View source for Factorize string into Lyndon words

{{task}}

Given a finite alphabet, we can lexicographically order all strings in the alphabet. If two strings have different lengths, then pad the shorter string on the right with the smallest letter. For example, we have 01 > 001, but 01 = 010. As we see, this order is a total preorder, but not a total order, since it identifies different strings.

A [[Lyndon word]] is a non-empty string that is ''strictly'' lower in lexicographic order than all its circular rotations. In particular, if a string is equal to a circular rotation, then it is not a Lyndon word. For example, since 0101 = 0101 (rotation by 2), it is not a Lyndon word. 

The first Lyndon words on the binary alphabet {0, 1} are:

: 0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00101, 00111, 01011, 01111, ...

Some basic properties:

* The only Lyndon word that ends with 0 is 0. 
* '''Proof.''' If s0 is a Lyndon word, and s is not empty, then s0 < 0s. If s contains 1 somewhere, then s0 > 0s. Therefore s has only 0. But then s0 = 0s, contradiction.
* The lexicographic order ''is'' a total order on the Lyndon words.
* '''Proof.''' For, the only way for two different strings s, s' to have the same lexicographic ordering is for one of them to pad to the other. We can assume that s00...0 = s'. If that is so, then s00...0 is a Lyndon word that ends with 0, so it is just 0, and so s is a Lyndon word that is also empty, contradiction.

The Chen–Fox–Lyndon theorem states that any string is uniquely factorizable into a sequence of Lyndon words non-decreasing in lexicographic order. Duval's algorithm computes this in O(length of input) time and and O(1) space.<ref>Duval, Jean-Pierre (1983), "Factorizing words over an ordered alphabet", ''Journal of Algorithms'', '''4''' (4): 363–381, doi:10.1016/0196-6774(83)90017-2</ref> See <ref name=":0">https://cp-algorithms.com/string/lyndon_factorization.html</ref> for a description of Duval's algorithm.

=={{header|ALGOL 68}}==
{{Trans|Python|with multiline output as with a number of other samples}}
<syntaxhighlight lang="algol68">
BEGIN # Factorize a string into Lyndon words - translated of the Python sample #

    PROC chen fox lyndon factorization = ( STRING s )[]STRING:
         BEGIN
            INT n  = UPB s;
            INT i := LWB s;
            FLEX[ 1 : 100 ]STRING factorization;
            INT f max := LWB factorization;
            WHILE i <= n DO
                INT j := i + 1, k := i;
                WHILE IF j <= n THEN s[ k ] <= s[ j ] ELSE FALSE FI DO
                    IF s[ k ] < s[ j ] THEN
                        k  := i
                    ELSE
                        k +:= 1
                    FI;
                    j +:= 1
                OD;
                WHILE i <= k DO
                    f max +:= 1;
                    IF f max > UPB factorization THEN
                        [ 1 : UPB factorization + 20 ]STRING new factorization;
                        new factorization[ 1 : UPB factorization ] := factorization;
                        factorization := new factorization
                    FI;
                    factorization[ f max ] := s[ i : i + j - k - 1 ];
                    i +:= j - k
                OD
            OD;
            STRING check := "";
            FOR i TO f max DO check +:= factorization[ i ] OD;
            IF check /= s THEN print( ( "Incorrect factorization", newline ) ); stop FI;
            factorization[ 1 : f max ]
         END # chen fox lyndon factorization # ;

    # Example use with Thue-Morse sequence                                     #
    STRING m := "0";
    TO 7 DO
         STRING m0 = m;
         FOR i FROM LWB m TO UPB m DO
             IF m[ i ]  = "0" THEN m[ i ] := "1" ELIF m[ i ]  = "1" THEN m[ i ] := "0" FI
         OD;
         m0 +=: m
    OD;

    []STRING m factors = chen fox lyndon factorization( m );
    FOR i FROM LWB m factors TO UPB m factors DO
        print( ( m factors[ i ], newline ) )
    OD

END
</syntaxhighlight>
{{out}}
<pre>
011
01
0011
00101101
0010110011010011
00101100110100101101001100101101
001011001101001011010011001011001101001100101101
001011001101
001
</pre>

== {{header|C++}} ==
Copied from <ref name=":0" />, under Creative Commons Attribution Share Alike 4.0 International License and converted to a runnable C++ program.<syntaxhighlight lang="c++">#include <iostream>
#include <vector>
#include <string>
#include <algorithm>

using namespace std;

vector<string> duval(string const& s) {
    int n = s.size();
    int i = 0;
    vector<string> factorization;
    while (i < n) {
        int j = i + 1, k = i;
        while (j < n && s[k] <= s[j]) {
            if (s[k] < s[j])
                k = i;
            else
                k++;
            j++;
        }
        while (i <= k) {
            factorization.push_back(s.substr(i, j - k));
            i += j - k;
        }
    }
    return factorization;
}

int main() {
    // Thue-Morse example
    string m = "0";
    for (int i = 0; i < 7; ++i) {
        string m0 = m;
        replace(m.begin(), m.end(), '0', 'a');
        replace(m.begin(), m.end(), '1', '0');
        replace(m.begin(), m.end(), 'a', '1');
        m = m0 + m;
    }
    for (string s : duval(m)) cout << s << endl;
    return 0;
}</syntaxhighlight>

{{out}}
<pre>
011
01
0011
00101101
0010110011010011
00101100110100101101001100101101
001011001101001011010011001011001101001100101101
001011001101
001
</pre>

=={{header|EasyLang}}==
{{trans|Julia}}
<syntaxhighlight>
proc lyndonfact s$ . .
   n = len s$
   s$[] = strchars s$
   i = 1
   while i <= n
      j = i + 1
      k = i
      while j <= n and strcode s$[k] <= strcode s$[j]
         if strcode s$[k] < strcode s$[j]
            k = i
         else
            k += 1
         .
         j += 1
      .
      while i <= k
         print substr s$ i (j - k)
         i += j - k
      .
   .
.
m$ = "0"
for i to 7
   for c$ in strchars m$
      if c$ = "0"
         m$ &= "1"
      else
         m$ &= "0"
      .
   .
.
lyndonfact m$
</syntaxhighlight>

=={{header|Java}}==
<syntaxhighlight lang="java">
import java.util.ArrayList;
import java.util.List;

public class FactorizeStringIntoLyndonWords {
 
    public static void main(String[] args) {
    	// Create a 128 character Thue-Morse word
    	String thueMorse = "0";
        for ( int i = 0; i < 7; i++ ) {
            String thueMorseCopy = thueMorse;
            thueMorse = thueMorse.replace("0", "a");
            thueMorse = thueMorse.replace("1", "0");
            thueMorse = thueMorse.replace("a", "1");
            thueMorse = thueMorseCopy + thueMorse;
        }
        
        System.out.println("The Thue-Morse word to be factorised:");
        System.out.println(thueMorse);
        
        System.out.println();
        System.out.println("The factors are:");
        for ( String factor : duval(thueMorse) ) {
        	System.out.println(factor);
        }
    }
    
    // Duval's algorithm
    private static List<String> duval(String text) {
        List<String> factorisation = new ArrayList<String>();
        int i = 0;
        
        while ( i < text.length() ) {
            int j = i + 1;
            int k = i;
            
            while ( j < text.length() && text.charAt(k) <= text.charAt(j) ) {
                if ( text.charAt(k) < text.charAt(j) ) {
                    k = i;
                } else {
                    k += 1;
                }
                
                j += 1;
            }
            
            while ( i <= k ) {
                factorisation.addLast(text.substring(i, i + j - k));
                i += j - k;
            }
        }
        
        return factorisation;
    }
    
}
</syntaxhighlight>
<pre>
The Thue-Morse word to be factorised:
01101001100101101001011001101001100101100110100101101001100101101001011001101001011010011001011001101001100101101001011001101001

The factors are:
011
01
0011
00101101
0010110011010011
00101100110100101101001100101101
001011001101001011010011001011001101001100101101
001011001101
</pre>

=={{header|jq}}==
{{Works with|jq}}

'''Works with gojq, the Go implementation of jq'''

'''Adapted from [[#Wren|Wren]]'''
<syntaxhighlight lang="jq">
def duval:
  . as $s
  | def a($i): $s[$i:$i+1];
  length as $n
  | { i: 0, factorization: []}
  | until (.i >= $n;
       .j = .i + 1
       | .k = .i
       | until (.j >= $n or a(.k) > a(.j);
            if a(.k) < a(.j) then .k = .i else .k += 1 end
            | .j += 1 )
       | until (.i > .k;
            .factorization += [$s[.i: .i+.j-.k]]
            | .i += .j - .k ) )
  | .factorization ;

# Thue-Morse example
def m:
  reduce range(0; 7) as $i ( "0";
    . as $m0
    | gsub("0"; "a")
    | gsub("1"; "0")
    | gsub("a"; "1")
    | $m0 + .) ;

m | duval | join("\n")
</syntaxhighlight>
{{output}}
As for [[#Wren|Wren]].


=={{header|Julia}}==
{{trans|Phix}}{{trans|Python}}
<syntaxhighlight lang="Julia">function chenfoxlyndonfactorization(s)
    n = length(s)
    i = 1
    factorization = String[]
    while i <= n
        j = i + 1
        k = i
        while j <= n && s[k] <= s[j]
            if s[k] < s[j]
                k = i
            else
                k += 1
            end
            j += 1
        end
        while i <= k
            push!(factorization, s[i:i+j-k-1])
            i += j - k
        end
    end
    @assert s == prod(factorization)
    return factorization
end

let m = "0"
    for i in 1:7
        m *= replace(m, '0' => '1', '1' => '0')
    end
    println(chenfoxlyndonfactorization(m))
end
</syntaxhighlight>{{out}}
<pre>
["011", "01", "0011", "00101101", "0010110011010011", "00101100110100101101001100101101", "001011001101001011010011001011001101001100101101", "001011001101", "001"]
</pre>

=={{header|MATLAB}}==
{{trans|Python}}
<syntaxhighlight lang="MATLAB">
clear all;close all;clc;
m = '0';
for i = 1:7
    m0 = m;
    m = strrep(m, '0', 'a');
    m = strrep(m, '1', '0');
    m = strrep(m, 'a', '1');
    m = strcat(m0, m);
end

factorization = chenFoxLyndonFactorization(m);
for index=1:length(factorization)
    disp(factorization(index));
end


function factorization = chenFoxLyndonFactorization(s)
    n = length(s);
    i = 1;
    factorization = {};
    while i <= n
        j = i + 1;
        k = i;
        while j <= n && s(k) <= s(j)
            if s(k) < s(j)
                k = i;
            else
                k = k + 1;
            end
            j = j + 1;
        end
        while i <= k
            factorization{end+1} = s(i:i + j - k - 1);
            i = i + j - k;
        end
    end
    assert(strcmp(join(factorization, ''), s));
end
</syntaxhighlight>
{{out}}
<pre>
    {'011'}

    {'01'}

    {'0011'}

    {'00101101'}

    {'0010110011010011'}

    {'00101100110100101101001100101101'}

    {'001011001101001011010011001011001101001100101101'}

    {'001011001101'}

    {'001'}
</pre>

=={{header|Phix}}==
{{trans|Python}}
<syntaxhighlight lang="Phix">
with javascript_semantics
function chen_fox_lyndon_factorization(string s)
    integer n = length(s), i = 1
    sequence factorization = {}
    while i <= n do
        integer j = i + 1, k = i
        while j <= n and s[k] <= s[j] do
            if s[k] < s[j] then
                k = i
            else
                k += 1
            end if
            j += 1
        end while
        while i <= k do
            factorization &= {s[i..i+j-k-1]}
            i += j - k
        end while
    end while
    assert(join(factorization,"") == s)
    return factorization
end function
-- Example use with Thue-Morse sequence
string m = "0"
for i=1 to 7 do
    m &= sq_sub('0'+'1',m)
end for
?chen_fox_lyndon_factorization(m)
-- alt:
printf(1,"\n%s\n",join(chen_fox_lyndon_factorization(m),"\n"))
</syntaxhighlight>
{{out}}
<pre>
{"011","01","0011","00101101","0010110011010011","00101100110100101101001100101101","001011001101001011010011001011001101001100101101","001011001101","001"}
 
011
01
0011
00101101
0010110011010011
00101100110100101101001100101101
001011001101001011010011001011001101001100101101
001011001101
001
</pre>

== {{header|Python}} ==
Duval's algorithm:<syntaxhighlight lang="python3">
def chen_fox_lyndon_factorization(s):
    n = len(s)
    i = 0
    factorization = []
    while i < n:
        j, k = i + 1, i
        while j < n and s[k] <= s[j]:
            if s[k] < s[j]:
                k = i
            else:
                k += 1
            j += 1
        while i <= k:
            factorization.append(s[i:i + j - k])
            i += j - k
    assert ''.join(factorization) == s
    return factorization
</syntaxhighlight>'''Example use with [[Thue-Morse|Thue-Morse sequence]]'''<syntaxhighlight lang="python3">
m='0'
for i in range(0,7):
     m0=m
     m=m.replace('0','a')
     m=m.replace('1','0')
     m=m.replace('a','1')
     m=m0+m

print(chen_fox_lyndon_factorization(m))
</syntaxhighlight>'''Output:'''<syntaxhighlight lang="text">
['011', '01', '0011', '00101101', '0010110011010011', '00101100110100101101001100101101', '001011001101001011010011001011001101001100101101', '001011001101', '001']
</syntaxhighlight>

=={{header|Raku}}==
{{trans|Julia}}
<syntaxhighlight lang="raku" line># 20240213 Raku programming solution

sub chenfoxlyndonfactorization(Str $s) {
   my ($n, $i, @factorization) = $s.chars, 1;
   while $i <= $n {
      my ($j, $k) = $i X+ (1, 0);
      while $j <= $n && substr($s, $k-1, 1) <= substr($s, $j-1, 1) {
         substr($s, $k-1, 1) < substr($s, $j++ -1, 1) ?? ( $k = $i ) !! $k++;
      }
      while $i <= $k {
         @factorization.push: substr($s, $i-1, $j-$k);
         $i += $j - $k
      }
   }
   die unless $s eq [~] @factorization;
   return @factorization
}

my $m = "0";
for ^7 { $m ~= $m.trans('0' => '1', '1' => '0') }
say chenfoxlyndonfactorization($m);</syntaxhighlight>
{{out}} Same as Julia example.
You may [https://ato.pxeger.com/run?1=fVJNToNAFE5ccoqvDSlMoA2sNGJt7-DGxGhCKYRpy6AzQ7Q27UXcdKGX8jS-4SeWxshiwrz3_b3J-_iU8bo6Hr8qnY2vvi_mqlogyVORlW-brViWIosTXUr-HmteCvdOS9iKYWcBKLZwbeHD5j7mPRzDlGCTJI-l8hFGBv2a801KWNxQTzQKnciKRNY1iePegxv6CFjUIlriqiWORqCQSkvXVoY2JnDITPO0vGrLnQ19f7L6JM9D25jN4BKsicQwGNDF87pI-360Zqb1qVn_PSbPlcqve17cGFFMmjv6pZGUNzWzjknv1Kw-ljxFJTapUvS6SF_wcHg8c6q1ZKorKc461t6y6LHtgmYaBsPIykqJp0vsTOlApsVEy1go1wkcTG_hhI5vjvo_cBhFUPH2v-WwCxY1i9TuU7dXPw Attempt This Online!]

=={{header|RPL}}==
Based on Duval's algorithm.
<code>THUEM</code> is defined at [[Thue-Morse#RPL|Thue-Morse]].
{{works with|HP|48G}}
 « DUP SIZE { } → s n res
   « 1
     '''WHILE''' DUP n ≤ '''REPEAT'''
        DUP 1 + OVER
        '''WHILE''' s OVER DUP SUB s 4 PICK DUP SUB DUP2 ≤ 5 PICK n ≤ AND '''REPEAT'''
           '''IF''' < '''THEN''' DROP OVER '''ELSE''' 1 + '''END'''
           SWAP 1 + SWAP
        '''END''' DROP2
        SWAP OVER - SWAP ROT
        '''WHILE''' DUP2 ≥ '''REPEAT'''
           'res' s 3 PICK DUP 7 PICK + 1 - SUB STO+
           3 PICK +
        '''END''' ROT ROT DROP2
      '''END''' 
      DROP res
 » » '<span style="color:blue">LYNDON</span>' STO     <span style="color:grey">''@ ( "string" → { "f1".."fn" } )''</span>
 
 « 2 7 ^ <span style="color:blue">THUEM</span> « →STR + » STREAM
   <span style="color:blue">LYNDON</span>
 » '<span style="color:blue">TASK</span>' STO 
{{out}}
<pre>
1: { "011" "01" "0011" "00101101" "0010110011010011" "00101100110100101101001100101101" "001011001101001011010011001011001101001100101101" "001011001101" "001" }
</pre>
'''Using local variables'''

The version below favors local variables to the stack, which makes it a little bit slower - but easier to read.
 « DUP SIZE 1 → s n ii
   « { }
     '''WHILE''' ii n ≤ '''REPEAT'''
        ii DUP 1 + → k j
        « '''WHILE''' s k DUP SUB s j DUP SUB DUP2 ≤ ii n ≤ AND '''REPEAT'''
             < ii k 1 + IFTE 'k' STO
             'j' 1 STO+
           »
          'j' k STO-
          '''WHILE''' ii k ≤ '''REPEAT'''
             s ii DUP j + 1 - SUB +
             'ii' j STO+
          '''END''' 
        »
      '''END''' 
 » » '<span style="color:blue">LYNDON</span>' STO     <span style="color:grey">''@ ( "string" → { "f1".."fn" } )''</span>

=={{header|Rust}}==
{{trans|Python}}
<syntaxhighlight lang="Rust">
fn chen_fox_lyndon_factorization(s: &str) -> Vec<String> {
    let n = s.len();
    let mut i = 0;
    let mut factorization = Vec::new();
    while i < n {
        let (mut j, mut k) = (i + 1, i);
        while j < n && s.as_bytes()[k] <= s.as_bytes()[j] {
            if s.as_bytes()[k] < s.as_bytes()[j] {
                k = i;
            } else {
                k += 1;
            }
            j += 1;
        }
        while i <= k {
            factorization.push(s[i..i + j - k].to_string());
            i += j - k;
        }
    }
    factorization
}

fn main() {
    let mut m = String::from("0");
    for _ in 0..7 {
        let m0 = m.clone();
        m = m.replace("0", "a");
        m = m.replace("1", "0");
        m = m.replace("a", "1");
        m = m0 + &m;
    }

    let factorization = chen_fox_lyndon_factorization(&m);
    println!("{:?}", factorization);
}
</syntaxhighlight>
{{out}}
<pre>
["011", "01", "0011", "00101101", "0010110011010011", "00101100110100101101001100101101", "001011001101001011010011001011001101001100101101", "001011001101", "001"]
</pre>

=={{header|Scala}}==
{{trans|Python}}
<syntaxhighlight lang="Scala">
object ChenFoxLyndonFactorization extends App {
  def chenFoxLyndonFactorization(s: String): List[String] = {
    val n = s.length
    var i = 0
    var factorization = List[String]()
    while (i < n) {
      var j = i + 1
      var k = i
      while (j < n && s.charAt(k) <= s.charAt(j)) {
        if (s.charAt(k) < s.charAt(j)) {
          k = i
        } else {
          k += 1
        }
        j += 1
      }
      while (i <= k) {
        factorization = factorization :+ s.substring(i, i + j - k)
        i += j - k
      }
    }
    assert(s == factorization.mkString)
    factorization
  }

  var m = "0"
  for (i <- 0 until 7) {
    val m0 = m
    m = m.replace('0', 'a').replace('1', '0').replace('a', '1')
    m = m0 + m
  }

  println(chenFoxLyndonFactorization(m))
}
</syntaxhighlight>
{{out}}
<pre>
List(011, 01, 0011, 00101101, 0010110011010011, 00101100110100101101001100101101, 001011001101001011010011001011001101001100101101, 001011001101, 001)

</pre>
=={{header|SETL}}==
{{trans|C++}}
<syntaxhighlight lang="setl">program lyndon_factorization;
    tm := "01101001100101101001011001101001100101100110"
        + "10010110100110010110100101100110100101101001"
        + "1001011001101001100101101001011001101001";

    loop for part in duval(tm) do
        print(part);
    end loop;

    proc duval(s);
        i := 1;
        fact := [];
        loop while i <= #s do
            j := i + 1;
            k := i;
            loop while j <= #s and s(k) <= s(j) do
                k := if s(k) < s(j) then i else k+1 end;
                j +:= 1;
            end loop;
            loop while i <= k do
                fact with:= s(i..i+j-k-1);
                i +:= j-k;
            end loop;
        end loop;
        return fact;
    end proc;
end program;</syntaxhighlight>
{{out}}
<pre>011
01
0011
00101101
0010110011010011
00101100110100101101001100101101
001011001101001011010011001011001101001100101101
001011001101
001</pre>

=={{header|Wren}}==
{{trans|Python}}
{{libheader|Wren-str}}
<syntaxhighlight lang="wren">import "./str" for Str

var duval = Fn.new { |s|
    var n = s.count
    var i = 0
    var factorization = []
    while (i < n) {
        var j = i + 1
        var k = i
        while (j < n && Str.le(s[k], s[j])) {
            if (Str.lt(s[k], s[j])) k = i else k = k + 1
            j = j + 1
        }
        while (i <= k) {
            factorization.add(s[i...i+j-k])
            i = i + j - k
        }
    }
    return factorization
}

// Thue-Morse example
var m = "0"
for (i in 0..6) {
    var m0 = m
    m = m.replace("0", "a")
    m = m.replace("1", "0")
    m = m.replace("a", "1")
    m = m0 + m
}
System.print(duval.call(m).join("\n"))></syntaxhighlight>

{{out}}
<pre>
011
01
0011
00101101
0010110011010011
00101100110100101101001100101101
001011001101001011010011001011001101001100101101
001011001101
001
</pre>