DENSITY OF CRITICAL F ACTORIZATIONS
TERO HARJU AND DIRK NOWOTKA
Abstract.We investigate the density of critical factorizations of infinte -怨恨的意思
pep小学英语四年级上册教案
quences of words.The density of critical factorizations of a word is the ratio
between the number of positions that permit a critical factorization,and the
number of all positions of a word.
We give a short proof of the Critical Factorization Theorem and show that the maximal number of noncritical positions of a word between two critical
ones is less than the period of that word.Therefore,we consider only words
of index one,that is words where the shortest period is larger than one half of
kayaktheir total length,in this paper.
On one hand,we consider words with the lowest possible number of critical points and show,as an example,that every Fibonacci word longer thanfive
has exactly one critical factorization and every palindrome has at least two
critical factorizations.
On the other hand,quences of words with a high density of critical points are considered.We show how to construct an infinite quence of words in four
letters where every point in every word is critical.We construct an infinite
quence of words in three letters with densities of critical points approaching
one,using square-free words,and an infinite quence of words in two letters
with densities of critical points approaching one half,using Thue–Mor words.
It is shown that the bounds are optimal.
Introduction
The Critical Factorization Theorem(CFT)[3,7]relates local periods with the global period offinite words.Let w=uv with|u|=p and z be the shortest suffix of w1u and prefix of vw2for suitable w1and w2,then z is the shortest repetition word of w at position p.The CFT states that in everyfinite word w there is a position p where the shortest repetition word z is as long as the global period d of w, moreover,p<d.The position p is called critical.Actually,we have at least one critical position in every d−1concutive positions in w.Consider the following example:
w=ab.aa.b
that has two critical positions2and4which are marked by dots.The period d of w equals3and w is of index1,since2d>|w|.The shortest repetition word in both critical positions is aab and baa,respectively.Note,that the shortest repetition words in the positions1and3are ba and a,respectively.The ratio of the number of critical positions and the number of all positions is called the density of critical positions.The density of w in our example is one half.
grippingDate:September12,2002.
教育培训机构排名1991Mathematics Subject Classification.68R15.
Key words binatorics on words,repetitions,critical factorization theorem, density of critical factorizations,Fibonacci words,Thue–Mor words.
1
2TERO HARJU AND DIRK NOWOTKA
The CFT is often claimed to be one of the most important results about words. However,it does not em to be well understood due to its little number of applica-tions and known implications.We investigate the frequenzy of occurences of critical factorizations in words in this paper to get a better understanding of critical points in general.We are concerned with words of a very low density,that is one or two critical factorizations in the whole word,and of an as high as possible density of critical factorizations.Prominent class of words are ud in our studies,namely, Fibonacci words,palindromes,and Thue–Mor words.
After we havefixed the basic notations in Section1,we give a technically im-proved version of a proof[5]of the Critical Factorization Theorem[3,7]and give a statement about the maximal distance between two critical points in a word.In Section2we show that Fibonacci words,which can be defined by palindromes[6], of length greater thanfive have exactly one critical position in contrast to the fact t
hat palindromes themlves have at least two critical positions.This result also implies immediately the two well-known facts that the period of a Fibonacci word is a Fibonacci number and that the Fibonacci word is not ultimately periodic,both proven differently in the literature.Section3contains the constructions of infinite quences of words in four letters with density one for every word,infinite quences of ternary words which has a limit of their densities at one,using square-free words [12,1,2],and infinite quences of binary words which has a limit of their densities at one half,using Thue–Mor words[11,10,1,2].We also show that the limits are optimal.
1.Preliminaries
In this ction wefix the notations for this paper.We refer to[8,4]for more basic and general definitions.
Let A be afinite nonempty alphabet and A∗be the monoid of allfinite words in A;the empty word is denoted byε.Let Aωdenote the t of all infinite words in A that have a beginning.An infinite word w∈Aωis called ultimately periodic if there exist two words u,v∈A∗such that w=uvω.Let w∈A∗in the following.The length of w is denoted by|w|and its i th letter is denoted by w(i).By definition |ε|=0.If w=w1uw2then u is called a factor of w.If w=uv then u and v are called prefix of w,denoted by u≤w,an
d suffix of w,denoted by v w, respectively,and let u=wv−1and v=u−1w.Note,thatεand w are both prefixes and suffixes of w.A word w is called bordered if there exists a word v=εsuch that w=vuv.A prefix u of a word w such that0<|u|<|w|which is also a suffix of w is called a border of w.A word w is called primitive if w=v k implies that k≤1.
An integer d,with1≤d≤|w|,is called a period of w if w(i)=w(i+d),for all1≤i≤|w|−d.The smallest period of w is denoted by∂(w)and it is also called the minimal period or the global period of w.We define the index ind(w)of a word w by
ind(w)= |w|∂(w) .
Note,that the index is often defined by ind(w)=|w|/∂(w)in the literature,how-ever,we u the integer part here,only.Let an integer p with1≤p<|w|be called position or point in w.Intuitively,a position p denotes the place between w(p) and w(p+1)in w.A word u=εis called a repetition word at position p if w=xy
DENSITY OF CRITICAL F ACTORIZATIONS3 with|x|=p and there exist x and y such that u x x and u≤yy .For a point p in w,let
英语口语学习软件
∂(w,p)=min |u| u is a repetition word at p
denote the local period at point p in w.Note,the repetition word of length∂(w,p) at point p is unbordered and∂(w,p)≤∂(w).A factorization w=uv,with u,v=εand|u|=p,is called critical if∂(w,p)=∂(w),and,if this holds,then p is called critical point,otherwi it is called noncritical point.Letη(w)denote the number of critical points in a word w.We shall reprent critical points of words by dots. For instance,the critical points of w=abaaba are2and4,and we show this by writing w=ab.aa.ba.In this example,∂(w)=3.
Let w=w(n)···w(2)w(1)denote the rever of w=w(1)w(2)···w(n).We call a word w a palindrome if w= w.
Let be an ordering of A={a1,a2,...,a n},say a1 a2 ··· a n.Then induces a lexicographic order on A∗such that
u v⇐⇒u≤v or u=xau and v=xbu with a b
where a,b∈A.A suffix v(prefix u)of w is called if v v (and u u)for any suffix v (prefix u )of w.We will identify orders on alphabets and their respective induced lexicographic orders throughout this article.Let −1 denote the inver order,say a n −1··· −1a2 −1a1,of .Letµ (w)and µ (w)denote the maximal suffixes of and −1,respectively,and let ν (w)andν (w)denote the maxi
女人英文名字mum prefixes of and −1,respectively. If the context is clear,we may writeµ ,µ ,ν ,andν forµ (w),µ (w),ν (w), andν (w),respectively.We only consider alphabets of size larger than one in the following.
The critical factorization theorem(CFT)was discovered by C´e sari and Vin-cent[3]and developed into its current form by Duval[7].
Theorem1(Critical Factorization Theorem).Every word w,with|w|≥2,has at least one critical factorization w=uv,with u,v=εand|u|<∂(w),i.e.,∂(w,|u|)=∂(w).
This theorem is a direct consquence from the following proposition which de-scribes one critical point in any word and will be technically more uful in the following.The proof of Proposition2is a technically improved version of the proof of the CFT by Crochemore and Perrin in[5].Note,thatµ (w)=µ (w)for any word w since they start with a different letter.
Proposition2.Let w be a word of length n≥2,and letβbe the shorter of the two suffixesµ (w)andµ (w).Then|wβ−1|is a critical point.
Proof.Assumeβ=µ (w)by symmetry.Letα=µ (w),so,α=u β.Let z be an unbordered repetition word at|wβ−1|.We show that|z|is a period of w,which will prove the claim.the day you went away伴奏
If w is a factor of z2,then obviously|z|is a period of w.If w=w1βw2for some w2=ε,thenβ −1βw2contradicts the choice ofβ.If wβ−1=yz,then, by the above,z≤β,sayβ=zβ ;but then z2β =zβ −1β=zβ implies that β=zβ −1β ;a contradiction.Conquently,β=zw and w=z1zw for a suffix z1of the unbordered word z.Therefore u is a suffix of z,and hence,u w is a suffix ofα.Conquently,u w α=u β,and so w β,which together with w −1β
4TERO HARJU AND DIRK NOWOTKA
implies that w ≤β.Thereforeβ=zw =w z ,and thusβ=z k z2for some z2≤z, which shows that|z|is a period of w.
The CFT follows since|wβ−1|<∂(w).For a different proof of the CFT by Duval,Mignosi,and Restivo,e Chapter8in[9].The next theorem justifies why we are only interested in words of index one in our investigation of the density of critical points.
Theorem3.Each t of∂(w)−1concutive points in w,where|w|≥2,has a critical point.
Proof.If w=u i u1,where u1≤u and∂(w)=|u|,then the maximal suffi any orders of A are longer than|u i−1u1|.Hence w has a critical point at point p, where p<∂(w).
Let p be any critical point of w=uv,where|u|=p,and let z be the smallest repetition word at position p.So,|z|=∂(w).
We need to show that if|v|≥∂(w),then there is critical point at p+k for 1≤k<∂(w).We have z≤v and∂(v)=∂(w).For,if∂(v)<∂(w),then z is bordered;a contradiction.Now,v has a critical point k such that we have k<∂(v)=∂(w).Clearly,this point p+k is critical also for w since the smallest repetition word at point p+k is a conjugate of z.Now,(p+k)−p=k<∂(w).
Maybe an even stronger motivation for considering only words of index one,is that in w k,with k≥3,the critical points of thefirst factor w are inherited by the next k−2factors w.That is,if w k=w1.w2w k−1,where|w1|is a critical point, then also|ww1|is a critical point of w k.
环球职业网校2.Words with Exactly One Critical Factorization
Every word longer than one letter has at least one critical factorization.We investigate words with only one critical factorization in this ction.Trivially,words of length two have no more than one critical point.We do not consider such cas but arbitrary long words.However,the following lemma limits our investigation to words in two letters.
Lemma4.A word w with only one critical factorization is binary,that is,it is over a two-letter alphabet.
Proof.Assume a word w contains the letters a,b,and c and has exactly one critical factorization.Let a b c.By symmetry,we can assume that|µ |<|µ |.Then p=|wµ−1 |is a critical point of w by Proposition2.Let a c b.Now,either |wµ−1 |or|wµ−1 |is a critical point p of w,again by Propposition2.But,p=p sinceµ begins with c andµ andµ begin with a and b,respectively.So,w has at least two critical points;a contradiction.
By Lemma4,we will only consider words in a and b in the rest of this ction. Let a b.Note,thatµ =µ andν =ν for any word sinceµ andµ start andν andν end with different letters.Proposition2straightforwardly leads to the following two facts.
Lemma5.If a word w has exactly one critical point,then either
w=ν µ andν ≤ν andµ µ
DENSITY OF CRITICAL F ACTORIZATIONS5 or
w=ν µ andν ≤ν andµ µ .
The inver of Lemma5does not hold in general.Consider w=aa.bb.abab which has two critical points,but we do have
ν =aa≤aabb=ν andµ =bbabab aabbabab=µ
and w=ν µ .
Proposition6.Every palindrome has at least two critical factorizations.
Proof.Let w be a palindrome.Assume w has exactly one critical point.By sym-metry,we can also assume thatµ µ .By the definition of maximal prefix and suffix and since w is a palindrome we have
µ (w)= ν ( w)= ν (w)andµ (w)= ν ( w)= ν (w)
where ν (w)and ν (w)denote the reversal ofν (w)andν (w),respectively.Now, ν ν ,and hence,ν ≤ν ,which contradicts Lemma5sinceν =ν in any ca.
Let us now consider the critical points of Fibonacci words.Fibonacci numbers are defined by
f0=1,f1=1,f k+2=f k+1+f k.
Fibonacci words are defined by
F1=a,F2=ab,F k+2=F k+1F k. Obviously,|F i|=f i.Let F=lim n→∞F n be the Fibonacci word.Obrve th
at F i≤F n,if1≤i≤n.It is also clear that all Fibonacci words are primitive.The following lemma will be ud to estimate the number of critical points in Fibonacci words.
Lemma7.We have that f n−2<∂(F n)≤f n−1for all n>2.
Proof.The cas for F3and F4are easily checked.Assume n> 4.Clearly,∂(F n)≤f n−1in any ca.If∂(F n)<f n−2,then F n−2is not primitive since
F n−2F n−2≤F n−2F n−3F n−2=F n,
a contradiction.If∂(F n)=f n−2,then F n≤F3n−2,and F n−2 F n implies that F n−1or F n−2is not primitive;a contradiction. Remark8.Fibonacci words have a clo connection to palindromes as the following properties show.Firstly,F n=αn d n where n≥3andαn is a palindrome and d n=a
b if n is even and d n=ba if n is odd.This result has been credited to Berstel in[6].Secondly,F n=βnγn,where n≥5andβn andγn are palindromes of length f n−1−2and f n−2+2,respectively,by de Luca[6].Moreover,de Luca shows that the two properties define the t of Fibonacci words.
Given Remark8and Proposition6,every palindrome has at least two critical factorizations,Theorem10is rather surprising.
6TERO HARJU AND DIRK NOWOTKA
Example9(Fibonacci words).We have
F2=a.b,F3=a.b.a,F4=ab.aa.b.
By the following Theorem,however,every Fibonacci word F n,with n>4has exactly one critical point,and that critical point is at position f n−1−1. Theorem10.A Fibonacci word F n,with n>4,has exactly one critical point p. Moreover,p is at position f n−1−1.
Proof.Let n≥7,and let p be a critical point of F n.Then p>f n−2,becau otherwi
F n−2F n−2≤F n−2F n−3F n−2=F n
implies that∂(w,p)≤f n−2contradicting Lemma7.Consider the factorization
F n=F n−2F n−3F n−2=F n−2F n−4F n−5F n−2.
Then p>f n−2+f n−4,since otherwi
F n−3F n−4F n−4F n−4=F n−2F n−4F n−4<F n−2F n−4F n−5F n−2=F n, implies|F n−3F n−4|<p≤|F n
−3F n−4F n−4|which gives∂(w,p)≤f n−4,a contra-diction.六级几分过
By induction we obtain
F n−2F n−4···F n−2i+1F n−2i F n−2i F n−2i
=F n−2F n−4···F n−2i+2F n−2i F n−2i
<F n−2F n−4···F n−2i+2F n−2i F n−2i−1F n−2
=F n
where1≤i≤ n2 −2,and
p> n
2
−2 i=1f n−2i.
So,we have
F n=F n−2F n−4···F3F2F n−2or F n=F n−2F n−4···F4F3F n−2
where p>f n−1−2and p>f n−1−3,respectively,and f n−1>p by Lemma7and Theorem3.So,p=f n−1−1or p=f n−1−2,that is,a critical point has to exist in the suffix
F2F n−2 F n or F3F n−2 F n
where the former ca gives the result.The latter ca leaves the possibilities a.b.aF n−2 F n.But since b F4,we have bab.aF n−2 F n and only the marked position is critical which proves the claim.
The following well known facts follow immediately from Theorem10. Corollary11.A Fibonacci word F n has the period f n−1,and the Fibonacci word F is not ultimately periodic.
The Fibonacci words are certainly not the only words with exactly one critical factorization.
