Python

Works with: Python version 3.x
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)))
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

[edit] Python Class based version

Only a slight change creates the following class-based implementation:

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

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.

s_of_n = S_of_n_creator(3)

[edit] Racket

#lang racket/base
 
(define (s-of-n-creator n)
(let ([count 0] ; 'i' in the description
[vec (make-vector n)]) ; store the elts we've seen so far
(lambda (item)
(if (< count n)
 ; we're not full, so, kind of boring
(begin
(vector-set! vec count item)
(set! count (+ count 1)))
 ; we've already seen n elts; fun starts
(begin
(set! count (+ count 1))
(when (< (random count) n)
(vector-set! vec (random n) item))))
vec)))
 
(define counts (make-vector 10))
 
(for ([iter (in-range 0 100000)]) ; trials
(let ([s-of-n (s-of-n-creator 3)]) ; set up the chooser
(for ([d (in-vector ; iterate over the chosen digits
(for/last ([digit (in-range 0 10)]) ; loop through the digits
(s-of-n digit)))]) ; feed them in
(vector-set! counts d (add1 (vector-ref counts d)))))) ; update counts
 
(for ([d (in-range 0 10)])
(printf "~a ~a~n" d (vector-ref counts d)))

Output:

0 29906
1 29863
2 29953
3 30111
4 29867
5 30157
6 29985
7 30325
8 30030
9 29803

[edit] REXX

/*REXX program using Knuth's algorithm S (random sampling n of M items).*/
parse arg trials size . /*obtain the arguments from C.L. */
if trials=='' then trials=100000 /*use default if not specified. */
if size=='' then size=3 /* " " " " " */
#.=0 /*a couple handfuls of counters. */
do trials /*OK, let's light this candle. */
call s_of_n_creator size /*create initial list of n items.*/
 
do gener=0 for 10 /*and then call SofN for each dig*/
call s_of_n gener /*call s_of_n with a single dig*/
end /*gener*/
 
do count=1 for size /*let's see what s_of_n wroth. */
_=!.count /*get a digit from the Nth item, */
#._=#._+1 /* ... and count it, of course. */
end /*count*/
end /*trials*/
 
say "Using Knuth's algorihm S for" comma(trials) 'trials, and with size='comma(size)":"
say
do dig=0 to 9 /*show & tell time for frequency.*/
say copies(' ',15) "frequency of the" dig 'digit is:' comma(#.dig)
end /*dig*/
exit /*stick a fork in it, we're done.*/
/*──────────────────────────────────S_OF_N_CREATOR subroutine───────────*/
s_of_n_creator: parse arg item 1 items /*generate ITEM number of items*/
do k=1 for item /*traipse through the 1st N items*/
 !.k=random(0,9) /*set the Kth item with rand dig.*/
end /*k*/
return /*out piddly work is done for now*/
/*──────────────────────────────────S_OF_N subroutine───────────────────*/
s_of_n: parse arg item; items=items+1 /*get "item", bump items counter.*/
c=random(1,items) /*should we replace a prev item? */
if c>size then return /*probability isn't good, skip it*/
_=random(1,size) /*now, figure out which previous */
!._=item /* ... item to replace with ITEM.*/
return /*and back to the caller we go. */
/*──────────────────────────────────COMMA subroutine────────────────────*/
comma: procedure; parse arg _,c,p,t;arg ,cu;c=word(c ",",1)
if cu=='BLANK' then c=' '; o=word(p 3,1); p=abs(o); t=word(t 999999999,1)
if \datatype(p,'W') | \datatype(t,'W') | p==0 | arg()>4 then return _
n=_'.9'; #=123456789; k=0; if o<0 then do; b=verify(_,' '); if b==0 then return _
e=length(_)-verify(reverse(_),' ')+1; end; else do; b=verify(n,#,"M")
e=verify(n,#'0',,verify(n,#"0.",'M'))-p-1; end
do j=e to b by -p while k<t; _=insert(c,_,j); k=k+1; end; return _

output when using the default input of: 100000 2

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

                frequency of the 0 digit is: 29,837
                frequency of the 1 digit is: 29,871
                frequency of the 2 digit is: 30,071
                frequency of the 3 digit is: 29,965
                frequency of the 4 digit is: 30,082
                frequency of the 5 digit is: 30,106
                frequency of the 6 digit is: 30,109
                frequency of the 7 digit is: 29,843
                frequency of the 8 digit is: 30,192
                frequency of the 9 digit is: 29,924

[edit] Ruby

Using a closure

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]}"}

Example

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

[edit] 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)
Output:
[30038, 29913, 30047, 30069, 30159, 30079, 29773, 29962, 30000, 29960]

[edit] 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
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

[edit] 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())
}

One run:

s3:=s_of_n_creator(3);
[0..9].pump(List,s3,"copy").println();
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:

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