Factorize string into Lyndon words: Difference between revisions
Factorize string into Lyndon words (view source)
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{{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.
Line 14 ⟶ 18:
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();
Line 37 ⟶ 114:
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}}==
Line 133 ⟶ 421:
end for
?chen_fox_lyndon_factorization(m)
-- alt:
printf(1,"\n%s\n",join(chen_fox_lyndon_factorization(m),"\n"))
</syntaxhighlight>
{{out}}
<
{"011","01","0011","00101101","0010110011010011","00101100110100101101001100101101","001011001101001011010011001011001101001100101101","001011001101","001"}
011
01
0011
00101101
0010110011010011
00101100110100101101001100101101
001011001101001011010011001011001101001100101101
001011001101
001
</pre>
==
Duval's algorithm:<syntaxhighlight lang="python3">
def chen_fox_lyndon_factorization(s):
Line 172 ⟶ 472:
</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}}==
Line 216 ⟶ 589:
["011", "01", "0011", "00101101", "0010110011010011", "00101100110100101101001100101101", "001011001101001011010011001011001101001100101101", "001011001101", "001"]
</pre>
=={{header|Scala}}==
Line 261 ⟶ 632:
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>
| |||