Cycle detection: Difference between revisions
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{{Draft task}}
;Task:
Detect a cycle in an iterated function using Brent's algorithm.
Line 28 ⟶ 31:
101,2,5,26,167,95
=={{header|11l}}==
{{trans|D}}
<syntaxhighlight lang="11l">F print_result(x0, f, len, start)
print(‘Cycle length = ’len)
print(‘Start index = ’start)
V i = x0
L 1..start
i = f(i)
V cycle = [0] * len
L 0.<len
cycle[L.index] = i
i = f(i)
print(‘Cycle: ’, end' ‘’)
print(cycle)
F brent(f, x0)
Int cycle_length
V hare = x0
V power = 1
L
V tortoise = hare
L(i) 1..power
hare = f(hare)
I tortoise == hare
cycle_length = i
^L.break
power *= 2
hare = x0
L 1..cycle_length
hare = f(hare)
V cycle_start = 0
V tortoise = x0
L tortoise != hare
tortoise = f(tortoise)
hare = f(hare)
cycle_start++
print_result(x0, f, cycle_length, cycle_start)
brent(i -> (i * i + 1) % 255, 3)</syntaxhighlight>
{{out}}
<pre>
Cycle length = 6
Start index = 2
Cycle: [101, 2, 5, 26, 167, 95]
</pre>
=={{header|8086 Assembly}}==
<syntaxhighlight lang="asm"> cpu 8086
org 100h
section .text
mov ax,3 ; Print the first 20 values in the sequence
mov bx,f
xor cx,cx
mov dx,20
call seq
mov ax,3 ; Use Brent's algorithm to find the cycle
mov bx,f
call brent
mov bp,ax ; BP = start of cycle
mov di,dx ; DI = length of cycle
mov dx,nl ; Print a newline
call prstr
mov dx,len ; Print "Length: "
call prstr
mov ax,di ; Print the length
call prnum
mov dx,ix ; Print "Index: "
call prstr
mov ax,bp ; Print the index
call prnum
mov dx,nl ; Print another newline
call prstr
mov ax,3 ; Print the cycle
mov bx,f
mov cx,bp
mov dx,di
jmp seq
;;; Brent's algorithm
;;; Input: AX = x0, BX = address of function
;;; Output: AX = mu, DX = lambda, CX, SI, BP destroyed
;;; The routine in BX must preserve BX, CX, DX, SI, BP
brent: mov bp,ax ; BP = x0
mov cx,ax ; CX = tortoise
xor dx,dx ; DX = lambda
mov si,1 ; SI = power
.lam: call bx ; Apply function (AX = hare)
inc dx ; Lambda += 1
cmp ax,cx ; Done yet?
je .mu
cmp dx,si ; Time to start a new power of two?
jne .lam
mov cx,ax ; Tortoise = hare
shl si,1 ; power *= 2
xor dx,dx ; lambda = 0
jmp .lam
.mu: mov ax,bp ; Find position of first repetition
mov cx,dx ; CX = lambda
.apply: call bx
loop .apply
mov cx,bp ; CX = tortoise
xor si,si ; SI = mu
.lmu: cmp ax,cx ; Done yet?
je .done
call bx ; hare = f(hare)
xchg ax,cx ; tortoise = f(tortoise)
call bx
xchg ax,cx
inc si
jmp .lmu
.done: mov ax,si
ret
;;; Function to use
;;; AX = (AX*AX+1) mod 255
f: mul al ; AX = AL*AL (we know anything mod 255 is <256)
inc ax ; + 1
div byte [fdsor] ; This is faster than freeing up a byte register
xchg al,ah ; Remainder into AL
xor ah,ah ; Pad to 16 bits
ret
;;; -- Utility routines --
;;; Print decimal value of AL. Destroys AX, DX.
prnum: mov dx,nbuf ; Number output buffer
xchg bx,dx
.dgt: aam ; Extract digit
add al,'0' ; Make ASCII digit
dec bx
mov [bx],al ; Store in buffer
xchg ah,al ; Continue with rest of digits
test al,al ; As long as there are digits
jnz .dgt
xchg bx,dx ; Put buffer pointer back in DX
prstr: mov ah,9 ; Print using MS-DOS call.
int 21h
ret
;;; Print DX values in sequence x, f(x), f(f(x))... starting at CX.
;;; AX = x0, BX = function. AX, CX, DX, SI destroyed.
seq: test cx,cx ; CX = 0?
jz .print
.skip: call bx ; Otherwise skip CX numbers
loop .skip
.print: mov cx,dx ; Amount of numbers to print
.loop: mov si,ax ; Keep a copy of AX (print routine trashes it)
call prnum
mov ax,si ; Restore x
call bx ; Find the next value
loop .loop
ret
section .data
fdsor: db 255 ; This 255 is used for the modulus calculation
db '...' ; Buffer for byte-sized numeric output
nbuf: db ' $'
ix: db 'Index: $'
len: db 'Length: $'
nl: db 13,10,'$'</syntaxhighlight>
{{out}}
<pre>3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95
Length: 6 Index: 2
101 2 5 26 167 95 </pre>
=={{header|Ada}}==
This implementation is split across three files. The first is the specification of a generic package:
<syntaxhighlight lang="ada">
generic
type Element_Type is private;
package Brent is
type Brent_Function is access function (X : Element_Type) return Element_Type;
procedure Brent(F : Brent_Function; X0 : Element_Type; Lambda : out Integer; Mu : out Integer);
end Brent;
</syntaxhighlight>
The second is the body of the generic package:
<syntaxhighlight lang="ada">
package body Brent is
procedure Brent (F : Brent_Function; X0 : Element_Type; Lambda : out Integer; Mu : out Integer) is
Power : Integer := 1;
Tortoise : Element_Type := X0;
Hare : Element_Type := F(X0);
begin
Lambda := 1;
Mu := 0;
while Tortoise /= Hare loop
if Power = Lambda then
Tortoise := Hare;
Power := Power * 2;
Lambda := 0;
end if;
Hare := F(Hare);
Lambda := Lambda + 1;
end loop;
Tortoise := X0;
Hare := X0;
for I in 0..(Lambda-1) loop
Hare := F(Hare);
end loop;
while Hare /= Tortoise loop
Tortoise := F(Tortoise);
Hare := F(Hare);
Mu := Mu + 1;
end loop;
end Brent;
end Brent;
</syntaxhighlight>
By using a generic package, this implementation can be made to work for any unary function, not just integers. As a demonstration two instances of the test function are created and two instances of the generic package. These are produced inside the main procedure:
<syntaxhighlight lang="ada">
with Brent;
with Text_Io;
use Text_Io;
procedure Main is
package Integer_Brent is new Brent(Element_Type => Integer);
use Integer_Brent;
function F (X : Integer) return Integer is
((X * X + 1) mod 255);
type Mod255 is mod 255;
package Mod255_Brent is new Brent(Element_Type => Mod255);
function F255 (X : Mod255) return Mod255 is
(X * X + 1);
lambda : Integer;
Mu : Integer;
X : Integer := 3;
begin
for I in 1..41 loop
Put(Integer'Image(X));
if I < 41 then
Put(",");
end if;
X := F(X);
end loop;
New_Line;
Integer_Brent.Brent(F'Access, 3, Lambda, Mu);
Put_Line("Cycle Length: " & Integer'Image(Lambda));
Put_Line("Start Index : " & Integer'Image(Mu));
Mod255_Brent.Brent(F255'Access, 3, Lambda, Mu);
Put_Line("Cycle Length: " & Integer'Image(Lambda));
Put_Line("Start Index : " & Integer'Image(Mu));
Put("Cycle : ");
X := 3;
for I in 0..(Mu + Lambda) loop
if Mu <= I and I < (Lambda + Mu) then
Put(Integer'Image(X));
end if;
X := F(X);
end loop;
end Main;
</syntaxhighlight>
{{out}}
<pre> 3, 10, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5
Cycle Length: 6
Start Index : 2
Cycle Length: 6
Start Index : 2
Cycle : 101 2 5 26 167 95</pre>
=={{header|APL}}==
{{works with|Dyalog APL}}
<syntaxhighlight lang="apl">brent←{
f←⍺⍺
lam←⊃{
l p t h←⍵
p=l: 1 (p×2) h (f h) ⋄ (l+1) p t (f h)
}⍣{=/2↓⍺} ⊢ 1 1 ⍵ (f ⍵)
mu←⊃{
(⊃⍵+1),f¨1↓⍵
}⍣{=/1↓⍺} ⊢ 0 ⍵ (f⍣lam⊢⍵)
mu lam
}
task←{
seq←{f←⍺⍺ ⋄ (⊃⍺)↓{⍵,f⊃⌽⍵}⍣(1-⍨+/⍺)⊢⍵}
⎕←0 20 ⍺⍺ seq ⍵ ⍝ First 20 elements
⎕←(↑'Index' 'Length'),⍺⍺ brent ⍵ ⍝ Index and length of cycle
⎕←(⍺⍺ brent ⍺⍺ seq⊢)⍵ ⍝ Cycle
}
(255 | 1 + ⊢×⊢) task 3</syntaxhighlight>
{{out}}
<pre>3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95
Index 2
Length 6
101 2 5 26 167 95</pre>
=={{header|BCPL}}==
<syntaxhighlight lang="bcpl">get "libhdr"
// Brent's algorithm. 'fn' is a function pointer,
// 'lam' and 'mu' are output pointers.
let brent(fn, x0, lam, mu) be
$( let power, tort, hare = 1, x0, fn(x0)
!lam := 1
until tort = hare do
$( if power = !lam then
$( tort := hare
power := power * 2
!lam := 0
$)
hare := fn(hare)
!lam := !lam + 1
$)
tort := x0
hare := x0
for i = 1 to !lam do
hare := fn(hare)
!mu := 0
until tort = hare do
$( tort := fn(tort)
hare := fn(hare)
!mu := !mu + 1
$)
$)
// Print fn^m(x0) to fn^n(x0)
let printRange(fn, x0, m, n) be
$( for i = 0 to n-1 do
$( if m<=i then writef("%N ", x0)
x0 := fn(x0)
$)
wrch('*N')
$)
let start() be
$( let lam, mu = 0, 0
// the function to find a cycle in
let f(x) = (x*x + 1) rem 255
// print the first 20 values
printRange(f, 3, 0, 20)
// find the cycle
brent(f, 3, @lam, @mu)
writef("Length: %N*NIndex: %N*N", lam, mu)
// print the cycle
printRange(f, 3, mu, mu+lam)
$)</syntaxhighlight>
{{out}}
<pre>3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95
Length: 6
Index: 2
101 2 5 26 167 95</pre>
=={{header|BQN}}==
<syntaxhighlight lang="bqn">_Brent← {F _𝕣 x0:
p←l←1
(I ← {p=l?
l↩1 ⋄ p×↩2 ⋄ 𝕩I𝕩 ;
l+↩1 ⋄ 𝕨I𝕩
}⍟≠⟜F) x0
m←0
{m+↩1 ⋄ 𝕨𝕊⍟≠○F𝕩}⟜(F⍟l) x0
l‿m‿(F⍟(m+↕l)x0)
}</syntaxhighlight>
{{out|Example use}}
<pre> (255|1+ט)_Brent 3
⟨ 6 2 ⟨ 101 2 5 26 167 95 ⟩ ⟩</pre>
=={{header|C}}==
{{trans|Modula-2}}
<syntaxhighlight lang="c">#include <stdio.h>
#include <stdlib.h>
typedef int(*I2I)(int);
typedef struct {
int a, b;
} Pair;
Pair brent(I2I f, int x0) {
int power = 1, lam = 1, tortoise = x0, hare, mu, i;
Pair result;
hare = (*f)(x0);
while (tortoise != hare) {
if (power == lam) {
tortoise = hare;
power = power * 2;
lam = 0;
}
hare = (*f)(hare);
lam++;
}
hare = x0;
i = 0;
while (i < lam) {
hare = (*f)(hare);
i++;
}
tortoise = x0;
mu = 0;
while (tortoise != hare) {
tortoise = (*f)(tortoise);
hare = (*f)(hare);
mu++;
}
result.a = lam;
result.b = mu;
return result;
}
int lambda(int x) {
return (x*x + 1) % 255;
}
int main() {
int x0 = 3, x = 3, i;
Pair result;
printf("[3");
for (i = 1; i <= 40; ++i) {
x = lambda(x);
printf(", %d", x);
}
printf("]\n");
result = brent(lambda, x0);
printf("Cycle length = %d\nStart index = %d\nCycle = [", result.a, result.b);
x0 = 3;
x = x0;
for (i = 1; i <= result.b; ++i) {
x = lambda(x);
}
for (i = 1; i <= result.a; ++i) {
if (i > 1) {
printf(", ");
}
printf("%d", x);
x = lambda(x);
}
printf("]\n");
return 0;
}</syntaxhighlight>
{{out}}
<pre>[3, 10, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5]
Cycle length = 6
Start index = 2
Cycle = [101, 2, 5, 26, 167, 95]</pre>
=={{header|C_sharp|C#}}==
This solution uses generics, so may find cycles of any type of data, not just integers.
<
// First file: Cycles.cs
Line 128 ⟶ 578:
}
</syntaxhighlight>
=={{header|C++}}==
<syntaxhighlight lang="cpp">struct ListNode {
int val;
ListNode *next;
ListNode(int x) : val(x), next(NULL) {}
};
ListNode* Solution::detectCycle(ListNode* A) {
ListNode* slow = A;
ListNode* fast = A;
ListNode* cycleNode = 0;
while (slow && fast && fast->next)
{
slow = slow->next;
fast = fast->next->next;
if (slow == fast)
{
cycleNode = slow;
break;
}
}
if (cycleNode == 0)
{
return 0;
}
std::set<ListNode*> setPerimeter;
setPerimeter.insert(cycleNode);
for (ListNode* pNode = cycleNode->next; pNode != cycleNode; pNode = pNode->next)
setPerimeter.insert(pNode);
for (ListNode* pNode = A; true; pNode = pNode->next)
{
std::set<ListNode*>::iterator iter = setPerimeter.find(pNode);
if (iter != setPerimeter.end())
{
return pNode;
}
}
}</syntaxhighlight>
=={{header|CLU}}==
<syntaxhighlight lang="clu">% Find a cycle in f starting at x0 using Brent's algorithm
brent = proc [T: type] (f: proctype (T) returns (T), x0: T)
returns (int,int)
where T has equal: proctype (T,T) returns (bool)
pow: int := 1
lam: int := 1
tort: T := x0
hare: T := f(x0)
while tort ~= hare do
if pow = lam then
tort := hare
pow := pow * 2
lam := 0
end
hare := f(hare)
lam := lam + 1
end
tort := x0
hare := x0
for i: int in int$from_to(1,lam) do
hare := f(hare)
end
mu: int := 0
while tort ~= hare do
tort := f(tort)
hare := f(hare)
mu := mu + 1
end
return(lam, mu)
end brent
% Iterate over a function starting at x0 for N steps,
% starting at a given index
iterfunc = iter [T: type] (f: proctype (T) returns (T), x0: T, n, start: int)
yields (T)
for i: int in int$from_to(1, start) do
x0 := f(x0)
end
for i: int in int$from_to(1, n) do
yield(x0)
x0 := f(x0)
end
end iterfunc
% Iterated function
step_fn = proc (x: int) returns (int)
return((x*x + 1) // 255)
end step_fn
start_up = proc ()
po: stream := stream$primary_output()
% Print the first 20 items of the sequence
for i: int in iterfunc[int](step_fn, 3, 20, 0) do
stream$puts(po, int$unparse(i) || " ")
end
stream$putl(po, "")
% Find a cycle
lam, mu: int := brent[int](step_fn, 3)
% Print the length and index
stream$putl(po, "Cycle length: " || int$unparse(lam))
stream$putl(po, "Start index: " || int$unparse(mu))
% Print the cycle
stream$puts(po, "Cycle: ")
for i: int in iterfunc[int](step_fn, 3, lam, mu) do
stream$puts(po, int$unparse(i) || " ")
end
end start_up</syntaxhighlight>
{{out}}
<pre>3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95
Cycle length: 6
Start index: 2
Cycle: 101 2 5 26 167 95</pre>
=={{header|Cowgol}}==
<syntaxhighlight lang="cowgol">include "cowgol.coh";
typedef N is uint8;
interface BrentFn(x: N): (r: N);
sub Brent(f: BrentFn, x0: N): (lam: N, mu: N) is
var power: N := 1;
var tortoise := x0;
var hare := f(x0);
while tortoise != hare loop
if power == lam then
tortoise := hare;
power := power << 1;
lam := 0;
end if;
hare := f(hare);
lam := lam + 1;
end loop;
tortoise := x0;
hare := x0;
var i: N := 0;
while i < lam loop
i := i + 1;
hare := f(hare);
end loop;
mu := 0;
while tortoise != hare loop
tortoise := f(tortoise);
hare := f(hare);
mu := mu + 1;
end loop;
end sub;
sub PrintRange(f: BrentFn, x0: N, start: N, end_: N) is
var i: N := 0;
while i < end_ loop
if i >= start then
print_i32(x0 as uint32);
print_char(' ');
end if;
i := i + 1;
x0 := f(x0);
end loop;
print_nl();
end sub;
sub x2_plus1_mod255 implements BrentFn is
r := ((x as uint16 * x as uint16 + 1) % 255) as uint8;
end sub;
PrintRange(x2_plus1_mod255, 3, 0, 20);
var length: N;
var start: N;
(length, start) := Brent(x2_plus1_mod255, 3);
print("Cycle length: "); print_i32(length as uint32); print_nl();
print("Cycle start: "); print_i32(start as uint32); print_nl();
PrintRange(x2_plus1_mod255, 3, start, length+start);</syntaxhighlight>
{{out}}
<pre>3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95
Cycle length: 6
Cycle start: 2
101 2 5 26 167 95</pre>
=={{header|D}}==
{{trans|Java}}
<syntaxhighlight lang="d">import std.range;
import std.stdio;
void main() {
brent(i => (i * i + 1) % 255, 3);
}
void brent(int function(int) f, int x0) {
int cycleLength;
int hare = x0;
FOUND:
for (int power = 1; ; power *= 2) {
int tortoise = hare;
for (int i = 1; i <= power; i++) {
hare = f(hare);
if (tortoise == hare) {
cycleLength = i;
break FOUND;
}
}
}
hare = x0;
for (int i = 0; i < cycleLength; i++)
hare = f(hare);
int cycleStart = 0;
for (int tortoise = x0; tortoise != hare; cycleStart++) {
tortoise = f(tortoise);
hare = f(hare);
}
printResult(x0, f, cycleLength, cycleStart);
}
void printResult(int x0, int function(int) f, int len, int start) {
writeln("Cycle length: ", len);
writefln("Cycle: %(%s %)", iterate(x0, f).drop(start).take(len));
}
auto iterate(int start, int function(int) f) {
return only(start).chain(generate!(() => start=f(start)));
}</syntaxhighlight>
{{out}}
<pre>Cycle length: 6
Cycle: 101 2 5 26 167 95</pre>
=={{header|Elixir}}==
{{trans|Ruby}}
<syntaxhighlight lang="elixir">defmodule Cycle_detection do
def find_cycle(x0, f) do
lambda = find_lambda(f, x0, f.(x0), 1, 1)
hare = Enum.reduce(1..lambda, x0, fn _,hare -> f.(hare) end)
mu = find_mu(f, x0, hare, 0)
{lambda, mu}
end
# Find lambda, the cycle length
defp find_lambda(_, tortoise, hare, _, lambda) when tortoise==hare, do: lambda
defp find_lambda(f, tortoise, hare, power, lambda) do
if power == lambda, do: find_lambda(f, hare, f.(hare), power*2, 1),
else: find_lambda(f, tortoise, f.(hare), power, lambda+1)
end
# Find mu, the zero-based index of the start of the cycle
defp find_mu(_, tortoise, hare, mu) when tortoise==hare, do: mu
defp find_mu(f, tortoise, hare, mu) do
find_mu(f, f.(tortoise), f.(hare), mu+1)
end
end
# A recurrence relation to use in testing
f = fn(x) -> rem(x * x + 1, 255) end
# Display the first 41 numbers in the test series
Stream.iterate(3, &f.(&1)) |> Enum.take(41) |> Enum.join(",") |> IO.puts
# Test the find_cycle function
{clength, cstart} = Cycle_detection.find_cycle(3, f)
IO.puts "Cycle length = #{clength}\nStart index = #{cstart}"</syntaxhighlight>
{{out}}
<pre>
3,10,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5
Cycle length = 6
Start index = 2
</pre>
=={{header|Factor}}==
This is a strict translation of the Python code from the Wikipedia article. Perhaps a more idiomatic version could be added in the future, although there is value in showing how Factor's lexical variables differ from most languages. A variable binding with <code>!</code> at the end is mutable, and subsequent uses of that name followed by <code>!</code> change the value of the variable to the value at the top of the data stack.
<syntaxhighlight lang="factor">USING: formatting kernel locals make math prettyprint ;
: cyclical-function ( n -- m ) dup * 1 + 255 mod ;
:: brent ( x0 quot -- λ μ )
1 dup :> ( power! λ! )
x0 :> tortoise!
x0 quot call :> hare!
[ tortoise hare = not ] [
power λ = [
hare tortoise!
power 2 * power!
0 λ!
] when
hare quot call hare!
λ 1 + λ!
] while
0 :> μ!
x0 dup tortoise! hare!
λ [ hare quot call hare! ] times
[ tortoise hare = not ] [
tortoise quot call tortoise!
hare quot call hare!
μ 1 + μ!
] while
λ μ ; inline
3 [ 20 [ dup , cyclical-function ] times ] { } make nip .
3 [ cyclical-function ] brent
"Cycle length: %d\nCycle start: %d\n" printf</syntaxhighlight>
{{out}}
<pre>
{ 3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95 }
Cycle length: 6
Cycle start: 2
</pre>
=={{header|FOCAL}}==
<syntaxhighlight lang="focal">01.10 S X0=3;T %3
01.20 S X=X0;F I=1,20;T X;D 2
01.30 D 3;T !" START",M,!,"LENGTH",L,!
01.40 S X=X0;F I=1,M;D 2
01.50 F I=1,L;T X;D 2
01.60 T !
01.70 Q
02.01 C -- X = X*X+1 MOD 255
02.10 S X=X*X+1
02.20 S X=X-255*FITR(X/255)
03.01 C -- BRENT'S ALGORITHM ON ROUTINE 2
03.10 S X=X0;S P=1;S L=1;S T=X0;D 2
03.20 I (T-X)3.3,3.6,3.3
03.30 I (P-L)3.5,3.4,3.5
03.40 S T=X;S P=P*2;S L=0
03.50 D 2;S L=L+1;G 3.2
03.60 S T=X0;S X=X0;F I=1,L;D 2
03.70 S H=X;S M=0
03.80 I (T-H)3.9,3.99,3.9
03.90 S X=T;D 2;S T=X;S X=H;D 2;S H=X;S M=M+1;G 3.8
03.99 R</syntaxhighlight>
{{out}}
<pre>= 3= 10= 101= 2= 5= 26= 167= 95= 101= 2= 5= 26= 167= 95= 101= 2= 5= 26= 167= 95
START= 2
LENGTH= 6
= 101= 2= 5= 26= 167= 95</pre>
=={{header|Forth}}==
Works with gforth.
<syntaxhighlight lang="forth">
: cycle-length { x0 'f -- lambda } \ Brent's algorithm stage 1
1 1 x0 dup 'f execute
begin 2dup <> while
2over = if
2swap nip 2* 0
2swap nip dup
then
'f execute rot 1+ -rot
repeat 2drop nip ;
: iterations ( x f n -- x )
>r swap r> 0 ?do over execute loop nip ;
: cycle-start { x0 'f lambda -- mu } \ Brent's algorithm stage 2
0 x0 dup 'f lambda iterations
begin 2dup <> while
swap 'f execute swap 'f execute rot 1+ -rot
repeat 2drop ;
: find-cycle ( x0 'f -- mu lambda ) \ Brent's algorithm
2dup cycle-length dup >r cycle-start r> ;
\ --- usage ---
: .cycle { start len x0 'f -- }
x0
start 1- 0 do 'f execute loop
len 0 do 'f execute dup . loop
drop ;
: f(x) dup * 1+ 255 mod ;
3 ' f(x) find-cycle
." The cycle starts at offset " over . ." and has a length of " dup . cr
." The cycle is " 3 ' f(x) .cycle cr
bye
</syntaxhighlight>
{{Out}}
<pre>
The cycle starts at offset 2 and has a length of 6
The cycle is 101 2 5 26 167 95
</pre>
=={{header|FreeBASIC}}==
===Brent's algorithm===
{{trans|Python}}
<syntaxhighlight lang="freebasic">' version 11-01-2017
' compile with: fbc -s console
' define the function f(x)=(x*x +1) mod 255
Function f(x As Integer) As Integer
Return (x * x +1) Mod 255
End Function
Sub brent(x0 As Integer, ByRef lam As Integer, ByRef mu As Integer)
Dim As Integer i, power = 1
lam = 1
' main phase: search successive powers of two
Dim As Integer tortoise = f(x0) ' f(x0) is the element/node next to x0.
Dim As Integer hare = f(f(x0))
While tortoise <> hare
If power = lam Then
tortoise = hare
power *= 2
lam = 0
End If
hare = f(hare)
lam += 1
Wend
' Find the position of the first repetition of length ?
mu = 0
tortoise = x0
hare = x0
For i = 0 To lam -1
' range(lam) produces a list with the values 0, 1, ... , lam-1
hare = f(hare)
Next
' The distance between the hare and tortoise is now ?.
' Next, the hare and tortoise move at same speed until they agree
While tortoise <> hare
tortoise = f(tortoise)
hare = f(hare)
mu += 1
Wend
End Sub
' ------=< MAIN >=------
Dim As Integer i, j, lam, mu, x0 = 3
brent(x0, lam, mu)
Print " Brent's algorithm"
Print " Cycle starts at position: "; mu; " (starting position = 0)"
Print " The length of the Cycle = "; lam
Print
j = f(x0)
Print " Cycle: ";
For i = 1 To lam + mu -1
If i >= mu Then
Print j;
If i <> (lam + mu -1) Then Print ", "; Else Print ""
End If
j = f(j)
Next
Print
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End</syntaxhighlight>
{{out}}
<pre> Brent's algorithm
Cycle starts at position: 2 (starting position = 0)
The length of the Cycle = 6
Cycle: 101, 2, 5, 26, 167, 95</pre>
===Tortoise and hare. Floyd's algorithm===
{{trans|Wikipedia}}
<syntaxhighlight lang="freebasic">' version 11-01-2017
' compile with: fbc -s console
' define the function f(x)=(x*x +1) mod 255
Function f(x As Integer) As Integer
Return (x * x +1) Mod 255
End Function
Sub floyd(x0 As Integer, ByRef lam As Integer, ByRef mu As Integer)
' Main phase of algorithm: finding a repetition x_i = x_2i.
' The hare moves twice as quickly as the tortoise and
' the distance between them increases by 1 at each step.
' Eventually they will both be inside the cycle and then,
' at some point, the distance between them will be
' divisible by the period ?.
Dim As Integer tortoise = f(x0) ' f(x0) is the element/node next to x0.
Dim As Integer hare = f(f(x0))
While tortoise <> hare
tortoise = f(tortoise)
hare = f(f(hare))
Wend
' At this point the tortoise position, ?, which is also equal
' to the distance between hare and tortoise, is divisible by
' the period ?. So hare moving in circle one step at a time,
' and tortoise (reset to x0) moving towards the circle, will
' intersect at the beginning of the circle. Because the
' distance between them is constant at 2?, a multiple of ?,
' they will agree as soon as the tortoise reaches index µ.
' Find the position µ of first repetition.
mu = 0
tortoise = x0
While tortoise <> hare
tortoise = f(tortoise)
hare = f(hare) ' Hare and tortoise move at same speed
mu += 1
Wend
' Find the length of the shortest cycle starting from x_µ
' The hare moves one step at a time while tortoise is still.
' lam is incremented until ? is found.
lam = 1
hare = f(tortoise)
While tortoise <> hare
hare = f(hare)
lam += 1
Wend
End Sub
' ------=< MAIN >=------
Dim As Integer i, j, lam, mu, x0 = 3
floyd(x0, lam, mu)
Print " Tortoise and hare. Floyd's algorithm"
Print " Cycle starts at position: "; mu; " (starting position = 0)"
Print " The length of the Cycle = "; lam
Print
j = f(x0)
Print " Cycle: ";
For i = 1 To lam + mu -1
If i >= mu Then
Print j;
If i <> (lam + mu -1) Then Print ", "; Else Print ""
End If
j = f(j)
Next
Print
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End</syntaxhighlight>
=={{header|Go}}==
{{trans|Python}}
{{trans|Wikipedia}}
Run it on the [https://play.golang.org/p/unOtxuwZfg go playground], or on [https://goplay.space/#S1pQZSuJci go play space].
<syntaxhighlight lang="go">
package main
import "fmt"
func brent(f func(i int) int, x0 int) (int, int) {
var λ, µ, power, tortoise, hare int
// Main phase: search successive powers of two.
power = 1
λ = 1
tortoise = x0
hare = f(x0) // f(x0) is the element/node next to x0.
for tortoise != hare {
if power == λ { // Time to start a new power of two.
tortoise = hare
power *= 2
λ = 0
}
hare = f(hare)
λ++
}
// Find the position of the first repetition of length λ.
µ = 0
tortoise, hare = x0, x0
for i := 0; i < λ; i++ {
// produces a list with the values 0,1,...,λ-1.
hare = f(hare)
// The distance between hare and tortoise is now λ.
}
// The tortoise and the hare move at the same speed until they agree.
for tortoise != hare {
tortoise = f(tortoise)
hare = f(hare)
µ++
}
return λ, µ
}
func f(i int) int {
return (i*i + 1) % 255
}
func main() {
x0 := 3
λ, µ := brent(f, x0)
fmt.Println("Cycle length:", λ)
fmt.Println("Cycle start index:", µ)
a := []int{}
for i := 0; i <= λ+1; i++ {
a = append(a, x0)
x0 = f(x0)
}
fmt.Println("Cycle:", a[µ:µ+λ])
}</syntaxhighlight>
{{out}}
<pre>Cycle length: 6
Cycle start index: 2
Cycle: [101 2 5 26 167 95]</pre>
=={{header|Groovy}}==
{{trans|Java}}
<syntaxhighlight lang="groovy">import java.util.function.IntUnaryOperator
class CycleDetection {
static void main(String[] args) {
brent({ i -> (i * i + 1) % 255 }, 3)
}
static void brent(IntUnaryOperator f, int x0) {
int cycleLength = -1
int hare = x0
FOUND:
for (int power = 1; ; power *= 2) {
int tortoise = hare
for (int i = 1; i <= power; i++) {
hare = f.applyAsInt(hare)
if (tortoise == hare) {
cycleLength = i
break FOUND
}
}
}
hare = x0
for (int i = 0; i < cycleLength; i++) {
hare = f.applyAsInt(hare)
}
int cycleStart = 0
for (int tortoise = x0; tortoise != hare; cycleStart++) {
tortoise = f.applyAsInt(tortoise)
hare = f.applyAsInt(hare)
}
printResult(x0, f, cycleLength, cycleStart)
}
static void printResult(int x0, IntUnaryOperator f, int len, int start) {
printf("Cycle length: %d%nCycle: ", len)
int n = x0
for (int i = 0; i < start + len; i++) {
n = f.applyAsInt(n)
if (i >= start) {
printf("%s ", n)
}
}
println()
}
}</syntaxhighlight>
{{out}}
<pre>Cycle length: 6
Cycle: 2 5 26 167 95 101 </pre>
=={{header|Haskell}}==
Most of solutions given in other languages are not total. For function which does not have any cycles under iteration (i.e. f(x) = 1 + x) they will never terminate.
Haskellers, being able to handle conceptually infinite structures, traditionally consider totality as an important issue. The following solution is total for total inputs (modulo totality of iterated function) and allows nontermination only if input is explicitly infinite.
<syntaxhighlight lang="haskell">import Data.List (findIndex)
findCycle :: Eq a => [a] -> Maybe ([a], Int, Int)
findCycle lst =
do l <- findCycleLength lst
mu <- findIndex (uncurry (==)) $ zip lst (drop l lst)
let c = take l $ drop mu lst
return (c, l, mu)
findCycleLength :: Eq a => [a] -> Maybe Int
findCycleLength [] = Nothing
findCycleLength (x:xs) =
let loop _ _ _ [] = Nothing
loop pow lam x (y:ys)
| x == y = Just lam
| pow == lam = loop (2*pow) 1 y ys
| otherwise = loop pow (1+lam) x ys
in loop 1 1 x xs</syntaxhighlight>
'''Examples'''
<pre>λ> findCycle (cycle [1,2,3])
Just ([1,2,3],3,0)
λ> findCycle ([1..100] ++ cycle [1..12])
Just ([1,2,3,4,5,6,7,8,9,10,11,12],12,100)
λ> findCycle [1..1000]
Nothing
</pre>
<pre>
λ> findCycle (iterate (\x -> (x^2 + 1) `mod` 255) 3)
Just ([101,2,5,26,167,95],6,2)
λ> findCycle (take 100 $ iterate (\x -> x+1) 3)
Nothing
λ> findCycle (take 100000 $ iterate (\x -> x+1) 3)
Nothing
</pre>
=={{header|J}}==
Line 134 ⟶ 1,316:
Brute force implementation:
<
r=. ~.@(,u@{:)^:_ y
n=. u{:r
(,(#r)-])r i. n
)</
In other words: iterate until we stop getting new values, find the repeated value, and see how it fits into the sequence.
Line 144 ⟶ 1,326:
Example use:
<
2 6</
(Note that 1 0 1 are the coefficients of the polynomial <code>1 + (0 * x) + (1 * x * x)</code> or, if you prefer <code>(1 * x ^ 0) + (0 * x ^ 1) + (1 * x ^ 2)</code> - it's easier and probably more efficient to just hand the coefficients to p. than it is to write out some other expression which produces the same result.)
=={{header|Java}}==
{{works with|Java|8}}
<
import static java.util.stream.IntStream.*;
Line 161 ⟶ 1,343:
static void brent(IntUnaryOperator f, int x0) {
int
int
for (int power = 1; ; power *= 2) {
int
for (int i = 1;
if (tortoise == hare
}
}
}
for (int i = 0; i < cycleLength; i++)
hare = f.applyAsInt(hare);
int cycleStart = 0;
for (int tortoise = x0; tortoise != hare; cycleStart++) {
tortoise = f.applyAsInt(tortoise);
hare = f.applyAsInt(hare);
Line 193 ⟶ 1,375:
.forEach(n -> System.out.printf("%s ", n));
}
}</
<pre>Cycle length: 6
Cycle: 101 2 5 26 167 95</pre>
=={{header|jq}}==
{{trans|Julia}}
{{works with|jq}}
'''Works with gojq, the Go implementation of jq'''
<syntaxhighlight lang="jq">def floyd(f; x0):
{tort: (x0|f)}
| .hare = (.tort|f)
| until( .tort == .hare;
.tort |= f
| .hare |= (f|f) )
| .mu = 0
| .tort = x0
| until( .tort == .hare;
.tort |= f
| .hare |= f
| .mu += 1)
| .lambda = 1
| .hare = (.tort|f)
| until (.tort == .hare;
.hare |= f
| .lambda += 1 )
| {lambda, mu} ;
def task(f; x0):
def skip($n; stream):
foreach stream as $s (0; .+1; select(. > $n) | $s);
floyd(f; x0)
| .,
"Cycle:",
skip(.mu; limit((.lambda + .mu); 3 | recurse(f)));
</syntaxhighlight>
'''The specific function and task'''
<syntaxhighlight lang="jq">
def f: (.*. + 1) % 255;
task(f;3)
</syntaxhighlight>
{{out}}
<pre>
{
"lambda": 6,
"mu": 2
}
Cycle:
3
10
101
2
5
26
167
95
</pre>
=={{header|Julia}}==
{{works with|Julia|0.6}}
Following the Wikipedia article:
<syntaxhighlight lang="julia">using IterTools
function floyd(f, x0)
local tort = f(x0)
local hare = f(tort)
while tort != hare
tort = f(tort)
hare = f(f(hare))
end
local μ = 0
tort = x0
while tort != hare
tort = f(tort)
hare = f(hare)
μ += 1
end
λ = 1
hare = f(tort)
while tort != hare
hare = f(hare)
λ += 1
end
return λ, μ
end
f(x) = (x * x + 1) % 255
λ, μ = floyd(f, 3)
cycle = iterate(f, 3) |>
x -> Iterators.drop(x, μ) |>
x -> Iterators.take(x, λ) |>
collect
println("Cycle length: ", λ, "\nCycle start index: ", μ, "\nCycle: ", join(cycle, ", "))</syntaxhighlight>
{{out}}
<pre>Cycle length: 6
Cycle start index: 2
Cycle: 101, 2, 5, 26, 167, 95</pre>
=={{header|Kotlin}}==
<syntaxhighlight lang="scala">// version 1.1.2
typealias IntToInt = (Int) -> Int
fun brent(f: IntToInt, x0: Int): Pair<Int, Int> {
// main phase: search successive powers of two
var power = 1
var lam = 1
var tortoise = x0
var hare = f(x0) // f(x0) is the element/node next to x0.
while (tortoise != hare) {
if (power == lam) { // time to start a new power of two?
tortoise = hare
power *= 2
lam = 0
}
hare = f(hare)
lam++
}
// Find the position of the first repetition of length 'lam'
var mu = 0
tortoise = x0
hare = x0
for (i in 0 until lam) hare = f(hare)
// The distance between the hare and tortoise is now 'lam'.
// Next, the hare and tortoise move at same speed until they agree
while (tortoise != hare) {
tortoise = f(tortoise)
hare = f(hare)
mu++
}
return Pair(lam, mu)
}
fun main(args: Array<String>) {
val f = { x: Int -> (x * x + 1) % 255 }
// generate first 41 terms of the sequence starting from 3
val x0 = 3
var x = x0
val seq = List(41) { if (it > 0) x = f(x) ; x }
println(seq)
val (lam, mu) = brent(f, x0)
val cycle = seq.slice(mu until mu + lam)
println("Cycle length = $lam")
println("Start index = $mu")
println("Cycle = $cycle")
}</syntaxhighlight>
{{out}}
<pre>
[3, 10, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5]
Cycle length = 6
Start index = 2
Cycle = [101, 2, 5, 26, 167, 95]
</pre>
=={{header|Lua}}==
Fairly direct translation of the Wikipedia code, except that the sequence is stored in a table and passed back as a third return value.
<syntaxhighlight lang="lua">function brent (f, x0)
local tortoise, hare, mu = x0, f(x0), 0
local cycleTab, power, lam = {tortoise, hare}, 1, 1
while tortoise ~= hare do
if power == lam then
tortoise = hare
power = power * 2
lam = 0
end
hare = f(hare)
table.insert(cycleTab, hare)
lam = lam + 1
end
tortoise, hare = x0, x0
for i = 1, lam do hare = f(hare) end
while tortoise ~= hare do
tortoise = f(tortoise)
hare = f(hare)
mu = mu + 1
end
return lam, mu, cycleTab
end
local f = function (x) return (x * x + 1) % 255 end
local x0 = 3
local cycleLength, startIndex, sequence = brent(f, x0)
print("Sequence:", table.concat(sequence, " "))
print("Cycle length:", cycleLength)
print("Start Index:", startIndex)</syntaxhighlight>
{{out}}
<pre>Sequence: 3 10 101 2 5 26 167 95 101 2 5 26 167 95
Cycle length: 6
Start Index: 2</pre>
=={{header|Mathematica}}/{{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">s = {3, 10, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5,
26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101,
2, 5, 26, 167, 95, 101, 2, 5};
{transient, repeat} = FindTransientRepeat[s, 2];
Print["Starting index: ", Length[transient] + 1]
Print["Cycles: ", Floor[(Length[s] - Length[transient])/Length[repeat]]]</syntaxhighlight>
{{out}}
<pre>Starting index: 3
Cycles: 6</pre>
=={{header|Modula-2}}==
{{trans|Kotlin}}
<syntaxhighlight lang="modula2">MODULE CycleDetection;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
TYPE IntToInt = PROCEDURE(INTEGER) : INTEGER;
TYPE Pair =
RECORD
a,b : INTEGER;
END;
PROCEDURE Brent(f : IntToInt; x0 : INTEGER) : Pair;
VAR power,lam,tortoise,hare,mu,i : INTEGER;
BEGIN
(* main phase: search successive powers of two *)
power := 1;
lam := 1;
tortoise := x0;
hare := f(x0); (* f(x0) is the element/node next to x0. *)
WHILE tortoise # hare DO
(* time to start a new power of two? *)
IF power = lam THEN
tortoise := hare;
power := power * 2;
lam := 0
END;
hare := f(hare);
INC(lam)
END;
(* Find the position of the first repetition of length 'lam' *)
mu := 0;
tortoise := x0;
hare := x0;
i := 0;
WHILE i < lam DO
hare := f(hare);
INC(i)
END;
(* The distance between the hare and tortoise is now 'lam'.
Next, the hare and tortoise move at same speed until they agree *)
WHILE tortoise # hare DO
tortoise := f(tortoise);
hare := f(hare);
INC(mu)
END;
RETURN Pair{lam,mu}
END Brent;
PROCEDURE Lambda(x : INTEGER) : INTEGER;
BEGIN
RETURN (x * x + 1) MOD 255
END Lambda;
VAR
buf : ARRAY[0..63] OF CHAR;
x0,x,i : INTEGER;
result : Pair;
BEGIN
x0 := 3;
x := x0;
WriteString("[3");
FOR i:=1 TO 40 DO
x := Lambda(x);
FormatString(", %i", buf, x);
WriteString(buf)
END;
WriteString("]");
WriteLn;
result := Brent(Lambda, x0);
FormatString("Cycle length = %i\nStart index = %i\nCycle = [", buf, result.a, result.b);
WriteString(buf);
x0 := 3;
x := x0;
FOR i:=1 TO result.b DO
x := Lambda(x)
END;
FOR i:=1 TO result.a DO
IF i > 1 THEN
WriteString(", ")
END;
FormatString("%i", buf, x);
WriteString(buf);
x := Lambda(x)
END;
WriteString("]");
WriteLn;
ReadChar
END CycleDetection.</syntaxhighlight>
=={{header|Nim}}==
Translation of Wikipedia Python program.
<syntaxhighlight lang="nim">import strutils, sugar
func brent(f: int -> int; x0: int): (int, int) =
# Main phase: search successive powers of two.
var
power, λ = 1
tortoise = x0
hare = f(x0)
while tortoise != hare:
if power == λ:
# Time to start a new power of two.
tortoise = hare
power *= 2
λ = 0
hare = f(hare)
inc λ
# Find the position of the first repetition of length λ.
tortoise = x0
hare = x0
for i in 0..<λ:
hare = f(hare)
# The distance between the hare and tortoise is now λ.
# Next, the hare and tortoise move at same speed until they agree.
var μ = 0
while tortoise != hare:
tortoise = f(tortoise)
hare = f(hare)
inc μ
result = (λ, μ)
when isMainModule:
func f(x: int): int = (x * x + 1) mod 255
let x0 = 3
let (λ, μ) = brent(f, x0)
echo "Cycle length: ", λ
echo "Cycle start index: ", μ
var cycle: seq[int]
var x = x0
for i in 0..<λ+μ:
if i >= μ: cycle.add x
x = f(x)
echo "Cycle: ", cycle.join(" ")</syntaxhighlight>
{{out}}
<pre>Cycle length: 6
Cycle start index: 2
Cycle: 101 2 5 26 167 95</pre>
=={{header|ooRexx}}==
<
x=3
list=x
Line 216 ⟶ 1,764:
End
Exit
f: Return (arg(1)**2+1)//255 </
{{out}}
<pre>
Line 224 ⟶ 1,772:
Cycle = 101 2 5 26 167 95
</pre>
=={{header|Perl}}==
{{trans|Raku}}
<syntaxhighlight lang="perl">use utf8;
sub cyclical_function { ($_[0] * $_[0] + 1) % 255 }
sub brent {
my($f, $x0) = @_;
my $power = 1;
my $λ = 1;
my $tortoise = $x0;
my $hare = &$f($x0);
while ($tortoise != $hare) {
if ($power == $λ) {
$tortoise = $hare;
$power *= 2;
$λ = 0;
}
$hare = &$f($hare);
$λ += 1;
}
my $μ = 0;
$tortoise = $hare = $x0;
$hare = &$f($hare) for 0..$λ-1;
while ($tortoise != $hare) {
$tortoise = &$f($tortoise);
$hare = &$f($hare);
$μ += 1;
}
return $λ, $μ;
}
my ( $l, $s ) = brent( \&cyclical_function, 3 );
sub show_range {
my($start,$stop) = @_;
my $result;
my $x = 3;
for my $n (0..$stop) {
$result .= "$x " if $n >= $start;
$x = cyclical_function($x);
}
$result;
}
print show_range(0,19) . "\n";
print "Cycle length $l\n";
print "Cycle start index $s\n";
print show_range($s,$s+$l-1) . "\n";</syntaxhighlight>
{{out}}
<pre>3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95
Cycle length 6
Cycle start index 2
101 2 5 26 167 95</pre>
=={{header|Phix}}==
Translation of the Wikipedia code, but using the more descriptive len and pos, instead of lambda and mu, and adding a limit.
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">function</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">255</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">brent</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">x0</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">pow2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">len</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">pos</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">tortoise</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x0</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">hare</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x0</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">tortoise</span><span style="color: #0000FF;">,</span><span style="color: #000000;">hare</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- (kept for output only)
-- main phase: search successive powers of two</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">tortoise</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">hare</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">pow2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">len</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">tortoise</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">hare</span>
<span style="color: #000000;">pow2</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">2</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">pow2</span><span style="color: #0000FF;">></span><span style="color: #000000;">16</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">len</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">hare</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">(</span><span style="color: #000000;">hare</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">hare</span>
<span style="color: #000000;">len</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000080;font-style:italic;">-- Find the position of the first repetition of length len</span>
<span style="color: #000000;">tortoise</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x0</span>
<span style="color: #000000;">hare</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">x0</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">len</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">hare</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">(</span><span style="color: #000000;">hare</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000080;font-style:italic;">-- The distance between the hare and tortoise is now len.
-- Next, the hare and tortoise move at same speed until they agree</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">tortoise</span><span style="color: #0000FF;"><></span><span style="color: #000000;">hare</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">tortoise</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tortoise</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">hare</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">(</span><span style="color: #000000;">hare</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">pos</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">len</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">len</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">pos</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">x0</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">3</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span><span style="color: #000000;">len</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">brent</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x0</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" Brent's algorithm\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #0000FF;">?</span><span style="color: #000000;">s</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" Cycle starts at position: %d (1-based)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" The length of the Cycle = %d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">len</span><span style="color: #0000FF;">})</span>
<span style="color: #0000FF;">?</span><span style="color: #000000;">s</span><span style="color: #0000FF;">[</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">..</span><span style="color: #000000;">pos</span><span style="color: #0000FF;">+</span><span style="color: #000000;">len</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
Brent's algorithm
{3,10,101,2,5,26,167,95,101,2,5,26,167,95}
Cycle starts at position: 3 (1-based)
The length of the Cycle = 6
{101,2,5,26,167,95}
</pre>
=={{header|PL/M}}==
<syntaxhighlight lang="plm">100H:
/* CP/M CALLS */
BDOS: PROCEDURE (FN, ARG); DECLARE FN BYTE, ARG ADDRESS; GO TO 5; END BDOS;
EXIT: PROCEDURE; CALL BDOS(0,0); END EXIT;
PRINT: PROCEDURE (S); DECLARE S ADDRESS; CALL BDOS(9,S); END PRINT;
/* THIS IS A HACK TO CALL A FUNCTION GIVEN ITS ADDRESS */
APPLY: PROCEDURE (FN, ARG) ADDRESS;
DECLARE (FN, ARG) ADDRESS;
INNER: PROCEDURE (X) ADDRESS; /* THIS SETS UP THE ARGUMENTS IN MEMORY */
DECLARE X ADDRESS;
GO TO FN; /* THEN WE JUMP TO THE ADDRESS GIVEN */
END INNER;
RETURN INNER(ARG);
END APPLY;
/* THIS IS ANOTHER HACK - THE SINGLE 0 (WHICH IS AN 8080 NOP) WILL BE
STORED AS A CONSTANT RIGHT BEFORE THE FUNCTION.
PL/M-80 DOES NOT ALLOW THE PROGRAMMER TO DIRECTLY GET THE ADDRESS
OF A FUNCTION, BUT THE ADDRESS OF THIS CONSTANT WILL BE RIGHT IN
FRONT OF IT AND CAN BE USED INSTEAD. */
DECLARE F$ADDR DATA (0);
/* F(X) FROM THE TASK */
F: PROCEDURE (X) ADDRESS;
DECLARE X ADDRESS;
RETURN (X*X + 1) MOD 255;
END F;
/* PRINT A NUMBER */
PRINT$NUMBER: PROCEDURE (N);
DECLARE S (7) BYTE INITIAL ('..... $');
DECLARE (N, P) ADDRESS, C BASED P BYTE;
P = .S(5);
DIGIT:
P = P - 1;
C = N MOD 10 + '0';
N = N / 10;
IF N > 0 THEN GO TO DIGIT;
CALL PRINT(P);
END PRINT$NUMBER;
/* PRINT F^M(X0) TO F^N(X0) */
PRINT$RANGE: PROCEDURE (FN, X0, M, N);
DECLARE (FN, X0, M, N, I) ADDRESS;
DO I=0 TO N-1;
IF I>=M THEN CALL PRINT$NUMBER(X0);
X0 = APPLY(FN,X0);
END;
CALL PRINT(.(13,10,'$'));
END PRINT$RANGE;
/* BRENT'S ALGORITHM */
BRENT: PROCEDURE (FN, X0, MU$A, LAM$A);
DECLARE (FN, X0, MU$A, LAM$A) ADDRESS;
DECLARE (MU BASED MU$A, LAM BASED LAM$A) ADDRESS;
DECLARE (TORT, HARE, POW, I) ADDRESS;
POW, LAM = 1;
TORT = X0;
HARE = APPLY(FN,X0);
DO WHILE TORT <> HARE;
IF POW = LAM THEN DO;
TORT = HARE;
POW = SHL(POW, 1);
LAM = 0;
END;
HARE = APPLY(FN,HARE);
LAM = LAM + 1;
END;
TORT, HARE = X0;
DO I=1 TO LAM;
HARE = APPLY(FN,HARE);
END;
MU = 0;
DO WHILE TORT <> HARE;
TORT = APPLY(FN,TORT);
HARE = APPLY(FN,HARE);
MU = MU + 1;
END;
END BRENT;
DECLARE (MU, LAM) ADDRESS;
/* PRINT THE FIRST 20 VALUES IN THE SEQUENCE */
CALL PRINT$RANGE(.F$ADDR, 3, 0, 20);
/* FIND THE CYCLE */
CALL BRENT(.F$ADDR, 3, .MU, .LAM);
CALL PRINT(.'LENGTH: $');
CALL PRINT$NUMBER(LAM);
CALL PRINT(.'START: $');
CALL PRINT$NUMBER(MU);
CALL PRINT(.(13,10,'$'));
/* PRINT THE CYCLE */
CALL PRINT$RANGE(.F$ADDR, 3, MU, MU+LAM);
CALL EXIT;
EOF</syntaxhighlight>
{{out}}
<pre>3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95
LENGTH: 6 START: 2
101 2 5 26 167 95 </pre>
=={{header|Python}}==
===Procedural===
Function from the Wikipedia article:
<syntaxhighlight lang="python">import itertools
def brent(f, x0):
# main phase: search successive powers of two
power = lam = 1
tortoise = x0
hare = f(x0) # f(x0) is the element/node next to x0.
while tortoise != hare:
if power == lam: # time to start a new power of two?
tortoise = hare
power *= 2
lam = 0
hare = f(hare)
lam += 1
# Find the position of the first repetition of length lam
mu = 0
tortoise = hare = x0
for i in range(lam):
# range(lam) produces a list with the values 0, 1, ... , lam-1
hare = f(hare)
# The distance between the hare and tortoise is now lam.
# Next, the hare and tortoise move at same speed until they agree
while tortoise != hare:
tortoise = f(tortoise)
hare = f(hare)
mu += 1
return lam, mu
def iterate(f, x0):
while True:
yield x0
x0 = f(x0)
if __name__ == '__main__':
f = lambda x: (x * x + 1) % 255
x0 = 3
lam, mu = brent(f, x0)
print("Cycle length: %d" % lam)
print("Cycle start index: %d" % mu)
print("Cycle: %s" % list(itertools.islice(iterate(f, x0), mu, mu+lam)))</syntaxhighlight>
{{out}}
<pre>
Cycle length: 6
Cycle start index: 2
Cycle: [101, 2, 5, 26, 167, 95]
</pre>
A modified version of the above where the first stage is restructured for clarity:
<syntaxhighlight lang="python">import itertools
def brent_length(f, x0):
# main phase: search successive powers of two
hare = x0
power = 1
while True:
tortoise = hare
for i in range(1, power+1):
hare = f(hare)
if tortoise == hare:
return i
power *= 2
def brent(f, x0):
lam = brent_length(f, x0)
# Find the position of the first repetition of length lam
mu = 0
hare = x0
for i in range(lam):
# range(lam) produces a list with the values 0, 1, ... , lam-1
hare = f(hare)
# The distance between the hare and tortoise is now lam.
# Next, the hare and tortoise move at same speed until they agree
tortoise = x0
while tortoise != hare:
tortoise = f(tortoise)
hare = f(hare)
mu += 1
return lam, mu
def iterate(f, x0):
while True:
yield x0
x0 = f(x0)
if __name__ == '__main__':
f = lambda x: (x * x + 1) % 255
x0 = 3
lam, mu = brent(f, x0)
print("Cycle length: %d" % lam)
print("Cycle start index: %d" % mu)
print("Cycle: %s" % list(itertools.islice(iterate(f, x0), mu, mu+lam)))</syntaxhighlight>
{{out}}
<pre>Cycle length: 6
Cycle start index: 2
Cycle: [101, 2, 5, 26, 167, 95]</pre>
===Functional===
In functional terms, the problem lends itself at first sight (see the Haskell version), to a reasonably compact recursive definition of a ''cycleLength'' function:
{{Trans|Haskell}}
{{Works with|Python|3.7}}
<syntaxhighlight lang="python">'''Cycle detection by recursion.'''
from itertools import (chain, cycle, islice)
from operator import (eq)
# cycleFound :: Eq a => [a] -> Maybe ([a], Int, Int)
def cycleFound(xs):
'''Just the first cycle found, with its length
and start index, or Nothing if no cycle seen.
'''
return bind(cycleLength(xs))(
lambda n: bind(
findIndex(uncurry(eq))(zip(xs, xs[n:]))
)(lambda iStart: Just(
(xs[iStart:iStart + n], n, iStart)
))
)
# cycleLength :: Eq a => [a] -> Maybe Int
def cycleLength(xs):
'''Just the length of the first cycle found,
or Nothing if no cycle seen.
'''
def go(pwr, lng, x, ys):
if ys:
y, *yt = ys
return Just(lng) if x == y else (
go(2 * pwr, 1, y, yt) if (
lng == pwr
) else go(pwr, 1 + lng, x, yt)
)
else:
return Nothing()
return go(1, 1, xs[0], xs[1:]) if xs else Nothing()
# TEST ----------------------------------------------------
# main :: IO ()
def main():
'''Reports of any cycle detection.'''
print(
fTable(
'First cycle detected, if any:\n'
)(fst)(maybe('No cycle found')(
showCycle
))(
compose(cycleFound)(snd)
)([
(
'cycle([1, 2, 3])',
take(1000)(cycle([1, 2, 3]))
), (
'[0..100] + cycle([1..8])',
take(1000)(
chain(
enumFromTo(0)(100),
cycle(enumFromTo(1)(8))
)
)
), (
'[1..500]',
enumFromTo(1)(500)
), (
'f(x) = (x*x + 1) modulo 255',
take(1000)(iterate(
lambda x: (1 + (x * x)) % 255
)(3))
)
])
)
# DISPLAY -------------------------------------------------
# showList :: [a] -> String
def showList(xs):
''''Compact stringification of a list,
(no spaces after commas).
'''
return ''.join(repr(xs).split())
# showCycle :: ([a], Int, Int) -> String
def showCycle(cli):
'''Stringification of cycleFound tuple.'''
c, lng, iStart = cli
return showList(c) + ' (from:' + str(iStart) + (
', length:' + str(lng) + ')'
)
# GENERIC -------------------------------------------------
# Just :: a -> Maybe a
def Just(x):
'''Constructor for an inhabited Maybe (option type) value.'''
return {'type': 'Maybe', 'Nothing': False, 'Just': x}
# Nothing :: Maybe a
def Nothing():
'''Constructor for an empty Maybe (option type) value.'''
return {'type': 'Maybe', 'Nothing': True}
# bind (>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b
def bind(m):
'''bindMay provides the mechanism for composing a
sequence of (a -> Maybe b) functions.
If m is Nothing, it is passed straight through.
If m is Just(x), the result is an application
of the (a -> Maybe b) function (mf) to x.'''
return lambda mf: (
m if m.get('Nothing') else mf(m.get('Just'))
)
# compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
def compose(g):
'''Right to left function composition.'''
return lambda f: lambda x: g(f(x))
# enumFromTo :: (Int, Int) -> [Int]
def enumFromTo(m):
'''Integer enumeration from m to n.'''
return lambda n: list(range(m, 1 + n))
# findIndex :: (a -> Bool) -> [a] -> Maybe Int
def findIndex(p):
'''Just the first index at which an
element in xs matches p,
or Nothing if no elements match.'''
def go(xs):
try:
return Just(next(
i for i, x in enumerate(xs) if p(x)
))
except StopIteration:
return Nothing()
return lambda xs: go(xs)
# fst :: (a, b) -> a
def fst(tpl):
'''First member of a pair.'''
return tpl[0]
# fTable :: String -> (a -> String) ->
# (b -> String) ->
# (a -> b) -> [a] -> String
def fTable(s):
'''Heading -> x display function -> fx display function ->
f -> value list -> tabular string.'''
def go(xShow, fxShow, f, xs):
w = max(map(compose(len)(xShow), xs))
return s + '\n' + '\n'.join([
xShow(x).rjust(w, ' ') + ' -> ' + fxShow(f(x)) for x in xs
])
return lambda xShow: lambda fxShow: (
lambda f: lambda xs: go(
xShow, fxShow, f, xs
)
)
# iterate :: (a -> a) -> a -> Gen [a]
def iterate(f):
'''An infinite list of repeated
applications of f to x.'''
def go(x):
v = x
while True:
yield v
v = f(v)
return lambda x: go(x)
# maybe :: b -> (a -> b) -> Maybe a -> b
def maybe(v):
'''Either the default value v, if m is Nothing,
or the application of f to x,
where m is Just(x).'''
return lambda f: lambda m: v if m.get('Nothing') else (
f(m.get('Just'))
)
# snd :: (a, b) -> b
def snd(tpl):
'''Second member of a pair.'''
return tpl[1]
# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.'''
return lambda xs: (
xs[0:n]
if isinstance(xs, list)
else list(islice(xs, n))
)
# uncurry :: (a -> b -> c) -> ((a, b) -> c)
def uncurry(f):
'''A function over a tuple
derived from a default or
curried function.'''
return lambda xy: f(xy[0], xy[1])
# concat :: [[a]] -> [a]
# concat :: [String] -> String
def concat(xxs):
'''The concatenation of all the elements in a list.'''
xs = list(chain.from_iterable(xxs))
unit = '' if isinstance(xs, str) else []
return unit if not xs else (
''.join(xs) if isinstance(xs[0], str) else xs
)
# MAIN ---
if __name__ == '__main__':
main()</syntaxhighlight>
{{Out}}
<pre>First cycle detected, if any:
cycle([1, 2, 3]) -> [1,2,3] (from:0, length:3)
[0..100] + cycle([1..8]) -> [1,2,3,4,5,6,7,8] (from:101, length:8)
[1..500] -> No cycle found
f(x) = (x*x + 1) modulo 255 -> [101,2,5,26,167,95] (from:2, length:6)</pre>
But recursion scales poorly in Python, and the version above, while good for lists of a few hundred elements, will need reworking for longer lists and better use of space.
If we start by refactoring the recursion into the form of a higher order (but still recursive) ''until p f x'' function, we can then reimplement the internals of ''until'' itself to avoid recursion, without losing the benefits of compositional structure:
Recursive ''until'':
<syntaxhighlight lang="python"># until :: (a -> Bool) -> (a -> a) -> a -> a
def until(p):
'''The result of repeatedly applying f until p holds.
The initial seed value is x.'''
def go(f, x):
return x if p(x) else go(f, f(x))
return lambda f: lambda x: go(f, x)</syntaxhighlight>
''cycleLength'' refactored in terms of ''until'':
<syntaxhighlight lang="python"># cycleLength :: Eq a => [a] -> Maybe Int
def cycleLength(xs):
'''Just the length of the first cycle found,
or Nothing if no cycle seen.'''
# f :: (Int, Int, Int, [Int]) -> (Int, Int, Int, [Int])
def f(tpl):
pwr, lng, x, ys = tpl
y, *yt = ys
return (2 * pwr, 1, y, yt) if (
lng == pwr
) else (pwr, 1 + lng, x, yt)
# p :: (Int, Int, Int, [Int]) -> Bool
def p(tpl):
_, _, x, ys = tpl
return (not ys) or x == ys[0]
if xs:
_, lng, x, ys = until(p)(f)(
(1, 1, xs[0], xs[1:])
)
return (
Just(lng) if (x == ys[0]) else Nothing()
) if ys else Nothing()
else:
return Nothing()</syntaxhighlight>
Iterative reimplementation of ''until'':
<syntaxhighlight lang="python"># until_ :: (a -> Bool) -> (a -> a) -> a -> a
def until_(p):
'''The result of repeatedly applying f until p holds.
The initial seed value is x.'''
def go(f, x):
v = x
while not p(v):
v = f(v)
return v
return lambda f: lambda x: go(f, x)</syntaxhighlight>
and now it all works again, with the structure conserved but recursion removed.
The Python no longer falls out of the tree at the sight of an ouroboros, and we can happily search for cycles in lists of several thousand items:
{{Works with|Python|3.7}}
<syntaxhighlight lang="python">'''Cycle detection without recursion.'''
from itertools import (chain, cycle, islice)
from operator import (eq)
# cycleFound :: Eq a => [a] -> Maybe ([a], Int, Int)
def cycleFound(xs):
'''Just the first cycle found, with its length
and start index, or Nothing if no cycle seen.
'''
return bind(cycleLength(xs))(
lambda n: bind(
findIndex(uncurry(eq))(zip(xs, xs[n:]))
)(lambda iStart: Just(
(xs[iStart:iStart + n], n, iStart)
))
)
# cycleLength :: Eq a => [a] -> Maybe Int
def cycleLength(xs):
'''Just the length of the first cycle found,
or Nothing if no cycle seen.'''
# f :: (Int, Int, Int, [Int]) -> (Int, Int, Int, [Int])
def f(tpl):
pwr, lng, x, ys = tpl
y, *yt = ys
return (2 * pwr, 1, y, yt) if (
lng == pwr
) else (pwr, 1 + lng, x, yt)
# p :: (Int, Int, Int, [Int]) -> Bool
def p(tpl):
_, _, x, ys = tpl
return (not ys) or x == ys[0]
if xs:
_, lng, x, ys = until(p)(f)(
(1, 1, xs[0], xs[1:])
)
return (
Just(lng) if (x == ys[0]) else Nothing()
) if ys else Nothing()
else:
return Nothing()
# TEST ----------------------------------------------------
# main :: IO ()
def main():
'''Reports of any cycle detection.'''
print(
fTable(
'First cycle detected, if any:\n'
)(fst)(maybe('No cycle found')(
showCycle
))(
compose(cycleFound)(snd)
)([
(
'cycle([1, 2, 3])',
take(10000)(cycle([1, 2, 3]))
), (
'[0..10000] + cycle([1..8])',
take(20000)(
chain(
enumFromTo(0)(10000),
cycle(enumFromTo(1)(8))
)
)
), (
'[1..10000]',
enumFromTo(1)(10000)
), (
'f(x) = (x*x + 1) modulo 255',
take(10000)(iterate(
lambda x: (1 + (x * x)) % 255
)(3))
)
])
)
# DISPLAY -------------------------------------------------
# showList :: [a] -> String
def showList(xs):
''''Compact stringification of a list,
(no spaces after commas).
'''
return ''.join(repr(xs).split())
# showCycle :: ([a], Int, Int) -> String
def showCycle(cli):
'''Stringification of cycleFound tuple.'''
c, lng, iStart = cli
return showList(c) + ' (from:' + str(iStart) + (
', length:' + str(lng) + ')'
)
# GENERIC -------------------------------------------------
# Just :: a -> Maybe a
def Just(x):
'''Constructor for an inhabited Maybe (option type) value.'''
return {'type': 'Maybe', 'Nothing': False, 'Just': x}
# Nothing :: Maybe a
def Nothing():
'''Constructor for an empty Maybe (option type) value.'''
return {'type': 'Maybe', 'Nothing': True}
# bind (>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b
def bind(m):
'''bindMay provides the mechanism for composing a
sequence of (a -> Maybe b) functions.
If m is Nothing, it is passed straight through.
If m is Just(x), the result is an application
of the (a -> Maybe b) function (mf) to x.'''
return lambda mf: (
m if m.get('Nothing') else mf(m.get('Just'))
)
# compose (<<<) :: (b -> c) -> (a -> b) -> a -> c
def compose(g):
'''Right to left function composition.'''
return lambda f: lambda x: g(f(x))
# concat :: [[a]] -> [a]
# concat :: [String] -> String
def concat(xxs):
'''The concatenation of all the elements in a list.'''
xs = list(chain.from_iterable(xxs))
unit = '' if isinstance(xs, str) else []
return unit if not xs else (
''.join(xs) if isinstance(xs[0], str) else xs
)
# enumFromTo :: (Int, Int) -> [Int]
def enumFromTo(m):
'''Integer enumeration from m to n.'''
return lambda n: list(range(m, 1 + n))
# findIndex :: (a -> Bool) -> [a] -> Maybe Int
def findIndex(p):
'''Just the first index at which an
element in xs matches p,
or Nothing if no elements match.'''
def go(xs):
try:
return Just(next(
i for i, x in enumerate(xs) if p(x)
))
except StopIteration:
return Nothing()
return lambda xs: go(xs)
# fst :: (a, b) -> a
def fst(tpl):
'''First member of a pair.'''
return tpl[0]
# fTable :: String -> (a -> String) ->
# (b -> String) ->
# (a -> b) -> [a] -> String
def fTable(s):
'''Heading -> x display function -> fx display function ->
f -> value list -> tabular string.'''
def go(xShow, fxShow, f, xs):
w = max(map(compose(len)(xShow), xs))
return s + '\n' + '\n'.join([
xShow(x).rjust(w, ' ') + ' -> ' + fxShow(f(x)) for x in xs
])
return lambda xShow: lambda fxShow: (
lambda f: lambda xs: go(
xShow, fxShow, f, xs
)
)
# iterate :: (a -> a) -> a -> Gen [a]
def iterate(f):
'''An infinite list of repeated
applications of f to x.'''
def go(x):
v = x
while True:
yield v
v = f(v)
return lambda x: go(x)
# maybe :: b -> (a -> b) -> Maybe a -> b
def maybe(v):
'''Either the default value v, if m is Nothing,
or the application of f to x,
where m is Just(x).'''
return lambda f: lambda m: v if m.get('Nothing') else (
f(m.get('Just'))
)
# snd :: (a, b) -> b
def snd(tpl):
'''Second member of a pair.'''
return tpl[1]
# take :: Int -> [a] -> [a]
# take :: Int -> String -> String
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.'''
return lambda xs: (
xs[0:n]
if isinstance(xs, list)
else list(islice(xs, n))
)
# uncurry :: (a -> b -> c) -> ((a, b) -> c)
def uncurry(f):
'''A function over a tuple
derived from a default or
curried function.'''
return lambda xy: f(xy[0], xy[1])
# until :: (a -> Bool) -> (a -> a) -> a -> a
def until(p):
'''The result of repeatedly applying f until p holds.
The initial seed value is x.'''
def go(f, x):
v = x
while not p(v):
v = f(v)
return v
return lambda f: lambda x: go(f, x)
# MAIN ---
if __name__ == '__main__':
main()</syntaxhighlight>
{{Out}}
<pre>First cycle detected, if any:
cycle([1, 2, 3]) -> [1,2,3] (from:0, length:3)
[0..10000] + cycle([1..8]) -> [1,2,3,4,5,6,7,8] (from:10001, length:8)
[1..10000] -> No cycle found
f(x) = (x*x + 1) modulo 255 -> [101,2,5,26,167,95] (from:2, length:6)</pre>
=={{header|Quackery}}==
<syntaxhighlight lang="quackery"> [ stack ] is fun ( --> s )
[ stack ] is pow ( --> s )
[ stack ] is len ( --> s )
[ fun put
1 pow put
1 len put
dup fun share do
[ 2dup != while
len share
pow share = if
[ nip dup
pow share
pow tally
0 len replace ]
fun share do
1 len tally
again ]
2drop
fun release
pow release
len take ] is cyclelen ( n x --> n )
[ 0 temp put
dip [ fun put dup ]
times [ fun share do ]
[ 2dup != while
fun share do
dip [ fun share do ]
1 temp tally
again ]
2drop
fun release
temp take ] is cyclepos ( n x n --> n )
[ 2dup cyclelen
dup dip cyclepos ] is brent ( n x --> n n )
[ 2dup
20 times
[ over echo sp
tuck do swap ]
cr cr
2drop
brent
say "cycle length is "
echo cr
say "cycle starts at "
echo ] is task ( n x --> )
3 ' [ 2 ** 1+ 255 mod ] task</syntaxhighlight>
{{out}}
<pre>3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95
cycle length is 6
cycle starts at 2
</pre>
=={{header|Racket}}==
I feel a bit bad about overloading λ, but it’s in the spirit of the algorithm.
<syntaxhighlight lang="racket">
#lang racket/base
;; returns (values lambda mu)
(define (brent f x0)
;; main phase: search successive powers of two
(define λ
(let main-phase ((power 1)
(λ 1)
(tortoise x0)
(hare (f x0))) ;; f(x0) is the element/node next to x0.
(cond [(= hare tortoise) λ]
;; time to start a new power of two?
[(= power λ) (main-phase (* power 2) 1 hare (f hare))]
[else (main-phase power (add1 λ) tortoise (f hare))])))
(values
λ
;; Find the position of the first repetition of length λ
(let race ((µ 0)
(tortoise x0)
;; The distance between the hare and tortoise is now λ.
(hare (for/fold ((hare x0)) ((_ (in-range λ))) (f hare))))
;; Next, the hare and tortoise move at same speed until they agree
(if (= tortoise hare) µ (race (add1 µ) (f tortoise) (f hare))))))
(module+ test
(require rackunit racket/generator)
(define (f x) (modulo (+ (* x x) 1) 255))
(define (make-generator f x0)
(generator () (let loop ((x x0)) (yield x) (loop (f x)))))
(define g (make-generator f 3))
(define l (for/list ((_ 20)) (g)))
(check-equal? l '(3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95))
(displayln l)
(let-values (([µ λ] (brent f 3)))
(printf "Cycle length = ~a~%Start Index = ~a~%" µ λ)))
</syntaxhighlight>
{{out}}
<pre>
(3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95)
Cycle length = 6
Start Index = 2
</pre>
=={{header|Raku}}==
(formerly Perl 6)
{{works with|Rakudo|2016-01}}
Pretty much a line for line translation of the Python code on the Wikipedia page.
<syntaxhighlight lang="raku"
my ( $l, $s ) = brent( &cyclical-function, 3 );
Line 253 ⟶ 2,815:
my $μ = 0;
$tortoise = $hare = $x0;
while ($tortoise != $hare) {
Line 263 ⟶ 2,823:
}
return $λ, $μ;
}</
{{out}}
<pre>3, 10, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, ...
Line 272 ⟶ 2,832:
=={{header|REXX}}==
===Brent's algorithm===
<
init= 3; $= init
do until length(
say ' original list=' $ ... /*display original number list*/
say '
say ' start index =' idx " ◄─── zero index" /* " " index " " */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Brent: procedure; parse arg x0 1 tort;
hare= f(x0)
do while
if pow==# then do; tort= hare
pow = pow + pow /*double the value of POW. */
hare= f(hare)
end /*
hare= x0
do
end
tort= x0
tort= f(tort)
hare= f(hare)
end /*mu*/
return # mu
/*──────────────────────────────────────────────────────────────────────────────────────*/
f: return ( arg(1) **2 + 1) // 255 /*this defines/executes the function F.*/</syntaxhighlight>
{{out|output|text= when using the defaults:}}
<pre>
original list= 3 10 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95 101 2 5 26 167 95 101 ...
numbers in list= 27
cycle length = 6
start index =
</pre>
===sequential search algorithm===
<
x= 3; $= x /*initial couple of variables*/
do until cycle\==0; x= f(x)
cycle= wordpos(x,
end /*until*/
say 'numbers
say '
say '
say 'cycle
/*──────────────────────────────────────────────────────────────────────────────────────*/
f: return ( arg(1) **2 + 1) // 255 /*this defines/executes the function F.*/</syntaxhighlight>
{{out|output|:}}
<pre>
original list= 3 10 101 2 5 26 167 95 101 ...
numbers in list= 9
cycle length = 6
start index = 2 ◄─── zero based
cycle sequence = 101 2 5 26 167 95
</pre>
===hash table algorithm===
This REXX version is a lot faster (than the sequential search algorithm) if the ''cycle length'' and/or ''start index'' is large.
<
!.= .; !.x= 1; x= 3; $= x /*assign initial value. */
do
if !.x\==. then leave /*A repeat? Then leave.*/
!.x=
end /*
say ' original list='
say 'numbers in list='
say ' cycle length ='
say ' start index =' !.x - 1
say 'cycle sequence =' subword(
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
f: return ( arg(1) **2 + 1) // 255 /*this defines/executes the function F.*/</
===robust hash table algorithm===
Line 357 ⟶ 2,926:
This REXX version allows the divisor for the '''F''' function to be specified, which can be chosen to stress
<br>test the hash table algorithm. A divisor which is <big> ''two raised to the 49<sup>th</sup> power'' </big> was chosen; it
<br>generates a cyclic sequence that contains over 1.5 million numbers.
<syntaxhighlight lang="rexx">/*REXX program detects a cycle in an iterated function [F] using a robust hashing. */
parse arg power . /*obtain optional args from C.L. */
numeric digits max(9, length(divisor) * 2 + 1) /*allow for the square plus one*/
say ' power =' power /*display the power to the term*/
say '
say
x=3;
do n=1+words($); x= f(x); $$=$$ x
if n//2000==0 then do; $=$ $$; $$=; end /*Is a 2000th N? Rejoin.*/
if !.x\==. then leave /*is this number a repeat? Leave*/
!.x= n
end /*n*/ /*N: is the size of $ list.*/
$= space($ $$) /*append residual numbers to $ */
if n<m then say ' original list=' $ ... /*maybe display the list to term.*/
say 'numbers in list=' n /*display number of numbers. */
say ' cycle length =' n - !.x /*display the cycle to the term. */
say ' start index =' !.x - 1 " ◄─── zero based" /*show the index.*/
if n<m then say 'cycle sequence =' subword($, !.x, n- !.x) /*maybe display the sequence*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
f: return ( arg(1) **2 + 1) // 255 /*this defines/executes the function F.*/</syntaxhighlight>
{{out|output|text= when the input (power of two) used is: <tt> 49 </tt>}}
<pre>
power = 49
divisor = {2**49}-1 = 562949953421311
original list= 3 10 101 2 5 26 167 95 101 ...
numbers in list= 9
cycle length = 6
start index = 2 ◄─── zero based
cycle sequence = 101 2 5 26 167 95
</pre>
Line 396 ⟶ 2,967:
There is more information in the "hash table"<br>
and f has no "side effect".
<
x=3; list=x; p.=0; p.x=1
Do q=2 By 1
Line 413 ⟶ 2,984:
Exit
/*-------------------------------------------------------------------*/
f: Return (arg(1)**2+1)// 255; /*define the function F*/</
=={{header|RPL}}==
Translation of the Brent algorithm given in Wikipedia.
{{works with|HP|48}}
≪ 1 1 0 → f x0 power lam mu
≪ x0 DUP f EVAL <span style="color:grey">@ Main phase: search successive powers of two</span>
'''WHILE''' DUP2 ≠ '''REPEAT'''
'''IF''' power lam == '''THEN''' <span style="color:grey">@ time to start a new power of two?</span>
SWAP DROP DUP
2 'power' STO*
0 'lam' STO
'''END'''
f EVAL
1 'lam' STO+
'''END'''
DROP2 x0 DUP <span style="color:grey">@ Find the position of the first repetition of length λ</span>
0 lam 1 - '''START'''
f EVAL '''NEXT''' <span style="color:grey">@ distance between the hare and tortoise is now λ</span>
'''WHILE''' DUP2 ≠ '''REPEAT''' <span style="color:grey">@ the hare and tortoise move at same speed until they agree</span>
f EVAL SWAP
f EVAL SWAP
1 'mu' STO+
'''END'''
DROP2 lam mu
≫ ≫ '<span style="color:blue">CYCLEB</span>' STO
≪ SQ 1 + 255 MOD ≫ 0 <span style="color:blue">CYCLEB</span>
{{out}}
<pre>
2: 6
1: 2
</pre>
=={{header|Ruby}}==
{{works with|ruby|2.0}}
<syntaxhighlight lang="ruby"># Author: Paul Anton Chernoch
# Purpose:
# Find the cycle length and start position of a numerical seried using Brent's cycle algorithm.
Line 455 ⟶ 3,057:
# Find mu, the zero-based index of the start of the cycle
tortoise, mu = x0, 0
while tortoise != hare
tortoise = yield(tortoise)
Line 471 ⟶ 3,071:
# A recurrence relation to use in testing
def f(x) (x * x + 1) % 255 end
# Display the first 41 numbers in the test series
puts (1..40).reduce([3]){|acc,_| acc << f(acc.last)}.join(",")
# Test the findCycle function
clength, cstart = findCycle(3) { |x| f(x) }
puts "Cycle length = #{clength}\nStart index = #{cstart}"</syntaxhighlight>
{{out}}
<pre>
3,10,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5
Cycle length = 6
Start index = 2
</pre>
=={{header|Scala}}==
=== Procedural ===
{{Out}}Best seen in running your browser either by [https://scalafiddle.io/sf/6O7WjnO/0 ScalaFiddle (ES aka JavaScript, non JVM)] or [https://scastie.scala-lang.org/kPCg0fxOQQCZPkOnmMR0Kg Scastie (remote JVM)].
<syntaxhighlight lang="scala">object CycleDetection extends App {
def brent(f: Int => Int, x0: Int): (Int, Int) = {
// main phase: search successive powers of two
// f(x0) is the element/node next to x0.
var (power, λ, μ, tortoise, hare) = (1, 1, 0, x0, f(x0))
while (tortoise != hare) {
if (power == λ) { // time to start a new power of two?
tortoise = hare
power *= 2
λ = 0
}
hare = f(hare)
λ += 1
}
// Find the position of the first repetition of length 'λ'
tortoise = x0
hare = x0
for (i <- 0 until λ) hare = f(hare)
// The distance between the hare and tortoise is now 'λ'.
// Next, the hare and tortoise move at same speed until they agree
while (tortoise != hare) {
tortoise = f(tortoise)
hare = f(hare)
μ += 1
}
(λ, μ)
}
def cycle = loop.slice(μ, μ + λ)
def f = (x: Int) => (x * x + 1) % 255
// Generator for the first terms of the sequence starting from 3
def loop: LazyList[Int] = 3 #:: loop.map(f(_))
val (λ, μ) = brent(f, 3)
println(s"Cycle length = $λ")
println(s"Start index = $μ")
println(s"Cycle = ${cycle.force}")
}</syntaxhighlight>
=== Functional ===
<syntaxhighlight lang="scala">
import scala.annotation.tailrec
object CycleDetection {
def brent(f: Int => Int, x0: Int): (Int, Int) = {
val lambda = findLambda(f, x0)
val mu = findMu(f, x0, lambda)
(lambda, mu)
}
def cycle(f: Int => Int, x0: Int): Seq[Int] = {
val (lambda, mu) = brent(f, x0)
(1 until mu + lambda)
.foldLeft(Seq(x0))((list, _) => f(list.head) +: list)
.reverse
.drop(mu)
}
def findLambda(f: Int => Int, x0: Int): Int = {
findLambdaRec(f, tortoise = x0, hare = f(x0), power = 1, lambda = 1)
}
def findMu(f: Int => Int, x0: Int, lambda: Int): Int = {
val hare = (0 until lambda).foldLeft(x0)((x, _) => f(x))
findMuRec(f, tortoise = x0, hare, mu = 0)
}
@tailrec
private def findLambdaRec(f: Int => Int, tortoise: Int, hare: Int, power: Int, lambda: Int): Int = {
if (tortoise == hare) {
lambda
} else {
val (newTortoise, newPower, newLambda) = if (power == lambda) {
(hare, power * 2, 0)
} else {
(tortoise, power, lambda)
}
findLambdaRec(f, newTortoise, f(hare), newPower, newLambda + 1)
}
}
@tailrec
private def findMuRec(f: Int => Int, tortoise: Int, hare: Int, mu: Int): Int = {
if (tortoise == hare) {
mu
} else {
findMuRec(f, f(tortoise), f(hare), mu + 1)
}
}
def main(args: Array[String]): Unit = {
val f = (x: Int) => (x * x + 1) % 255
val x0 = 3
val (lambda, mu) = brent(f, x0)
val list = cycle(f, x0)
println("Cycle length = " + lambda)
println("Start index = " + mu)
println("Cycle = " + list.mkString(","))
}
}
</syntaxhighlight>
{{out}}
<pre>
Cycle length = 6
Start index = 2
Cycle = 101,2,5,26,167,95
</pre>
=={{header|Sidef}}==
{{trans|Raku}}
<syntaxhighlight lang="ruby">func brent (f, x0) {
var power = 1
var λ = 1
var tortoise = x0
var hare = f(x0)
while (tortoise != hare) {
if (power == λ) {
tortoise = hare
power *= 2
λ = 0
}
hare = f(hare)
λ += 1
}
var μ = 0
tortoise = x0
hare = x0
{ hare = f(hare) } * λ
while (tortoise != hare) {
tortoise = f(tortoise)
hare = f(hare)
μ += 1
}
return (λ, μ)
}
func cyclical_function(x) { (x*x + 1) % 255 }
var (l, s) = brent(cyclical_function, 3)
var seq = gather {
var x = 3
{ take(x); x = cyclical_function(x) } * 20
}
say seq.join(', ')+', ...'
say "Cycle length #{l}.";
say "Cycle start index #{s}."
say [seq[s .. (s + l - 1)]]</syntaxhighlight>
{{out}}
<pre>3, 10, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, ...
Cycle length 6.
Cycle start index 2.
[101, 2, 5, 26, 167, 95]</pre>
=={{header|Visual Basic .NET}}==
{{trans|C#}}
<syntaxhighlight lang="vbnet">Module Module1
Function FindCycle(Of T As IEquatable(Of T))(x0 As T, yielder As Func(Of T, T)) As Tuple(Of Integer, Integer)
Dim power = 1
Dim lambda = 1
Dim tortoise As T
Dim hare As T
tortoise = x0
hare = yielder(x0)
' Find lambda, the cycle length
While Not tortoise.Equals(hare)
If power = lambda Then
tortoise = hare
power *= 2
lambda = 0
End If
hare = yielder(hare)
lambda += 1
End While
' Find mu, the zero-based index of the start of the cycle
Dim mu = 0
tortoise = x0
hare = x0
For times = 1 To lambda
hare = yielder(hare)
Next
While Not tortoise.Equals(hare)
tortoise = yielder(tortoise)
hare = yielder(hare)
mu += 1
End While
Return Tuple.Create(lambda, mu)
End Function
Sub Main()
' A recurrence relation to use in testing
Dim sequence = Function(_x As Integer) (_x * _x + 1) Mod 255
' Display the first 41 numbers in the test series
Dim x = 3
Console.Write(x)
For times = 0 To 39
x = sequence(x)
Console.Write(",{0}", x)
Next
Console.WriteLine()
' Test the FindCycle method
Dim cycle = FindCycle(3, sequence)
Console.WriteLine("Cycle length = {0}", cycle.Item1)
Console.WriteLine("Start index = {0}", cycle.Item2)
End Sub
End Module</syntaxhighlight>
{{out}}
<pre>3,10,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5,26,167,95,101,2,5
Cycle length = 6
Start index = 2</pre>
=={{header|Wren}}==
Working from the code in the Wikipedia article:
<syntaxhighlight lang="wren">var brent = Fn.new { |f, x0|
var lam = 1
var power = 1
var tortoise = x0
var hare = f.call(x0)
while (tortoise != hare) {
if (power == lam) {
tortoise = hare
power = power * 2
lam = 0
}
hare = f.call(hare)
lam = lam + 1
}
tortoise = hare = x0
for (i in 0...lam) hare = f.call(hare)
var mu = 0
while (tortoise != hare) {
tortoise = f.call(tortoise)
hare = f.call(hare)
mu = mu + 1
}
return [lam, mu]
}
var f = Fn.new { |x| (x*x + 1) % 255 }
var x0 = 3
var x = x0
var seq = List.filled(21, 0) // limit to first 21 terms say
for (i in 0..20) {
seq[i] = x
x = f.call(x)
}
var res = brent.call(f, x0)
var lam = res[0]
var mu = res[1]
System.print("Sequence = %(seq)")
System.print("Cycle length = %(lam)")
System.print("Start index = %(mu)")
System.print("Cycle = %(seq[mu...mu+lam])")</syntaxhighlight>
{{out}}
<pre>
Sequence = [3, 10, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101, 2, 5, 26, 167, 95, 101]
Cycle length = 6
Start index = 2
Cycle = [101, 2, 5, 26, 167, 95]
</pre>
=={{header|zkl}}==
Algorithm from the Wikipedia
<
# main phase: search successive powers of two
power:=lam:=1;
Line 515 ⟶ 3,402:
}
return(lam,mu);
}</
<
{{out}}
<pre>
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