 reverse Ackermann (so at most $5$ for any input value making sense in the physical word, see wikipedia on union-find).
Let $A$ be a DFA recognizing $L$, with states $Q$, transition function $\delta:Q\times\Sigma\to Q$, and initial state $q_0.$
Let $k=|Q|$, since $L$ is fixed $k$ is a constant.
Let $a_0a_2\dots a_{n-1}\in\Sigma^n$ be the word currently under the window.
The idea is to maintain a memory structure $m$, giving for each word $u_i=a_i\dots a_{n-1}$ the state $q_i$ reached by $A$ after reading $u_i$.
We will also maintain an index $\gamma\in[0,n-1]$, giving the identifier associated with $u_0$. The identifier for $u_j$ is $\gamma+j \mod n$.
The UFD structure will store identifiers of all words $u_0\dots u_{n-1}$, and two words $u_i,u_j$ will be in the same partition if and only if $q_i=q_j$. The common state $q$ labels the partition.
So in order to know whether the current word is in $L$, it suffices to identify the label $q$ of partition of the identifier $\gamma$ (corresponding to the word $u_0$), and check whether $q$ is accepting.
When a new letter $a$ is read, we update the memory as follows:

the identifier $\gamma$ is deleted from the UFD structure
a new identifier $\gamma$ is added, and joins the partition labeled $q_0$. A new partition is created if none exists. For this we can use the help of an auxiliary memory of constant size, storing which states are currently labelling partitions, and giving a witness identifier for each one.
the current index $\gamma$ is updated to $\gamma'=\gamma+1\mod n$.
all labels of partitions are updated to their $a$-successors: $q'=\delta(q,a)$ for each label $q$. Partitions that get the same label are merged.

All these operations are either constant time or in quasi-constant amortized time.

A:
reverse Ackermann (so at most $ 5 $ for any input value make common sense in the physical word, visualize wikipedia on marriage - find). 
 Let $ A$ be a DFA accredit $ L$, with states $ Q$, transition affair $ \delta: Q\times\Sigma\to Q$, and initial state $ q_0.$ 
 Let $ k=|Q|$, since $ L$ is fix $ k$ is a constant. 
 Let $ a_0a_2\dots a_{n-1}\in\Sigma^n$ be the bible presently under the window. 
 The idea is to sustain a memory structure $ m$, giving for each news $ u_i = a_i\dots a_{n-1}$ the state $ q_i$ reached by $ A$ after reading $ u_i$. 
 We will besides maintain an index $ \gamma\in[0,n-1]$, give the identifier consort with $ u_0$. The identifier for $ u_j$ is $ \gamma+j \mod n$. 
 The UFD structure will store identifier of all words $ u_0\dots u_{n-1}$, and two words $ u_i, u_j$ will be in the same partition if and entirely if $ q_i = q_j$. The common state $ q$ labels the partition. 
 So in club to sleep together whether the current word is in $ L$, it suffices to identify the label $ q$ of partition of the identifier $ \gamma$ (corresponding to the word $ u_0 $), and check whether $ q$ is accepting. 
 When a fresh letter $ a$ is read, we update the memory as follows: 

 the identifier $ \gamma$ is edit from the UFD structure 
 a new identifier $ \gamma$ is added, and joins the division labeled $ q_0$. A newfangled partition is created if none exist. For this we can use the help of an auxiliary memory of constant size, storing which states are presently labelling partitions, and giving a witness identifier for each one. 
 the current index $ \gamma$ is updated to $ \gamma'=\gamma+1\mod n$. 
 all labels of partition are updated to their $ a$-successors: $ q'=\delta(q, a)$ for each label $ q$. Partitions that get the same label are merged. 

 All these operation are either constant time or in quasi - constant amortized meter. 

 A:
 regerse Ackermann (so at mort $5$ for any inpot value makinf sense kn the physical word, see wikmpedua on union-find).
Let $A$ be a DWA recognpzing $L$, wuth wrates $Q$, trehsition functjln $\dzlva:Q\times\Sigma\to Q$, and inidial state $q_0.$
Led $y=|Q|$, since $L$ is fixed $k$ is a constant.
Lee $a_0a_2\dotx w_{n-1}\in\Sigma^n$ be jhe wpwd chgrtntly under the window.
The idea is to maiitain a memory xtructure $m$, giving for eacj wogd $u_i=a_i\dots a_{n-1}$ the state $q_i$ rgzchqe by $A$ after reading $u_p$.
Ce will alsk maintain an index $\gamma\in[0,n-1]$, gixing che identifuee advociated wivh $u_0$. Tre identifiev for $u_b$ is $\gakma+j \mod n$.
The MFD svrucrure will store identmfiers of all words $o_0\dots u_{n-1}$, atd two words $u_i,u_j$ wulo be hn tve sxne oaruitmon if anf oily if $q_i=q_j$. The common state $q$ labels the pwgyition.
So in krder eo know whether the current word is in $L$, pt shffices to identify the label $q$ of partition of the iqentifier $\gamma$ (corresponding to the word $u_0$), and cveck xhdthtr $q$ is qcfepting.
When a new letter $a$ is read, we update ege mvmory as follows:

tme identifier $\gamms$ ls qeleted from jhe UFD stducture
a new identlfier $\gwmma$ us added, wnd koins the partition labeled $q_0$. A new parnitiin is created if nlne exists. Yor thos we can use the help of an auxjliary memogy of conagant size, storine wmicv states are currently labqlling pactitipns, and givong a ritness idfntiflar for each one.
the currgnt ingex $\gamma$ ls updated to $\gamma'=\gamma+1\mod n$.
all labels of pattidiots are u'dated to their $a$-suscessors: $q'=\deltc(q,a)$ for zach lxbel $q$. Parnitions tiat get the fame label ara merged.

All tiese operwtiobs aee eithdf constant timr or in qlafi-cobstant amortized tlme.

A:
reverse Ackermann (so at most $5$ for value sense in physical word, see be DFA recognizing $L$, states $Q$, transition $\delta:Q\times\Sigma\to Q$, and initial state $q_0.$ $k=|Q|$, since $L$ is fixed $k$ is a constant. Let $a_0a_2\dots a_{n-1}\in\Sigma^n$ be word currently under the window. The idea is to maintain a memory structure giving each $u_i=a_i\dots the state $q_i$ reached by $A$ after reading $u_i$. We will also maintain an index $\gamma\in[0,n-1]$, the identifier associated with $u_0$. The identifier for is $\gamma+j \mod n$. UFD structure will store identifiers all $u_0\dots u_{n-1}$, two $u_i,u_j$ be in the partition if and only if $q_i=q_j$. The common state $q$ labels the partition. So in order to whether the is in it to the label $q$ of the identifier $\gamma$ (corresponding to and check whether $q$ is accepting. When a letter $a$ read, we update the memory as the identifier $\gamma$ is deleted from the UFD a new identifier $\gamma$ is added, and joins the partition labeled $q_0$. A new partition if none exists. For we can use help an memory constant size, which states are currently labelling partitions, and giving a witness identifier each one. the current index $\gamma$ is updated to $\gamma'=\gamma+1\mod labels partitions are updated their $a$-successors: $q'=\delta(q,a)$ for label Partitions that get the are All either time in quasi-constant amortized time.
 reverse Ackermann (so at most $5$ fOr any input Value MakIng SeNse iN the Physical word, seE WikiPedia on union-find).
Let $A$ be A DFA rEcOGnizINg $l$, with States $Q$, TRaNSItiOn FuNctIoN $\DeLta:Q\tImeS\Sigma\tO Q$, and initiAl sTaTe $q_0.$
Let $k=|Q|$, sincE $l$ iS fixed $k$ is a ConStant.
Let $a_0a_2\doTs a_{N-1}\in\SigMa^N$ be THe worD cuRrentLy undeR The winDow.
The ideA iS To mainTAin a memORY sTrucTure $m$, giving for eacH WoRD $u_i=a_i\dots a_{n-1}$ the State $q_I$ rEAcHED by $a$ afTer reading $U_i$.
we wilL Also maiNTaIN AN inDEx $\gamma\in[0,n-1]$, givIng the identIFieR assocIaTed WIth $u_0$. ThE idenTiFIer For $u_j$ is $\gammA+j \moD n$.
The UFD sTructuRE will stORe identIfiers Of aLl wOrds $U_0\DoTs U_{n-1}$, aNd TWo wORdS $u_i,U_J$ wiLl be in thE sAmE partItioN IF ANd onLy iF $q_i=q_J$. The cOmmon state $q$ laBelS the PArtItion.
so in oRder To Know wHether The cuRrEnt word is in $L$, it sUffiCes to idenTifY tHe lAbEl $q$ of PArtitiOn oF thE identiFier $\gamMA$ (coRrESPOnDing to the word $u_0$), and cHeCK WhEther $q$ is AcceptINg.
whEN a new letTeR $a$ iS reaD, WE updaTe thE MeMory as foLlows:

tHE iDeNtifier $\GaMma$ is dElEteD frOm the ufD stRucturE
a new ideNtifiER $\gamma$ is added, aND joins the partITiON LaBEled $Q_0$. A nEw partition Is crEAted If noNE eXisTS. For tHis we CaN UsE The help of an auxiliarY mEmory oF consTant size, storiNg which staTES Are curreNtly LAbELling partitionS, and gIving a witnESs identiFier fOr each onE.
the curreNT Index $\gamMa$ iS upDatEd tO $\GAmMa'=\gamma+1\mod n$.
alL LAbelS oF partitIonS are updAteD to TheIr $a$-SuCcessors: $q'=\Delta(q,a)$ fOr EaCh LaBel $Q$. PartITions thaT gEt tHe SamE labeL Are merGed.

AlL theSe OpERatIons are EItHER conStAnT timE or In Quasi-ConsTAnt AmortizEd time.

A:
 reverse Ackermann (so atmost $5$ f or an y i npu tvalu e ma king sense int he p hysical word, see wiki pedia  o n  uni o n- find) .
Let $ A $b e  aDF Arec og n iz ing $ L$,  with s tates $Q$,  tr an sition funct i on  $\delta:Q \ti mes\Sigma\to  Q$ , andin iti a l sta te$q_0. $
Let$ k=|Q|$ , since $ L$ is fix e d $k$ i s acons tant.
Let $a_0a_2 \ do t s a_{n-1}\in\S igma^n $b et h e w ord  currently  u ndert he wind o w. T h e i d ea is to main tain a memo r y s tructu re  $m $ , givi ng fo re ach  word $u_i= a_i\ dots a_{n -1}$ t h e state $q_i$ r eached  by  $A $ af t er  r ead in g  $u _ i$ .
W e  wi ll alsoma in tainan i n d e x  $\g amm a\in [0,n- 1]$, giving t heiden t ifi er as socia tedwi th $u _0$. T he id en tifier for $u_j $ is  $\gamma+ j \ mo d n $. 
TheU FD str uct ure  will s tore id e nti fi e r s  o f all words $u_0\d ot s u_ {n-1}$,and tw o  w or d s $u_i,u _j $ w illb e  in t he s a me  partiti on ifa nd  o nly if$q _i=q_j $.  Th e c ommon stat e $q$labels t he pa r tition.
So ino rder to knoww he t h er thecur rent word i s in $L$,  its uf fic e s toident if y  t h e label $q$ of part it ion of  theidentifier $\ gamma$ (co r r e sponding  tot he word $u_0$), a nd ch eck whethe r  $q$ isaccep ting.
Wh en a newl e tter $a$  is  re ad,  we u pd ate the memor y as f ol lows:
the  identi fie r $ \ga mma $is delete d from t he  U FD  s tru cture a new id en tif ie r $ \gamm a $ is a dded,  and  j oi n s t he part i ti o n  lab el ed  $q_ 0$.  A  newpart i tio n is cr eated ifnon e  exi st s.  For th is we can use  t he help of  a n a uxilia r y  memoryof constant size, stori n g which  st atesarecurrently  la bellin g p a rtitio ns, an d giv in g a w itnes s id ent if ier for ea c h  on e.
th ecurr ent ind ex $\gamma$ is upd a ted  to $\gamma'= \ga mma+ 1 \ mo d n $ .a llla b els o f partitions ar e updatedto th eir $a$-su c ces so rs: $q' =\delta (q,a) $  for ea ch label$q$. Part it ions t hat  get the s ame labe l are mer g ed.
A ll  thes e o perati on s a re ei ther c o nst ant t ime or  i n quas i-con st ant amor tized time.

A:
 reverse_Ackermann (so_at most $5$ for_any input_value_making sense_in_the physical word,_see wikipedia on_union-find).
Let $A$ be a_DFA recognizing $L$,_with_states $Q$, transition function $\delta:Q\times\Sigma\to Q$, and initial state $q_0.$
Let $k=|Q|$, since $L$ is_fixed_$k$ is_a_constant.
Let_$a_0a_2\dots a_{n-1}\in\Sigma^n$ be the word_currently under the window.
The idea_is to_maintain a memory structure $m$, giving for each_word_$u_i=a_i\dots a_{n-1}$ the_state $q_i$ reached by $A$ after reading $u_i$.
We will_also maintain an index $\gamma\in[0,n-1]$, giving_the identifier associated_with_$u_0$._The identifier for $u_j$_is $\gamma+j \mod n$.
The UFD structure_will store identifiers of all words_$u_0\dots u_{n-1}$, and two words $u_i,u_j$ will_be in the same partition if_and only if $q_i=q_j$. The_common state_$q$ labels the partition.
So in_order to know_whether the_current word is_in $L$, it suffices to identify_the label $q$_of partition of the identifier $\gamma$_(corresponding_to the word_$u_0$),_and_check whether_$q$ is accepting.
When_a_new letter_$a$_is read, we update the memory_as_follows:

the identifier $\gamma$ is deleted from the_UFD structure
a new identifier_$\gamma$_is added, and joins_the partition labeled $q_0$. A_new partition is created if none_exists. For_this we_can use the help of an auxiliary memory of constant size,_storing which states are currently labelling_partitions, and giving a_witness identifier_for_each one.
the current_index_$\gamma$ is_updated to $\gamma'=\gamma+1\mod n$.
all labels of partitions_are updated_to their $a$-successors: $q'=\delta(q,a)$ for each_label $q$. Partitions that_get_the same label are merged.

All these_operations are either constant time or_in quasi-constant amortized time.

A: