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
- 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_creatorthat given the maximum sample size, returns a functions_of_nthat takes one parameter,item. - Function
s_of_nwhen 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;
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++
<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:
30052 29740 30197 30223 29857 29688 30095 29803 30098 30247
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>
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>(setf *random-state* (make-random-state t))
(defun make-selector (n)
(let ((i 0) res)
(lambda (&optional (x nil x-p))
(if (and x-p (< (random (incf i)) n))
(if (< (length res) n) (push x res) (setf (elt res (random n)) x)))
res)))
- test
(loop repeat 100000
with freq = (make-array 10 :initial-element 0)
do (let ((f (make-selector 3)))
(mapc f '(0 1 2 3 4 5 6 7 8 9)) (mapc (lambda (i) (incf (aref freq i))) (funcall f)))
finally (prin1 freq))</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 (uniform(0.0, 1.0) < (cast(double)n / i))
sample[uniform(0, n)] = item;
return sample;
};
}
void main() {
enum nRuns = 100_000; size_t[10] bin;
foreach (trial; 0 .. nRuns) {
auto sofn = sofN_creator(3);
int[] sample;
foreach (item; 0 .. bin.length)
sample = sofn(item);
foreach (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(int n) {
size_t i; int[n] sample = void; static rng = Xorshift(0);
int[] next(in int item) {
i++;
if (i <= n)
sample[i - 1] = item;
else if (uniform(0.0, 1.0, rng) < (cast(double)n / i))
sample[uniform(0, n, rng)] = item;
return sample[0 .. min(i, $)];
}
}
void main() {
enum nRuns = 100_000; size_t[10] bin;
foreach (trial; 0 .. nRuns) {
SOfN!(3) sofn;
foreach (item; 0 .. bin.length - 1)
sofn.next(item);
foreach (s; sofn.next(bin.length - 1))
bin[s]++;
}
writefln("Item counts for %d runs:\n%s", nRuns, bin);
}</lang>
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:
->kas 0: 29941 1: 29963 2: 29941 3: 30005 4: 30087 5: 29895 6: 30075 7: 30059 8: 29962 9: 30072 ->
J
Note that this approach introduces heavy inefficiencies, to achieve information hiding.
<lang j>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
)</lang>
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 30099 1 29973 2 29795 3 29995 4 29996 5 30289 6 29903 7 29993 8 30215 9 29742</lang>
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>
Objective-C
Uses blocks <lang objc>#import <Foundation/Foundation.h>
typedef NSArray *(^SOfN)(id);
SOfN s_of_n_creator(int n) {
NSMutableArray *sample = [NSMutableArray arrayWithCapacity:n];
__block int i = 0;
return [[^(id item) {
i++;
if (i <= n) {
[sample addObject:item];
} else if (rand() % i < n) {
[sample replaceObjectAtIndex:rand() % n withObject:item];
}
return sample;
} copy] autorelease];
}
int main(int argc, const char *argv[]) {
NSAutoreleasePool *pool = [[NSAutoreleasePool alloc] init];
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([NSNumber numberWithInt:i]);
[bin addObjectsFromArray:sample];
}
NSLog(@"%@", bin);
[bin release];
[pool release];
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
<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
<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
<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>
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
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