Learning Balls of Strings with Correction Queries
19
0
0
Texte intégral
(2)
(3)
(4) !#"$!%&' ( ) * +! ( ? ,.-0/214365879;:=<>/21? @BA.CEDFCHG+IJALK A(GMG N0IJCED;NHK OFA RP Q CESUTVDXWYASZN\[4TU]E^FA.GMN P;_ A.N`DFa Q OFGMTZb>cMCEdFOFA _ N`DFCYWYA(c P `N D;WXeFGMf.WFf(GMTUKhg!N`DicjTUDFT k2l mnlpo(q>rtsvuwq> x.yiz\x.{v|\o }>sil ~o }>r o.;{i LyHm}>rMm =k2x(`qto;Jq> L;J{iljq>mn} =~zYljq>Vo(.; ¢¡ (£Hqn¡.qt¤¦o.¥¨rM§Ex `©.. .i¤L¤¢Lª.«(¬¢h¤Bo ¥nqn©.qtª.o.® Lx.{=o`;Ezio.{ Ro(¯Yx.qto(}>x.q>l±°wyi¯YlMqn}w²!yEq>l {³HJ{HLlMq>mn}>´4µLlpo.{v|¶xL{i{HlM} ·¸q>Á yi© lÂ` ¹EÃy ÄnÅ.ª(q>£Hx.¡.ºUl ¢mn .mn®¦l yHÄÆq+¤¢»! .l ¢{H Lx(¤E¼Ç}w©³R¥®¢o.ª(yE£`qto.®`m Çpi£i¢ÇL½.ÈL.«.L¬L£iÇ(Eio.ÉL{¢Ê(®H}n¾É ¿¦ L}>®HÇ.l {i (£L{H£HlL ¦HÀH¥Æqt¤ o.{Hr l 1. 2. 2. 2. 1. 2. 0Ë ÌÍ Î Ï Ð=Ñ.ÎLÒÓ yHq>{i 4}>silJ·LEÔ m Ó o.{io¸Õ2{i .yH{¶¹El l xLzYl ¹o({¶o(rM}>l+lpo(q>{H{i zio(qtoL¹E L~\³¯=o.mnl ¹ xL{ }>sHlhyimnlx(º&o.{×Öwqto.r lLRo.¯Hl}>xvo({imØ!lMq¸¹HÙFlMq>l {¢}4mnx.qn} x(ºÚ¢yiljq>lMm `x.º;Ø&sirjs}>sHlw¯Yl m}ÛE{HxpØ&{o(q>lw}>sHlw~lM~¯YlMq>mnsHzho.{=¹lpÚ¢yH.o.l {Hr l Ú¢yiljq>lMm ¢°JxpØlMlMqp¢zEqto.rM}>rpo(Yl `V¹El {irMl&}>lM{=¹HmB}>x4mnsixpØ}>sio(}'ºF}>sHl2ºUx.q>~lMq!o(q>l x(ºU}>l {Üop.o.o.¯ilL³}>sim4m¸yHmny=o. {ix.}4}>sHlhrpo.mnlx(º}>silo(}n}>lMqpRx\ .lM}4q>xLyi{i¹ }>äåsHmzEq>xL¯Hå lM~\Ø!l\zHq>x.zYxLmnl¶}>x yimnlvo {ilMØÝÛ`{=¹Üx.º+Ú`yHlMq>l m 'rpo.lp¹ßÞ>à.átá>ânÞ ã ê à(sHæl {8çMèioëâjá mâj}nq>é>`{HØ& simhrtsmnyiدHlJ~m}>}nyi}>¹Elp¹ }>{xë}>}>sHsHl2l\ºVqtÖwo.~qto(lMr Ø!lLx.q>Ûhl }>x(siºRlM (qhqto.mn~sil¶~.o(o.}>rp¹io(o Y}>l {HmhºZljq>}l {iº+r l.} ¯Yl x.{i Lm+}>x0}>sil±}to(q> Llj}4Vo({i .y=o. .lL;x.q4mnsHlzHq>x.zYxLmnl m±or x.ä qnq>lMrM}>xL{³;}>s=o }mJ}>x m>o ¢Fom}nq>{i x.º}>sil±o.{i .y=o( LlrMx.mnl¸}>x}>sil¸Ú¢yilMqn ì!íápí í=}>sHl¸l ¹H}¹Em}to({ir l. sHl {hØl&{¢}nq>x`¹HyirMlwo{HxL{H¾îm}to({=¹io qt¹r o.mnm'x.º;o.{H Ly=o( Ll m ¢}>sio(}x.ºF}>x.zYxLxL .r o. ¯io.mx(º=m}nq>{i .m ê l!mnsHxpØ×}>s=o(}R}>simr o.mnmBm¦{ix(}¦lpo(q>{io.¯Hl{¸Õw{H Lyi{³Ô mBï×ðñ ~x`¹El L¯HyH}}>sio(}B}mBlpo q>{=o.¯HlØ&}>shoJ{Hlpo(q{`yi~¸¯YlMq¦x(ºYr x.qnq>lMrM}>xL{Ú¢yiljq>lMm À{=o(¢;ØlrMxL{=¹EyirM}4mnlMlMqto.lMòEzYlMq>~l {¢}>mJØ&}>s×o({×Öwqto.rMlmn~yHo(}>{i 0}>sil ¯Yl siopEx.yHqwx.º'ohs`yi~o.{¶óBòEzYlMqn}p=o.{i¹\mnsixpØô}>sio(}wxLyEq+o( .x.q>}>si~$mwq>l mnm}to({`} }>xo(zizEq>x òE~o(}>l+o.{imØ!lMq>m õ¶ö`÷;ø¸ù Ï.úRÍLû uwqto.~~o }>rpo..{HºUlMq>l {Hr lL(Öwqto(r l';lpo(q>{H{i H ²!x.qnq>l rM}>x.{hüJyHlMq>l m ó¹H} Ó m}to.{Hr lLi»o.mJx(º`}nq>{i .m ý þ 12ÿY9F/ RÿF<>/21 ± CECH^ =DFC
(5) OFC
(6) \NED
(7)
(8) "!# wA.GjA$ =TVDF]ib¶C&%¸I+N&'=SUCED)(+*±D;WôWFC,HCE^ iD;
(9) C - O;A(/ D .!0
(10) 12#3"4"56
(11) !0798:; :=<>: !$ +NEb '³CHGj?D (@*%¨ A AHA.NEGMbNE]EC ^;btEc D8WYA.K.NEWYALb N`]EC P ECH$^ JCE^;SUW O; N FHAO;NEW cjC K CED³bt^FSVc4A(D³;K YK SUCEdRA.WYTZNEb±N`D;$W G OF)C H b G CP OFB CWYTZKpcjTUCED³N`GMTVALb TVD CEG WYA(G¸cjCX]HA cN`D;Ib wA.GMb±cjCXbj^;K KO Ji^FALb>cMTVCHD;Mb L *JchcMO;N¢chcjNT \A P HCE#^ v N XO; N FHADFA.A.WYALW sim&Øx.q>Û0Øo(mwmnyHzizYx.qn}>l ¹¶{\z=o qn}w¯¢}>silH q>xL .qto(~~lx.º¦}>sHl4óyEq>xLzYlpo({ë²!xL~~yE¾ {i}¢FyH{=¹ElMq}>siPl OR)ð Q0R=Tð SVUwlM}Ø!x.q>Ûx.ºóBòEr l l {Hr lL³nH!¾½(L½p¾. WXX(·Esim4ziyH¯irpo(}>x.{ xL{H0q>Zl Yil rM}>m}>sil4o(yH}>sHx.q>m ÔHElMØ&m ?.
(12) cjOFTZbTUD"%¨CEGZ vN¢cjTUCEDTUDCEG WYA(GhcMC d³N`GjcjTZK TUd;N¢cMA\cjC Ji^FT ALbN`D;WK(C \d³A(cjTVcjTUCED;bCEGM]HN`D;TUbjA.W '>$%îN ¶CH^;b vNE]HN TVD;A.b±WF^FGjTUDF]vcMOFAbjC^ E \A(G b P 'F^Yc '³ALK(NE^;btAC&% 794C:; :VJi^FALb>cMTVCHD;b P HCE^ ¶TU]EOicd³Cibjbj2T ';2S XO; N FH A \TUbMbjA.WXcMOFVA FHA(ZG ³GMbtc¸dFGMT AL?G OC(LML L4C
(13) +NEWFN Yb A FHA(GZHcMOFTUDF] O;NEbK O;NEDF]EALW &CE^D;N`cj^FG N`SUNS Ü^;bjAcMOFA G A ' P SUNE^F D;K KO ECE^;G %îN FECE^;GjTVcjA\btALN`P G K O8A.DF]ETUDFA P c idR A D EMA > JCEG WFb %¨CHSVSU
(14) C
(15) ßSUTVTD =b×NED;WDFC`cMAWF
(16) C D cMOFA N`D;bI JA(G b L nDcjOFTZb×d;NEGtcMTUK a ^FSUNEG K.NEbjA P HCE^ WYTZP bMK
(17) C FEA.MG LML LtcjO;N`cDF C =TUDF] &C %0I+N '>=SUCED +NEbK.N`SUSVALW Z&?
(18) ;&"! 'F^YK c D V M
(19) ?
(20) !H b8GMA(TU]EDFALWcjOFA.GjA vN`D K A.DHcM^FGMTVALbÜN`]HTC L *4]iN`TUD P cMOFA WF N .!0?
(21) 1NE3"4"56
(22) !0798:; :=<>: ! JNHb '³CHGjD TUb+D;C`c4K SUA.NEG P 'F^Yc4TVcTUb+ALNEIb \cjCvK OFALK \cjO;N`c .!0?
(23) 1N 3>4"56): ?: < : !E JNHb 'RCEGMD TV D "! P=_ ^FNS L # P C ECH^ JCE^;SUW¶dFGMC ';N 'FNS TVD\cMCYWF N cMOFA430]EGMA.N`c&K C \dRA cMTcMTVCHD;b&C %³cMOFA4d;NHb>c L nD;WYA.A.W P cjOFA8NHKpcM^;N`S±bjA.NEGMK OßA(D;]ETUDFA.bvNEGjAÜN 'FSVAXcMC dFGMCEdRCHbjA &!0!I:;7%$
(24) T OFA(D N H A wCHGMW TUb DFC`c %¨GMMA Ji^FA.DHc L2g+OFCibtAÜK(CEGMGjALKpcjTUCED³b¶NEGj+A \CHbtcëC %cjA(D GMA(SUTUN 'FSUA '³ALK(N`^³btA cjO;AÜGj A %¨A(GMA(D;K(A WYTUK cjTUCED;NEIG TZEb 'F^FTUSVc %¨GjC cjOF+A 'FTUSUSVTUCED;b\C % J A 'ßd;N`]HA.b¶TUD;WY'A &YALWôNESVS
(25) C FHA(G¶cjOF+A JCEGMSUW L [4A(D³K A cMOFA G A 'XN`D³WvcjO;AbjA.NEGMK OvA.DF]ETUDFA.b+NEGjA¸dFSU N =TUDF]cMOFAd;N`GjcwC %'N`D 2T \dRA(IG %¨A.Kpc 'F^Yc d³
(26) C JA(IG %¨P ^F)S (¸GMNHK SUA P &N ';SVAcjCXGMA cj^;GjD GjA.SVMA F¢N`DichWFC=K(C^ \A(DicMb %¨GMC N GMA(SU A F¢N`DiPc Ji^FA(ZG P 'F^Yc N`SZbtC±cjCK CHGjGMA.K c!N`Dbj^;btdRA.K c JH^;A(GZL* nD0CEcjOFA.G JCEG WFb P bj^;K O¶N`D0CHGMNHK SUA2TZbN&'FSUA2cMCN`D;bI wA.G cjEC O;N¢c JAbtO³N`SUSBK(N`SU6S
(27) !0!I: 7+$ & , T: !$ :0 0L g+OFAÜbtA(ctcMTVDF] Tc btA.S %CYK(K(^FGMbTVD CEcjOFA.GëK TUGMK(C^ vb>c N`D;K(A. b G OFA.D N O=C^ vN`D '³A.TVD;] TUb NEIb EA.W8cjC×d;Gj
(28) C F=TZWYAvWFN¢c N TUD N`DTUDHcMA(G NEK cjNT FEAvbtTVcj^³N¢cjTUCED P N`DN`SVcjA.GjD;N`cjNT FEA¶cjC×O³ N FiTUDF] cjO;A OiT^ \NED A &=dRA(GjcXSUN '³A.SVSUTVD;] O=^F]EA Ji^;N`DicMTcMTVALbëC %WFN¢c N K.N`D 'RA cjCßO; N FEA8cjOFA SVALN`GMDFTUDF] b Ybtcj A NHb .- JH^;A(ZG 0/&cjO;A¸O=C^ vN`D 'A &YdRA(Gjc OFCcjO;A(DëCHDFNS vSUN '³A.SUbwcjO;CHbjA±TVcj A vbJcMO;N¢cN`GMA GjA JH^;TVGMA.W L14MA FEA(GjcjO;A(SUA.bMb P NEbMbtC^ \TUDF]vcMO;N¢ccjOF2A (¸GMNHK SUATZb¸NëOiT^ \NEDÜ'A &YdRA(Gjc¸TUDicjGMC=WF^;K ALb DF A K(CED;btcjG N`TUDicMMb L*(¸D0cjOFAJCEDFAwO;N`D³W P TcTUb'TUD;K CHD;K A.2T F¢&N ';SVAwcjCNEIb 43C&165
(29) ?879$ &1;D=C^ 'RA(G C&% Ji^FA.GjTUA.b Rg+O;A(GMAV vN +'³A\DFC K O;N`D³K AcjC ]EA cA(DFCH^F]EO N`D;bI JA(G b4TUD8GMA.NEbjCED³N&'FSUAcjTN \AL # C¶cjOFAhSUA.N`GMDFTUDF]\N`SU]ECEGMTVcjC O bjOFCE^;SUW N`NT N¢c \TUDF2T \TZbtTUDF]\cMOFAhDiT^ 'RA(G&C % Ji^FA(GMTVALb+ A FHA(D T % JA 0^³b>cJd; N %¨CHG2TVc TVcjOëN wCHGMbjA+cjNT \A¸K C \dFSVA &YTc L"(¸DvcMOFA±CEcjOFA.G&O;N`D³W P NOiT^ \NED -¨CEG\M A FEA(D cjOF+A G A ':/TZb %îNESVSUNT 'FSVA L&g+OFA(GM;A %¨CHGjAcjOFAXSUA.N`GMDFTUDF] NESV]HCEGMTcMCO btO;CE^FSZW NE2T N`c SVALN`GMDFTVD;] %¨^FD³KpcjTUCED³bCEGhSZN`DF]H^;N`]HA.b±cjO;N`c0N`GMAGMC 'F^³b>c %¨GMC K(CEGMGjALKpcjTUCED³b4cMO;N¢Pc v N ÜDFCEc '³ATZWYA.NES P cMO=^;b4N`d;dFGj*C &YNT vN¢cjA L nD cMOFAë N '³
(30) C FHA G A 'A &YN \dFSVA cjO;AëWYTZb>c N`D;K(Av^;btALKW '>8cjOFA btALN`G K O8A(D;]ETUDFA\cj.C ³D;W N×K SUCHbjA.btc0b>cMGjTUDF]×TZbN F¢N`GMTUNEDichC %+P cjOF,A :8$ 7 8$ 7
(31) : O;TUK O \A.NHbt^FGMA.bcMOFA \TUDF2T C^ DiT^ 'RA(GëC %0WYA(SUA cMTVCHD P TUD;bjA(GjcjTUCED CHGëbtT^ ';b>cMTcM^YcjTUCED CEdRA(G N¢cjTUCED³bvDFA(ALWYA.W cjC cMGMNED;b %¨CHIG CEDFAbtcjGMTVDF]¶TUDHcMC¶NEDFC`cMOFA(<G ;= P D*> L=g+O;TUbWYTZbtcMN`D³K AN`D;W F¢N`GMTUNEDHc b OFA(GMA¸ALNEK O A(SU A \A(DicMNEIG CEdRA(G N¢cjTUCED vN XO;N FHANëWFT@?A(GMA(Dic JA(TU]EOicO;N FEA 'RA(A.D8^;bjA.W8TVD# \NEDA³A(SZWFbTVD;K(SV^³WYTVD;] Q C \dF^Yc N¢cjTUCED³N`SFIJTVCHSVCH] B; > P @'N`DF]H^;N`]H4A C×CYWYA.SVSUTUDF2] ; >³N`D³9W DN`ctcjA.Gj2D EALK CH]EDFTVcjTUCEFD ; G>=L nD.HG N& E vN¢cMTUK.N`S nD"%¨A.GjA.D;K AB-%HI J/ P TVc4TUb vNETVDFSNë^³btALWëTUD D =TUD;WFbC%'d;NEd³A.GMbML (¸D cjOFAÜCHDFAÜO;NED;W P btM A FEA(G N`S wCHIG YbvN`NT N¢c WF A FEA.SVCHdFTUDF] b>c N`D;W;N`G Wôd;GjC ';&N 'FTUSUTUbtcjTZK HI \cjA.K O;DFTNJi^FALbTUD CEG WYA.GvcjCôSVALN`GMDôcMOFA# JA(TU]EOic bC&%hcMOFA8ALWYTc CEdRA(G N¢cMTVCHD;bK; L P !M> L4(¸D cjOFA¶C`cMOFA(GO;N`D³W P cjOFA¶A.WYTVcWYTUbtcMNED;K A¶D;N`cj^FG N`SU2S XN`d;d³ALN`G b±TUD bjd³ALK NT ³OK HI 4dFGMC 'FSU A vb P TUD d;N`GjcjTZK ^FSZN`G O;A(D\CEDFA +N`Dic bcjCSUA.NEGjD¶SUNEDF]E^³N`]EALb %¨GjC ÝDFCHTUbIWFN`cMNO; P P > L nD\cjOFASZN¢cjcjA(G K(NEbjA P wA 0^;btcwGM A vN`ZG cMO;N¢cwcjOFAhK SZNEbMbjA.bC %BSUNEDF]E^;NE]EALb&b>cM^;WYTUA.W\cMOFA(GMA±NEGjA4D;C`cJWF'A ;DFALW %¨CESUSVC
(32) TUDF]¸cjOFA Q OFC vb 0[±TVA.GMNEGMK OL* nD;WFA(A.W P A FEA.D0cMOFAA.NHbtTUA.btcSVAMFEA.SYC&%;cMOFTZbOFTUA(G N`G K O> P cjOFA K SZNEbMbv&C %GMA(]E^;SUNEGSUNEDF]E^³N`]EALb P TZbD;C`c N`c N`SUS±GMC ';^;b>cvcMC DFCHTUbjA P btTUD;K AÜcjOFA d;N`GMTc %¨^FD;KpcMTVCHD;b2-9 O;TUK O K(NED,'RAvWYA'³DFA.WNEbGMA(]H^FSUNEGhSUNEDF]E^;NE]EALb/±N`GMADFCEcSVALN`GMD;N&'FSUA¶TVD cjO;A dFGjALbtA.D;K A+C %RDFCHTUbjQA ;= R> L nD¶cjOFTZbd;NEd³A.G2N`SZbjC P TUD¶CHGMWFA(GcjC N FHCETZWcMOFTUb2WY@T SK(^FSc P JAbtO;NESVS D. th.
(33) K CED³btTZWYA(GCHDFNS bjd³ALK TZN`S ;DFTVcjA0SUNEDF]E^;NE]EALb cjO³N¢c±bjA(A A(SUA \A(Dic N`GZvcjCE%¨CHGI vN`S¦SZN`D;]E^;NE]EA cjOFA.CEGMA cjTZK TZN`D³b P ';^YcwN`GMAGMA(SUA F¢N`Dic %¨CEG&cjCEdRCEP SUCE]HTUbtcMb2N`D;W\K C \dFSVA&V%¨CHG2K C 'FTUD;N`cjCEGMTZN`SUTUbtcMb cjOFA
(34) 1 1 V 07 !$ <
(35) 0L nD cMOFAXd;N`dRA(G wA cjO=^;bvbtcj^³W" cjO;A×dFGjC 'FSUA C&%TUWFA(DicjT2%9=TVDF]';NESVSZbV%¨GjC K CHGjGMA.K a cjTUCED Ji^FA(GMTUA.Mb LCG VA P '³A.]ETUD ' GM A F=TZbtTVcjTUDF]\cjO;AA.WYTVcWFTUbtcMNED;K A0NED;WXTUDHcMGjCYWY^³K TUDF]\cjOFA ';N`SUSUb TVD # A.Kpc L DCL³g+OFA.D wAbtOF
(36) C TUD # A.K Mc L \cjO³N¢c '³N`SUSUb¸N`GMADFC`cSVALN`GMD;&N 'FSUPA TVcj#O *±DF]ESU^FTUD H b ¶MA 'RA(G btO;TVd¶N`D;WMA Ji^FNT F¢N`SUA(D;K(A Ji^FA.GjTUA.b P cjO;N`cTZbN`DFCEcjOFA.GGjALNEbjCEDcMC¸TUDicjGMC=WF^;K AwcjOFAK(CEGja GjALKpcjTUCED Ji^FA.GjTUA.Mb L&G A¸bjOF
(37) C ßTVD # ALKpc L cMO;N¢c ';N`SUSUbwN`GMASVALN`GMD;&N ';SVA TVcjOëNSVTUDFA.NEG&D=C^ 'RA(G &C %¦bt^;K O Ji^FA.GjTUA.Mb L`g+OFA(D\TUD # A.K Mc LG P JA+btcj^³"W cMOFA4'A ?A.K cjNT FEA.DFA.bMb&C %CE^FGwN`SU]ECEGMTVcjCO %¨GMC N`D 'A &Yd³A.GjNT \A(DicMNESBdRCETUDHcC % F=TU A P bjOF
(38) C TUDF]0cMO;N¢c4TVc4TZPb !I 07 P TUDXd;N`GjcjTZK ^;SUNEG OFA(D cjO;A N`D;bI wA.GMbJC%cjO;A9(¸GMNHK SUANEGjAN`d;dFGjC*&YTN vN¢cjA LYe'TUD;NESVSN P JAK CHD;K SU^;WYAhTUD # A.KpcL LCL 1 j:/ wÿY9;<j1432: :ô- 143 3&7: *4/ D &1 3C4T: 7 Σ TZb¸BN ;DFTVcjA¶DFCEDFMA \dYc ×btA(c¸&C %2Ib 'RCESZb¸K.N`SUSVALW 12: 7 7 :;! 0L * 07=!$ >< w = TZb!N`D ³DFTcMAwbjMA Ji^FA.D;K A&C %FSVA(ctcMA(G b LMG A GjTVcjA Σ %¨CEGcjOFAJbtA(c&C %;NESVSib>cMGjTUDF]ibB
(39) C FEA.G a ...a E N ; D W %¨CHG&cjO;A4 A \dYc vbtcjGMTVDF)] LE@BA(c 'RA4cMOFA4SUA(DF]EcjOC % N`D;W cjO;A4D=C^ 'RA(Gw&C % Σ C=K.K ^FGMGMA(λD;K(A.b!&C % a TUD w LG A4bM N cjO³N¢c2|w| Nhb>cMGjTUDF] u TZbN 0T; :, w"?:+C |w| % v P WYA.DFC`cMA.W u v P T % u = a . . . a NED;W×cMOFA(GMA0A &=TZbtc u , . . . , u ∈ Σ bt^;K OÜcjO;N`c v = u a u . . . a u L * 12&><T <>:±TZb2NED ¶bjC^ ';bjA c L ⊆ Σ Li@BA(c IN 'RAcMOFA¸bjA c2C %¦DFCEDvDFA.]HN`cjNT FEA4TUDicjA(]HA(G b LHeFCEG N`SUS k ∈ IN P SVA(c Σ = {w ∈ Σ : |w| = k} N`D;W Σ = {w ∈ Σ : |w| ≤ k} L g+OFA :$ 7 $ 07
(40) ?: w ) TUbwcjO;A \TUDFT2 ^C D=^C 'RA(GJC&% :Z8$ 7 3T: !I
(41) 7%$
(42) T DFA.A.WYALW cjCcjG N`D;b %¨CEZG w TUDHcMC wd(w, ;='> L8C×CEGMAJdFGjALK TZbtA.2S P wAbM N cjO³N¢c w !I: 5 !$ 7 :0 7 w $ 07 : 3 P T J % ( A V T j c F O . A B G * / GjTVctcMA(D w → ` N ; D W -I :;12: 7%$
(43) / P CHG2- D / − w-J$ P C :;!07%$
(44) / CE2 w = uav w = uv w = uv N`D;W w = uav G . / ` N ³ D W - 0T0 07%$ 7 "7%$
(45) / P O;A(GMA P w = uav w = ubv N`D³W a 6= b Li@BA(c −→ WYA(DFCEcjA k GM A GMTcMTVD;]b>cMA(d;Mb LEg+OFAALWYTcJWYTUbtcMNED;K A u, v ∈ Σ P a, b ∈ Σ TUb4cjOFA \TUDFNT 0C^ k ∈ IN bj^;K OXcjO³N¢c w −→ w L < P d(abaa, aab) = 2 btTUD;K(A d(w, w ) NED;W cjOFA0GjMA GjTVcjTUDF]v&C % TUDHcMC K.N`DFDFCEc 'RA0NEK OFTU A FHA.$W TVcjO abaa −N`$ − SVALbjbwcjO³→ D aaa D0b>cM→ A(d;Mbaab L04CEcjTZK A¸cjO;N`c d(w, w ) K(NEDabaa 'RAK(C \dFaab ^YcMA.W TVDëcjNT ¶A O (|w| · |w |) ' ¶ALN`D;b+C %CW iD³&N \TUKdFGMCE]HGMN E ¶TUDFA ] ; D>=L g+OFPA %¨CESUSU
(46) C TVDF]btcMNED;WFNEGMW dFGMCEdRA(Gjc btcMN`cjA.b4cjO;N`c d(w, w ) TZb¸N`c±SUA.NEbtc4cMOFAD=C^ 'RA(G JU x(}w{¶ó!²|\ &C %TVD;bjA(GjcjTUCED³bwD;A(A.WFA.W cjC\MA Ji^;NESVTZbjA¸cMOFASUA(DF]EcjO; b . . . . 1.
(47) . . . . ∗. n. a. 1. n. ∗. 0. ∗. n. ∗. k. 0 1 1. ≤k. ∗. 0. 0. . 0. 0. n n. 0. 0. 0. k. ∗. k. 0. 0 . 0. 0. 0.
(48)
(49) 0 ∗' 0
(50) |w| − |w0 |
(51) )(#
(52) !I:+*
(53) : ! w, w ∈ Σ d(w, w ) ≥
(54) $ ,.
(55) &! w0 w/ d(w, w0 ) =
(56) |w| − |w0 |
(57) w w0 <*) ! +"$#%&.
(58) !
(59) 1 1. '. UJx(}2{¶ó!²|\. c!TZb wA.SVSVa iD;
(60) C DcjO;N`c'cMOFA+A.WYTVc&WYTZb>c N`D;K(AwTZbN 7 : 7 !$ ;= '> btCTVcK(CEDFEAM=bcjC cjO;A b>cMGj^;K cj^FGMAC&%'NA7E: 7=!$ %3C :;LFg+OFA.GjA %¨CEGMA P Tc+TZb+D;N¢cM^FGMNES;cMCTUDicjP GMCYWY^;K A ';NESVSZbJC&%'btΣcjGMTVD;]Hb P btTUD;K A\TUD $N + N P cjO;A.bjA\NEGjAcjO;A\bjTN ¶d;SVALb>cSZN`DF]H^;N`]HA.b¸CEDFAE vN ÜK CHD;btTZWYA.GML¦g+OFA
(61) 1 1 Z: T7=!I: &?$!I $ WYA.DFC`cMA.W P TZb4cMOFA¶bjA chLC&%2NE< SVSbtT cj% GMTVDF]ib O;CHbjA WYTUbtcMNED;K oA∈TZb&ΣN`c ¶Cib>c r %¨GjC or ∈B IN(o)P = {w ∈ ΣB (o): d(o, P %¨CHGwΣN`SU=S {a, b} PL ≤ r} cjOFA.D B (ba) = {a, b, aa, ba, bb, aba, baa, bab, bba} NED;W w) g+OFA2SZN¢cjcjA(GA &YN \dFSVAwTVSUSU^;b>cMGMN`cjALbBcMOF6A %îNHKpc!cjO;N`c'cMOFAJDiT^ B 'R(λ) A(G&C =%³btΣcjGMTUDF]HbTVD¶N ';rNESV∈S=K(INNED '³AA &Yd³CHDFA(DicjTZN`S TcMO×cjOFA0GMNHWYTV^³ b L &=dRA(GMNT ¶A.DicMN`SUNS -îbjA(Ag'N 'FSV A M/ wA0K(SVALN`GM2S ëDFC`cMTUK(A cjO;N`c %¨CHGvKN &YALW SUA(DF]EcjOß&C %¸K(A(DicjGMA P cjO;AX N FHA(G N`]HAvDiT^ 'RA(G\&C %hb>cMGjTUP DF]HbTZVb \CHGjA cjO;NED JU x(}w{¶ó!²|\ ∗. ∗. r ∗. r. 1. . r. 10. ≤r.
(62) c TUK(A$';TV]H]EA(G O;A(DôcjOFA×G NEWYTU^;b\TZb\TVD³K GMA.NEbjA.W ' L&g+O;TUbK(C V'FTVD³N¢cjCHGjTZN`SJA'&YdFSUCHbjTUCED C=K.K ^FG b\NEb\bjCiCHDôNHb |Σ| ≥ 2 P N`SVcjO;CE^F]HO JA SUA.N FHAëCHd³A.D cMOFA JH^;A.btcjTUCEDßC&% ;D³WYTVD;]N ]EA(D;A(G N`S %¨CEZG 0^FSZN8cjO;N`c wCH^FSUWô]ETNFEA cMOFA+FECESU^C \AXC&%N`D ';NESVS B (o) L g+OFTUbGjAM \NEGI r. Ð=Ì ö Ò ÕRljqto. Ll{`yH~¯YlMqx(º&m}nq>{i Lm4{×o\¯io.Ø&}>s×o½0lj}n}>lMq>mo(zis=o(¯YlM}p²!l {¢}nq>l m±o(q>l qto({=¹ExL~ m}nq>{i .m ilpo(rjs¶r xL~zHyH}to(}>x.{¶m2¹HxL{Hlx lMqw½.hrMl {¢}nq>l m
(63)
(64)
(65) ! ( " # $ % & ' * .. UJx(}&{\ó'²|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bjA.b2cjOFAd;GjC 'FSVAM C&%'cjOFAhGMA(dFGMA.bjA(DicMN`cjTUCEDëbMK OFAM ¶AcMO;N¢c JAbjOFCE^;SUW ^;bjA¸cMC\SVALN`GMDcjO;A ';N`SUSUbMLBI+NEbjTUK.N`SUS2 JA\DFA(ALW8GMA(d;GjALbtA.DHc N¢cMTVCHD;b O;CHbjA¶bjT A\TUbhGMA.NHbtCHD;N&'FSUA OFTZK O TZbhDFCEc cjOFAvK(NHbtA¶C&%+N`D PA'&YO;NE^;b>cMT2FHAA.D=^C \A(G N¢cjTUCED L *±D NEScMA(GMD; N¢cMT2FHAGMA(d;GjALbtA.DHc N¢P cMTVCHD8K CH^FSUW,'RA '> :;7 :;!7 $ $ 07%$ 10 :$ 7 : &"7 87E&7 -32546 /¶bjTVD³K A '³N`SUSUb¶N`GMB A ;D;TcMA NED;WcjO=^;b¶GMA(]H^FSUNEG SUNEDF]E^;NE]EAL b Li[4
(66) C J A FEA.G P 'A &YdRA(GM2T \A(Dic bJbjOF
(67) C cMO;N¢c+cjO;AK(CEGMGjALbtdRCED;WFTVDFV] \TVDFNT 0T^ 72546 TUb &C %cjA.DX'A &Yd³CHDFA(DicMTUNES TcMO r -9'F^Fc±SVTUDFALN`G TcMO |o| /<;N D*> L"*4]HNETVD P N %¨CEZG \NESdFGjC=C %'&C %!cjO;TUb dFGjCHd³A.Gtc TZbN K O;N`SUSUA(DF]HTVDF] K C V'FTUD;N¢cMCEGMTUNESdFGMC 'FSU A P 'F^YchTVD N`D ×K(NHbtATcbtA. A vb±cjO;N`c 2546N`GMAhDFC`cGMA.NHbtCHD;N&'FSUAGjA.dFGjALbtA.DicMN¢cMTVCHD;bwC&% ';N`SUSUbML (¸D\cjOFA4CEcjOFA.G2O;NED;W P > O ¶DFCEc2GjA.dFGMA.bjA(DiccjOFA ';N`SUS B (o) '0cMOFA±d³N`TUG (o, r) Tc btA.S %( nD;WYA(ALW P TVcMbbtT AëTUb L C×CHGjA.
(68) C FEA(G WYA.K(TUWFTVDF# ] OFA cMOFA(G CHGDFC`cTUb 2T E \A.WYTZN¢cMA (¸DFA×CEDF|o|NS +O³NElog b\cMrC -M/K C \dF^FP cjA d(o, w) N`D;W - D /vwK OF∈ALK B (o)OFA cMOFA(GëcjOFTZb WYTUbtcMNED;K A8TUb OFTZK O TUb NEK OFTU A F¢&N ';SVA×TUD cjNT \A Lwe!TUD;N`SU2S P TVcëTZb + log r) d³Cibjbj2T ';SVA %¨CEG≤N ';rNEPSVS=cMCO³ N FEA+btMA FEA(G N`S=bj^;K OGMA(d;GjALbtA.DHc ON¢cM(|o| TVCHD; b ·LM|w| % cjOFA.D B (a) = P ΣbjA = {a} ¨ % H C ' G U T ; D t b M c E N ; D. K A F ' Y ^ ! c j c ; O U T ' b U T ! b 4 N j b ³ d L A. K Z T ` N i S ( K E N MG OFA(D0cjOFAJN`SUdFO; N '³A(c P B (λ) = {λ, a, aa, aaa} O;NEbN`c4SVALNEbtc DSUA cjcjA(G b P (o, r) TZb4N¶^FDFT Ji^FAhcjO=^;b
(69)
(70) :$ Z&1GjA.dFGjALbtA.DicMN¢cMTVCHD&C % B (o) r. r. 2. 3. 9":;9=< 8 :. UJx(}&{\ó'²|\. #%1>. |Σ| ≥ 2.
(71) ?. Br1 (o1 ) = Br2 (o2. 794C: . )'. o1 = o 2. &?. r. r1 = r 2. . 794C: | + r = |o | + r L nD;WYA(ALW P SVA(c 1F> B (o ) = B (o ) ' D&|o c j F O . A D GMCEd L< P bjC o w ∈ B (o ) P cjO=^;b w ∈ Σ P d(o , o w) = |w| = r $ : L ±
(72) C. %¨GjC D&GMCEd L L o# w ∈ B (o ) P o )≤r d(o w, o ) ≥ |o w| − |o | C JA¸WYALWY^;K(A4cjO³N¢c Prd(o≥ |ow, w| = L + g F O A M b & N \. 4 A M G . A H N btCHDFTUDF]P=TVA.SUWFb r − |o | 7 4T:;!Z::K $ 07 @GBD E<IH−J> |o | = |o | + L
(73) ? 0T4 |o | + r ≥ |o | + r 7 4T
(74) 7 |w| = r +|o | ML o w|Σ|
(75) ≥ N 2 o 6ow6L* nD;oWYA.A.W P bj^FdFdRCHbjA&cjwO³N¢∈c oΣ btcMNEGtc b TcMON`D a NED;W¶SVA(c b ∈ Σ \ {a} O G A4WYA;DFA w = b o NED;W¶]EA c2cjOFAGMA.bj^FSVcML 8 P9" ;9Q< R9QBS *±bMbtC^ \A cjO³N¢c L&g+O;A(DßA(TVcjOFA.G CHG L # ^Fd;d³CibtA cjO;A btALK CED³WëK(NHbtA P TcMOFCE^Fc+SUoCHbMbJ6=C&%!o]EA(D;A(G N`SUTc L=I Q oSZN`TN 6 D oP cMP OFA(GMAhoA'&YTZ6b>c boNbtcjGMTVD;] w bj^;K O cjO;N`9c -*/ |w| = r + |o | P -=D / o w N`D³W -+ / o 6 w L)*±b o w P JAWYA.WF^;K AcMO;N¢c ?. !I A@CBED1 <. #. r2. 1. ! .
(76) -. /. 1. 2. 2. 1. 1. '. 2. 2. /. 1. ! 2. 1. 2. 1. -. /. 1. . 1. 1. 2. 1. 2. 2. 2. 2. . 1. r1. 1. 2. '. ∗. 2. 1. 1. 2. 2. r1. 1. 1. 1. 2. 1 1. ' -. 1. 1. 1. 2. r2. 1. 2. 1. 1. r1. r1. 1. !. 1. 2. 2. 1. 1.
(77) ' D&GMCEd L LMC×CEGMA(
(78) C FHA(G NEb wA2]EA c L ¸DvP cMOFA¸CEcjOFA.GwO³N`D;W P P P btC wA¸]EA c P P ±b JAWYA.WF^;K AhcMO;N¢c bjC P cP jO=^;b ¦g+OFTZbTUbT \d³CibjbjT ;SVA P bjC NP `D;WÜcjOFP A.D JU x(}w{¶ó!²|\ P Q SUNET [4A(D³K A GMA(d;GjALbtA.DHcMTVD;]0cMOFPA ';N`SUS 'ëcjOFAd;NETVG (o, r) TZb+GjALNEbjCED;N'FSVA LY[4C
(79) JA FEA.G P Tc4TZb wCHGtcMOëP DFCEcjTZK TUDF]¶cjO;N`c 4<>:P';NEBSVSZb (o) P OFCibtAhG NEWYTUT¦N`GMA¸ D;C`cd³CH2S =DFC \TUNESVNS GMA(SZN¢cjALWvcMC cjOFA¸SVA.DF]`cMO&C %cjO;A¸K A.DicjGMA.b -Z: < P r > 2 / P TUSUS³dRCHbjAcjGMTZK >¶dFGMC 'FSU A vb2&C %K C \dFSVA &YTc L CEDFA¶DFA.A.WFb¸cjC+ vN`D;TVdF^;SUN`cjA\N¢chSVALNEbtc eFCEGhTUD;btcMN`D³K A cMC SUA.NEGjD8cjOFAE';NESVS CEDFA0btcjGMTUDF]v&C %P SUA(DF]EcjO r L³g+OFA.Gj A %¨CEGMBA P (λ) TUDXcjO;=A Σ%¨CESUSU
(80) C P TVDF] P wA TVSUS'NE2S + N YbK CED³btTZWYA(G < M ';N`SUSUb+CHDFSN1. C &GjCHd. d(o |w| = r1 +|o1 | 1 , w) = |w|−|o1 |
(81) d(o1 , w) =
(82) r'1 # L( V
(83)
(84) w ∈ Br1 (o1 ) o2 6 w d(o , w) > |w| − |o |
(85) 2
(86) 2 D L LC* |w| = r + |o |
(87)
(88) d(o2 , w) > r1 + |o1 | − |o2 | = r2 1 1 L N 2' w 6∈ Br2 (o B o1 = o 2 2) r1 (o1 ) 6= Br2 (o2 ) ' 2
(89) L. r1 = r 2 t u r. . |o|. ≤r. r. 9. ") ! ".
(90) 1 3T4C: 7. Σ. $. . : #
(91) :5 0:K: 3T
(92) 1@5
(93) ?7 $
(94) 1 q() ' 56: 85V794C&7
(95) 1 1 B (o) +*
(96) :;!
(97) > < M 2$ r ≤ q(|o|) : 7 GB (Σ) : ?&7 :794C: ;: 7
(98) 1 1 qr < M$
(99) 1 1 q. #%$ *. q. -07 9;1±<j143 t: 9F/ ±7B9;<t7:
(100) ^FA(GZÜSUA.N`GMDFTUDF]XTUb0NXd;N`G NEWYTU] TVDicjGMCYWY^;K(A.W,'>K*4DF]HSV^;TVD ;= 8>=LB[±A(G \CYWYA(S6'FGjTUDF]ibN @BA.NEGjD;A(G -¨O;MA /NED;WvN`D (¸G NEK SUQA -îbjOFAM/TVDicjCd;SUN LEg+OFA4]HCHN`SYC%cjOFA±@A.NEGjDFA.GTUbcjCTZWYA(DicMT %9 cjOFAGMA(dFGMA.bjA(Dic N¢cjTUCED&C %N`D¶^FCD =DFC
(101) D¶SUNEDF]E^;NE]EA P '0bj^C'C \TVctcMTVDF]VJH^;A(GMTVALbcMChcjOFA (¸G NEK(SVA L g+OFASUN`ctcMA(G =DF
(102) C 4bJcMOFAhcMNEGj]HA c4SZN`DF]H^;N`]HANED;WXN`D;Ib wA.GMbJdFGMCEdRA(GMNS vcjC\cMOFA Ji^FA(GMTUA.Qb -$ : P btOFA\WYC=A.b¸DFCEc¸SUTV*A /0LC×CEGMA(
(103) C FHA(G P cjO;A¶@A.N`GMDFA.G±TZb 'RCE^;D;W 'X'A SK TUA(D³;K ÜK CHD;btcjG N`TUDHc b -*/ [4AëK.N`DCEDFNS bjC^ 'C \TVc¶NXdRCENS =DFC \TZN`S&D=C^ V'³A.G&C % Ji^FA(GMTUA.Bb -îTVDcMOFAëbjT Av&C %+cjOFAcMN`GM]EA(c GjA.dFGjALbtA.DicMN¢cMTVCH:D /!N`D;.W -=D /!cMOFA¸ N F¢NETVSZ&N 'FSUA+
(104) C FEA.GMNESVS=cM2T \A ^;b>c '³A¸d³CH2S =DFC ¶TZN`S;TVD\cMOFA±bjT A &C %!cjOFAhc N`GM]EA c+GMA(dFGMA.bjA(DicMN`cjTUCED $ L IwA(c wA.A(D8cjOFAvWYNT ?RA.GjA.DicK(C V'FTVD³N¢cjTUCED³b¸C % Ji^FA.GjTUA.b P CEDFA P K.N`SUSVALW 6 - C×TUDFNT vN`SU2S *WYMA Ji^;N`cjA4g'A.NEK O;A(G / P TZb2bj0^ SK TUA(DiccjCSUA.NEGjD 2 46 ;N 8> LEg JC =TVD;W;b&&C % Ji^FA(GMTUA.b&NEGjA+^;bjA.W . . . !. . 9 ") ! " H % :;7 : 12 E 12
(105) <&TM< :; & ∗
(106) P>;7 4T: : . E: E: ! I4"$ 3 , T:;!$ :; <' 794C: :Z&!0?: ! 0T'7 $ 7 @ 07=!$ >Σ< w ∈ ΣL∗ ∈7 #<7 4T: !I 12:: !
(107) C 056: ! ' : ?
(108) 7 : (w) ' $ E:$ 7 4T:;! A$ w ∈ L ' &! ! $ w ∈/ L :>;K794C: Z
(109) : @: , 0$ * &12: ?: ,;T: ! $ :0 ' 7 4T: :
(110) !0 :;! 0T79$ 7 - 7 4T: !Z: 3 !I:0 : T7
(111) 7%$
(112) / #12
(113) <&TM< : 7 9 7 C 4 : !I ;1N": #: ! &C 056: ! ' :;
(114) 7 :% $& ' $ P:'$ 794C: !(') * $ K = L '
(115) ! K∈< 7=! $ ><$:;1N
(116) >< $ >< 7 794C: 57 7 :;7=! $ V8 $ ,: !I:; : K(K) $ 4 L K 6= L 4* SVcjOFCH^F]EO+ ßNED;W$, O;N EF A¶A.btcMNF' SVTZbjOFA.W8cjOFAMv btA.S2HF A.bhNEbhNXb>c N`D;W;N`G W8K C F' TUD;N`a cjTUCED P cMOFA(GMANEGjA¸GjALN`SR]EGMCE^FD³WFbwcjC ³' A.SVTUA HF A±cMO;N¢c-$&8N`GMA¸cjC=CdRC
(117) J A(GI¨% ^FSRcjC¶A'&YTUbtc+CEG+A HF A(D ³' A btTN0 ^FSZN¢cMA.W L ±* b¶bt^;]E]EALb>cMA.W TUD ;N 8>6J A v N K³' A N&F' SUAvcjC bt^C³' b>cMTcM^YcjA cjOFAM TVcjOôN GMNED;WYC WYGMN C&% b>cMGjTUDF]ibcjO;N`cN`GMA¶cjOFA.D bt^C'T¶ TVctcMA.W NHb. - ;8 7 3 61 $ >0< / F' ^YccjO;A(GMA N`GMA@v N`D K.NEbjA.bV OFA(GMA bMN&\ dFSUTUDF] TZb¶DFC`cvdRCHbMbtTNF' SUA NHbcjOFAXGMA(SUA ¢F N`DicvWYTUbtcjP GMTNF' ^YcjTUCEDôTUb ^FDC= DFC
(118) D N`D;W0/CEGvTUD;NHK(K ALbjbjTNF' SVA ;=MG>=2L IJALbtTZWYA.b P J A TVSUS±D;C`cëK CHD;bjTUWYA.G N`D;W1$, ³' ALK(N`^³btAcjO;A WFC¶D;C`c4OFA.SVd cjC\SUA.NEGjD$;' N`SUSZb 7 7. 2. sil}>~lhrMxL~zilMòE}yHmny=o(ër x.{ir lMq>{Hm+}>sil}>~lhmnzYlM{`}o(ºU}>ljq±q>lMr l `{H \lpo(rjs {ilMØ lMò`¾ o.~zHl.`o.{i¹}to(Ûl m'}>sHlJl {H .}>sx.º³}>sil2{EºZx(q>~o(}>xL{q>lM}>yHq>{Hlp¹¯¢}>sil+Öwqto.r lw{¢}>x¸o.r rMxLyi{¢}"3 s`yim (xLyEqBr xL{Hm}nqto.{¢}BmBmnx.~lMØ&s=o }Bm}nq>xL{i .lMq¯iyE}¦{ix(}q>l m}nq>rM}>lL.mn{ir lØ!l!ºZx¢r yHmBxL{ .x¢x`¹ ¯=o(m2xL{H¢ G.
(119) % . 007E: # ; : 7 m, n ∈ IN
(120) B |Σ| ≥ 2 o∈ T . : 5
(121) 1 > <
(122) ! $ 9 7 " 4 7 7 T 4
(123) 7 $ : T7%$ 0 :; : *
(124) :;!5E
(125) 1 1 ≤m,n B = {B5 r$(o) 794 $&: r ≤
(126) m, Σ ∗ , |o| ≤ n} ≤m,n ?::0
(127) !$ 165 ;:; Ω(|Σ|n ) , T:;!$ :; $ 794C:56
(128) ! 07Z
(129) : . 9":;9=< 8 :. H. . ? !I ¸eFCESUSUC
(130) TVDF] ;NML> JA WYALbjK(GjTN'RA N`D/*W"FEA.GMbMN`GZ# O;C# vN`TUDicMN`TUD;b¶N8bjA c C&%±NESVS d³CibjbjT2';SVA ';NESVSZbML *JccMOFA P 'RA(]ETUDFDFTUDF] P S = B Lw[±A(GëNED;b JA(GcjCcjO;A8AMJi^FTNSF¢N`SUA(D;K(A JH^;A(ZG L = B (o) TZbNXK CE^;DHcMA(GMA'&FN& \dFSUA o LB[4A(GNED;bI wA.G±cMC cMOFA \A V'³A.GMbjOFTUdKJi^FA.GI TZb (! L *JcA.NEK O b>cMA(d P cjOFA *W"FHA(G bjNEGIXA(SUTN ¶TUD;N`cjA.b vN`D '³N`SUSUbC&% 'F^FcCEDFSN×CEDFA o &C %2K A.DicjGMA o N`D;WXG NEWFTV^;b 0 L *±bcMOFA(GMANEGjA O(|Σ| ) bj^;K O '³N`SUSUb±TVD B S P TZWYA(DicjT2%9=TVD;] cjOFMA GMMA Ji^FTUGjALb Ω(|Σ| ) Ji^FA(GMTVAL b L t u cbjOFCH^FSU# W '³A\DFCEcjALWÜcMO;N¢chT %JcjO;A\@A.NEGjDFA.G¸TZb]ENT FEA.D8CED;A¶btcjGMTVD;] %¨GMC cMOFEA ';N`SUS O;A K(N`D¶SUA.N`GMD0^³btTUDF]hN¸dRCENS =DFC \TZN`S=D=C^ 'RA(GC % L
(131) G AbtO³N`SUSFbtA.A2cMO;N¢c
(132) !0!Z:Z 7%$
(133) A, T:;P !$ :; - / P TUDicjGMC=WF^;K AL@ W 'RA(SU
(134) C P N`SUSU
(135) C cjCv]HA cGjCH^FD;WëcjOFALbtAd;GjC 'FSVMA vb . ≤m,n. r. n. ≤m,n. n. UJx(}&{\ó'²|\. mnsix(qn}>l {. û Ø&i¯Yl. 9 ") ! " % :;7 Z: 0:K: 12&><T <>:&? P 07 !$ < 0T'7 $ 7=7 :P5P7 4T: :Z&!0?: ! 7 E794C: !I 12: 0V:;! &LC 056:;! ' :;
(136) 7 :Z (w) ' $ w :'$ 794C: ! $ w ∈ L ' &!
(137) !0!I: +7 $ & 5 N! N7 $ &7 7 $ $ 7 07 :$ 7 $ 07 &?: 0 ' 794C&7 $ $ 07 !$ < !I 7 w w (w) =L
(138) :Pw 07=∈/!$ >L< {w0 ∈ L : d(w,ww0∈) $L Q7 $ $ 7 07 } . 4C`cMTUK(AëcMO;N¢c\cMOFA K(NED A.NHbtTUNS '³A×bj2T ^FSUN`cjAL W =DF
(139) C TUDF] cjOFA cMNEGj]HA c\SUNEDF]E^³N`]EA L ×CEGMA(C
(140) FHA(G P JA O;N FHA bjA(A.D TUD cMOFAXTVDicjGMCYWY^;K cjTUCED cMO;N¢c¶cjO;A D;N`cj^FG N`SUSN A&YTUbtcvTVDôGjALN`SVa wCHGjSZW\N`dFdFSUTZK(N¢cMTVCHD;bbj^;K ONEbcMOFA±bjA.NEGMK O¶A(DF]HTVD;A.bC%RcMOFA G A ' L>*4SZbtC P JA4K(NED¶D;C`cjA4cjO;N`c ×N`GMAGMA(SUA F¢N`Dic %¨GMC 6NK CH]EDFTVcjTNFEAdRCETUDic2C&% F=TUA QHg+O;A(GMATUb2]EGMC
(141) TVDF]hAMFiTZWYA.D;K AcMO;N¢cJN K OFTVSZW×NEMK Ji^FTVGMA.b+O;TUbD³N¢cjNT FEASZN`D;]E^;NE]EAh^;bjTVDF]\cMOFAK CHGjGMA.K cjTUCED;b+C %!O;TUb ¶CEcjOFA.G ;=*!M> LC*4D³W SUNHb>c 'F^Fc¸DFCEc4SVALNEbtc P NEb JA(SUS'NHb ';N`SUSZbGjA.2S CED×NvWYTZb>c N`D;K(A P cjO;N`c %¨CEGMA.bjO;NEWF
(142) C 4bwD;TUK(A SVALN`GMDFTVD;]GjALbt^;Sc b L nD;WYA.A.W P bj^FdFdRCHbjA4cjO³N¢c wAN`GMA±DFCEc JCEGZ =TVDF]P TVcjOëcjOFAA.WFTcWYTZb>c N`D;K(A N`D;W b>cMGjTUDF]ib 'F^FccjOFA &^³K SUTUWYALN`D×WYTZb>c N`D;K(AN`D;W WYTZb Yb+TUD cjOFAdFSZN`D;A LCG AK(NED dFGMCYK A.A.W TVD. \b>c N`]HA.bwPcjCvSUA.NEGjDXN\WYTUbI C%K(A(DicjGMA O NED;WëG NEWFTV^;b R TVcjO L -*/ G Ab>c N`Gjc 'bjA(SUA.K cjTUDF] D±dRCETUDHc b %îN`G %¨GMC A.NHK OC`cMOFA(G+N`D;WCE^Yc btTZWYA±C A, %BcjO;BA WYTZb , wEA +N`DiccjC×TZWYA.DHcMT %9 L'@CiC =TVDF$] %¨CHG cjO; A O;N`dFO³N NEGMWF2S '¸NHb =TUDF]+cjC+cMOF4A (¸Gja NEK(SVA T %bt^;K O CEGbj^;K O N8d³CHTVDic\TUbTUD cjO;A WYTZb TZb±A.DFCE^F]HO nDicM^FTcM2T FHA(NS wAN`GMA]ECEa TUDF]±cMC ³D;WcMOF A TVcjO FEA.IG P%¨MA Ji^FA(GMTUA.Mb L - D / G AvNEIb cjOFB A (¸GMNHK SUAcMC ]H2T FHAK CHGjGMA.K a cjTUCED³b±cMC A NED;W B L Q CED;K(A(GMDFTUDF] A P cjOFA (¸GMNHK SUAGMA cM^FGMD;bN\d³CHTVDic TUD;btTZWYAhcMOFA0WYTZb NHb4K SUCHbjANEbdRCHbMbjT2'FSUAhcjC L SVALN`GMS2 cjO;TUb d³CHTVDic TVSU S '³AN`cwcMOFAhTVDicjA.CGMbjA.K cjTUCEDC %cMOFAbtA.P ] \A(Dic [OA] NED;WvcMOFA '³CH^FAD;WFQ NEIG vK(TVG K P SUA¸C % cjOFA\c N`GM]EA cWYTZIb ?L'@BNT E A TZbjA P SUA c D '³A\cMOFAëK(CEGMGjALKpcjTUCED &C % B L)-+ / G AWYGM N $cjOFAvSUTUDFA.b N`D³W TcMON4GM^FSUA(MG Lg+O; A TUDHcMA(G btALKpcTVD Lg+O;A( D JA+WYGM N cMOFA+K TUGMK(SVA TcMON (AC) K C \d;NHbjMb L
(143) G (BD) A4]EA(ccjOFA4G NEWFTV^;b 'V ¶ALNEbj^FGMTVDF]cMOFA±WYOTZb>c N`D;K(A '³A(c wA.A(D O N`D;W C LH[±A(D;K(A P Tc+TUbJALNEIb 0cMCSUA.NEGjD\cMOFA '³N`SUSUbJ&C % IR TVcjO L0±
(144) C P %¨CYK ^³btTUDF]¶CED ';NESVSZbw&C %'btcjGMTVD;]Hb P JA \N ëOFCEdRAcjO;N`ccjOFAdFGMA F=TUCE^;bNEdFdFGMCHNHK OTUb4]EC=CYW NED;WcMGIcjCvGMA(dFGMCYWY^;K ATVcML C. 0. A. C. B. D. O. UJx(}&{\ó'²|\. 2. L.
(145) j: ±:=<t1± 3 /29F9F7 RÿF<>/w1±: þ 471wÿF< 4<t1±3 nDcjO;TUb+bjA.KpcMTVCHD P wAdFGMCEdRCHbjA±N`D N`SU]ECEGMTVcjOC cjO;N`cJSUA.NEGjD³b6';NESVSZbw^;btTUDF]\N 16$ :
(146) !D=^C 'RA(G C&% L"*KpcM^;N`SU2S P wAN`GMA±]ECHTVD;]cjC %¨CESUSV
(147) C cMOFAdFGjMA F=TVCH^;bwNESV]HCEGMTcMCO P cMO;N¢c+SUA.N`GMD;b6';N`SUSUb C&% ' btALN`G K OFTVD;$] 'RCEG WYA.Gd³CHTVDic bTVD CHGMWFA(GcMCÜWYA.WY^³K AvcMOFAëK A.DicjGMA L ±A FEA.GtcMOFA(SUA.bMb P cjOFAIRdFGMC ';SVMA TUb \CHGjAK C \dFSVTZK(N`cjALW TVcj+O ';NESVSZb&C %b>cMGjTUDF]i b L e'TUG b>c O;A(DßCEDFAXbtT^ 'C \Tc bN btcjGMTUDF] CH^YcMbjTZWYA C %KN ';NESVSJcjC cMOFA (¸GMNHK SUA btOFA×btcjTUSVS N`D;bI wA.GMb P TVcjO Nb>cMGjTUDF] cjO;N`c '³A.SVCHDF]Hb¶cMC cjOFAÜK(TVG K SUA×WYA(SUT2 \TVcjTUDF]cjO;A ';N`SUS L P nD CEcjOFA.G wCHGMWFb P TVc TVSUSb>cMTVSUS '³A¶d³CibjbjNT 'FSVA0cjC K TUGMK(C^ vbjK(GjNT '³ATc L¦[4
(148) C J A FEA.G P NëbtcjGMTVD;]ë&C %cMA(DÜO³NEbN SVCEcC&%WFT@?A(GMA(DicdRCHbMbtTN'FSUAXK(CEGMGjALKpcMTVCHD;b P K CHDicjG N`GMTVSNcjC O;N`cO;N`dFdRA(D³b\TVDßcjO;A×dFSUNEDFAL < P cMOFA dRCHbMbtNT 'FSUA K CHGjGMA.K cjTUCED;+ b %¨CHG×cjO;AôbtcjGMTVDF] aaaa 5 N! N7 cjOFA ';N`SUS B (bb) N`GMA LLI vWYA;DFTVcjTUCEDC%¦cjOFA cjO;A {aa, aab, aba, baa, aabb, abab, abba, baab, baba, bbaa} (¸GMNHK SU A TUSVS&K OFC=CibtA¶CEDFA¶&C %wcjOFMA N`ZG 'FTVcjG N`GMTVNS P dRC`cjA.DicjTZN`SU2S ×cMOFEA wCHGMbtc¸CHDF+A 5 N! N7 BP cjO;A @BA.NEGjD;A(GH bwd³CHTVDicC&% F=TVAM PL 4A FHA(GjcjOFA.SVALbjb cjOFA (¸G NEK(SVA H bwd³CEcjA.DHcMTUNES vN`SUA FECHSVA.D;K A¸TUb+SUTN a TcMA.@W '>cjOFA %¨CHSVSU
(149) C TVD;]GMA.bj^FSVc P cjO;N`c±K O;N`G P NEK cjA.GjTZbtALbcMOFAbtA(c+C % &1 1cMOFAhd³Cibjbj2T ';SVAK CHGjGMA.K a cjTUCED;b %¨CHG4Nvb>cMGjTUDF] . . 2. . . . 2. . % : 7 r(o) : #Z&1 16
(150) m @ 07 !$ <$ N7 m 6∈ Br(o) ?4T:;7 4T: :;7 O3T $ 12:
(151) !0!I: 7+B $ &C , $ ,:K 7=165
(152) ? d(z, m) = m {z ∈ Σ ∗ : d(o, z) = r. 9":;9=< 8 :. d(o, m) − r}. . TVD;T2 ^C SVA.DF]`cMO wU x.}2{\ó'²|\ !I ¸@BA c k = d(o, m) NED;WK CHD;bjTUWYA.GN±WFA(GMT2F¢N¢cMTVCHD ¨% GjC o cMC m C&%C\ L)¸ * b m 6∈ B (o) P JA]EA(c k > r P bjC\cjOFTZb¸WFA(GMT2F¢N¢cMTVCHDXd;NEbMbtALbJcjOFGMCE^;]EO×NbtcjGMTVDF] k o− →m bj^;K O×cMO;N¢c o −→r rw0 −k−r ¦L @BA ch^;bhWYA;DFAcjOFA\btA(c w0 w) = Nr `D³W d(w, m) = k − r}−¦L →I+NEmbjTUK.N`SUS2 P w0 ∈ W P btC W 6=W∅ L =C×CE{wGMA(C
(153) ∈HF A(ΣG P ∗W: d(o, L ⊆ B e'TUD;NESVSN P SUA c Z WYA(D;C`cjAcjOFA0bjA c4C% NESVScjOFAdRCHbMbjT2F' SUAK CHGjGMA.K cjTUCED;b+C% m LTG A0K(SUNET2 rcj(o) O;N`c L nD³WYA(ALW SUA c NED;W L % cjOFA.D w TZb±NbtcjGMTVD;] Z=W m) > d(w, m) P C&% Br (o) cjO;N`cTUbP btcjGMTZKpzcjSN∈8 ZK(SVCibtA.G¸cMwC ∈mWcjO;NED d(z, j b C . K ` N F D F D E C P c ' ³ A X N K CHGjGMA.K cjTUCED C&% m L P z z (¸DvcjOFA¸C`cMOFA(GwO;N`D;W P T2% j c F O . A D NED;WvTVc P d(z, m) < d(w, m) d(o, m) ≤ d(o, z) + d(z, m) P ¨% CESUSVC
(154) 4 b2cjO;N`c d(o, z) ≥ d(o, C L * ¸ b N`D;W m) − d(z, m) > d(o, m) − d(w, m) d(o, m) = k #. w ¸ A E ] ( A c. F O Z T K ë O Z T w b N T. \ ³ d i C j b j b N T ' F V S A t b U T ; D ( K A L C P P d(w, m) = k − r d(o, z) > r z ∈ Z ⊆ Br (o) L. nD8K(CED;bjAMi J F ^ . A ; D. K A M A F E ( A Z G . t b j c M G U T F D ] U T b ¢ N ¸ c j c F O ¶ A j b N \ A P d(z, m) = d(w, m) = k − r WYTUbtcMNED;K A ¨% GjC m NEb0A EF A.GI K CHGjGMA.K cjTUCED z ∈ Z P cMO=^;b W ⊆ Zw L ∈C×WCHGjA.C
(155) EF A(G P J AO;N EF A bjC bjC z) ≥ r L?±* b z ∈ Br (o) P d(o, m) ≤ d(o, z) + d(z, m) P w A\WYA.WF^;K AcjO;N`c d(o, z) = r L CÜCEkGMA(≤C
(156) EF d(o, A.G P w z)A¶+O;N kEF −A¶b>rc P N¢cMA.W×d(o, cMO;N¢c d(z, m) = k − r P bjC w AK CHD;K SU^;WYAcMO;N¢c Z ⊆ W L U x(}2{¶ó!²|\ t u J [4A(GMA±TZb+N]HA(C\ A(cjGMTUK±TVDicjA.Gjd;GjA(cMN¢cMTVCHDvC&B% cMOFAGjALbt^FSVcJN³' C
(157) HF AiL @BA c+^;b+WYA;DFA¸cjOFAV :=< 7 :;)7 N`D³WÜcMOFA ' $ !I 12: Cr (o) = {w ∈ [o, m] = {w ∈ Σ ∗ : d(o, w) + d(w, m) = d(o, m)} L ; + g F O . A T C L \. t b M c ¢ N M c . A J b M c ; O ¢ N 4 c \ N > b M c j G U T F D ] U T ± b NdRCHbMbjT2F' SUAhK CEGMGMA.KpcMTVCHDëC% m $ , Σ ∗ : d(o, w) = r} z L ¦ + g F O A % î E N. K ¸ c M c ; O ¢ N c ; O H N ¸ b j b A F H ( A G ` N ' S ³ d i C j b bjT2;' SVA0K(CEGMGjALKpcMTVCHD;b±bjOFC
(158) 4 bcMO;N¢c zcjOF∈ A]H[o,A(Cm] \ A(∩cjGZv CrC&(o) % Σ ∗ TUbHF A(GZ WYTN?RA.GjA.m DHc¨% GMC cjO;N`c4C&% IR2 L # A.K(CED;WYSN F ' ^;TVSZWYTUDF]cjOFAK A.DicjGMAJC³% N ;' N`SUS>¨% GjC $ btcjGMTVDF]ib!CED0TVcMbdRA(GMTUdFOFA(GZ TZbWYT@SK(^FSc ¨% CEG+N¢c+SVALNEbtcD P GjALNEbjCED;bM"L (¸DëcjOFACEDFA¸O;N`D³W P (Σ ∗, d) TZbJNV\ A cMGjTZK±bjd;NEK(A TVcjO DFCHF A.K cjCEG btd;NHK ANEbNEDX^FD;WYA.GjSN= TVDF]vbtcjGM^;KpcM^FGMAFL g+OFTZbTZb+cjOFAbMN&\ AbtcjCEGZë NHb+T % J A J A(GMAcjGZ= TVDF]¶cMC SVALN`GMDcjOFAëWFTUbIY bC&% cMOFAëd;SUNEDFA TVcjO B ^;btc¶N8K C\ d;NHbjb F' ^YcD;CÜGj^;SVA.GMLML L (¸D cMOFAëCEcjOFA.G O;N`D;W P cjOFAd;GjC F' SVAM TZb ¨% CEGZ\ NESVSN$4T
(159) !I ;= P> ?. !.
(160) . %$ * : C 0 : $ 7 : :;7P E 07=! $ ><& W = {w , . . . , w } ' 0 ?8$ >< z ∈ Σ ∗ 1 n 7 4T
(161) 7479$ :$ 79$ :; P
(162) ! max '
(163) !I: N P 4C&!II3 !I 12:7 w∈W d(z, w) w∈W d(z, w) g+OFA(GMA;¨% CHGjA wA ^;b>c¶b>cM^;W" cjOFA '³N`SUSUbTUD \CEGMAWYA c N`TUSwNED;W vN& EA¶cMOFA 'RA.btcdRCHbta btTNF' SUAë^;bjAëC% cjPOFA bjC8NHb0DFCEc0cMC#'F^FTUSZWcjOFAXK A.DicjGMA.bP%¨GMC bMK G N¢c K O L G A 'RA(]HTVD ' WYTUbtcjTUDF]H^FTUbjOFTUDF]¶cjOFASUCED;]EA.btc4b>cMGjTUDF]ib+C&!% NED ;' N`SUS% 9 ") ! " % 4C:M3 3T:;!&!I: ! @Z&1 1 ' : ?&7 : ' $ 7 4T:P : 7. max
(164) 1 1 7 4T: 07 ! $ <
(165) 794C&7 Z: 12&><E7 B (o)
(166) ?E&!I: B r (o) 7E K $ 7V07A12: B><r794 (o) max B (o) = {z ∈. 9":;9=< 8 :. . r. r. Br (o) : ∀w ∈ Br (o), |w| ≤ |z|}.. JA ]EA c P SUA c Σ = {a, b} L *¸b B (ba) = {a, b, aa, ba, bb, aba, baa, bab, bba} P ¢ L + g F O h A > b M c j G U T F D H ] w b & C % ` N M G 4 A j G M A v. ` N Z G . N F ' U S A ³ ' ALK(NE^;btA B cjOFAMXNE(ba) GjA¶N`=SUS ';{aba, ^FTVSVc%¨baa, GMC bab, cMOFA¶bba} K(A(DicjGMA o '×WYCHTVDF] r BTVD;bjA(Gjcj(o)TUCED³b L # CëO;N F=TUDF]CHDFAbtcjGMTVD;] CEDFA bj2T \dFNS H³O;NHbcMC]E^FALbjb4cjOFATVD³btA.GtcMA.WXSUA ctcMA(G b±NED;W×WYA(SUA cMAcMOFA cjC w∈B ;D;W No E]HNETV(o) D L@P G A]HA Mc . <. . 1. max 1. UJx(}&{\ó'²|\. max r. max r. <*) ! +". H. %. $,. (o w.
(167) . d(o, w) = |w| − |o| = r). . !I ¸@BA c'^;bNHbjbj^C \AcjO;N`c o w NED;W d(o, w) = |w|−|o| = r LL g+OFA.D w ∈ B (o) L @BA c r ³' AJN4b>cMGjTUDF]P N7 |w0 | > |w| Lg+OFA.D P '<D&GMCEd L P d(o, w0 ) ≥ |w0 |−|o| > |w|−|o| w0 btC w0 6∈ Br (o) 4L g+O;A(GMA;¨% CEGMA P w ∈ Brmax(o) L Q CEDFHA(G btA.S2 P SUA c w ∈ Brmax(o) =N`D³rW P BL g+OFA(D cjO;A(GMAA&=TZbtcMbhNëGMA GMTVcjTUDF] WFA(GMT2F¢N¢cMTVCHD k L %JN WYA(SUA cMTVCHD kN`dF= dRA.d(o, NEGMb4NEw)SVCHDF≤]\cjrO;TUb¸WYA(GMT2¢F N`cjTUCED P cjOFA.D+¨% CEGM]EA cjcjTUDF]vTVc¸SUA.NEW;b+cjCëNoDFA −→ wWYA.GjTNF¢N¢cjTUCED$%¨GMC cjC btC ¶ A¶b>cMGjTUDF] w0 OFCHbjA0SUA(DF]EcjO TUb |w| + 1 N`D;W O;CHbjAWYTZb>c N`D;K(AcMC o TZb ≤ k P btC o L +% N×bj^C;' btcjTVcj^YcMTVCHD N`dFdRA.NEGMbNESVCHDF]ëcMOFAëWYA.GjTN¢F N¢cMTVCHD bjN N`D ' N w 6∈ Brmax (o) cjOFA.D E A(A.dFTUDF]hcjOFA a N`D³WVv N&= TUDF]hcjO;A4TVD³btA.GtcMTVCHD¶C% NhD;A b N`SZbtChSUA.NEP W;bcjC0NWFaA(GMT2¢F N¢cMTVCHbD P ¨% GjC o cjCÜbjC\ Avb>cMGjTUDF] w0 OFCHbjA\SVA.DF]`cMO TZb |w| + 1 NED;WWYTUbtcMNED;K A\cMC o TZb ≤ k P btC BL g+O;A(GMA;¨% CEGMA CHDFS2× TUD;bjA(GjcjTUCED;bCYK(K(^FGN`SUCEDF]ëcjOFAvWYA.GjTN¢F N¢cMTVCHD btC L w 6∈ Brmax (o) @NEbtcjSN P T2% k < r P cjOFA.D WYCHPTVD;] CHDFAV\ CHGjA¶TVD³btA.GtcMTVCHD8SUA.NEW;b4cMCXN WYA.GjTN¢F N¢cjTUCED P ¨% GjC o o wcjC btC ¶ A w0 OFCibtA±SVA.DF]`cMOvTUb |w| + 1 NED;WvWYTZb>c N`D;K(AcjC o TZb ≤ k P bjC w 6∈ Brmax(o) =L [4A.D;K A L k=r t u # A H F A(G N`S+b>cMGjTUDF]Hb¶C% N`GMA@HF A(GZ TVD"¨% CHGIv N`cjTNEF AL nD;WFA(A.W SVA(c ³' A×N`D N`GZF' TcMGMNEGIv SUA cjcjA.GMFL g+OFA.D BcjOFrmax Abtcj(o) GMTVD;] ar o = a . . . a o bjN`cjTZb³A.b o aP r o N`D³aW ∈d(o,Σ ar o) = ?. Uwx.}&{\ó'²|\. w ∈ Brmax (o).
(168) PrcjO;N`jb Cc a o ∈NESVSUBC
(169) ^;b(o)cMCëL ]EC×A(CEc±GMOFA(CHC
(170) FHSUWXA(G CP % T2%³CED;A %¨ =GMCDF C
(171) 4N`b DrXP CEb>cMDFGjA+TUDFK(] NED0A.NHbtTUSNWYA.WY^³K A 'oXLbG +AN`d;K SZdFN`TVTND; ] a o w∈B (o) cjOFASUA cjcjA(G bQ-îbjA(A *±SV]HCTL*/0L eFCEGTUD;btcMNED;K A P B (bb) = {aabb, abab, abba, abbb, baab, baba, babb, bbaa, bbab, L E^FDFD;TVDF]+$# 6
(172) CED cMOFA×b>cMGjTUDF] w = baba NED;W G NEWFTV^;b bbba, bbbb} M c M G E N ; D b %¨CHIG vb P N¢cA.NEK O SUC=CEd P cjOFA i SUA cjcjA(G&C % w cMC8N`D a cjO;N`cTZbdF^Fc¶N¢ccjOFA r = 2 '³A.]ETUDFDFTUDF]N`D;W cMOFA(D×bjC^ 'C \Tc b4TVccjC\cMOFA (¸G NEK SUA L c K(CE^FDicMbJcMOFAD=C^ V'³A.G&C %!cjNT ¶ALbcMOFTUb cjG N`D;b %¨CEZG vN¢cjTUCED TUbNHK(K(A(dYcMA.W LCG A]EA( c (w ) w K O;NEDF]EALb c i w w D;C 0 1 baba aaba baba H A.b 1 2 baba abba H A.b 2 3 abba aabb r. | {z } r. max r. r. max 2. th. 0. P. 0. max r.
(173) . .
(174) . ö Ï ö om}nq>{i w = x . . . x ∈ B (o) E}>silJqto.¹Hyim r Í Ï ö }>sHl+r l {¢}nq>l x(º}>silJ¯=o. B (o) ûû cø← 0;ö i ← 1;ú aù ←ox ½ ûû ÕwmnmnyHc~<l wr = x . . Î. x ³ö o({=¹¶lM} w = ax . . . xö ú x . . . x û Q w ← w ;c ← c + 1 û ö iú ←ø i +(w ö 1) = W X û Õ2mnmnyi~l w = x . . . x o.{=¹ Ï ö Î Ï x . . . x B 1:. ! P<. $ 6 . #. .
(175) . . . . max r. n. 1. r. n. 0. n. 1. 0. 1. OFA(D. G. i−1 i+1. 1. n. 0. n. r+1. n.
(176) btcjCHd;b TcMO .1 < V
(177) !0!I: 7
(178) :
(179) ?. c=2=r. P. $ # 6 *. #. . w = aabb. N`D³W GMA cM^FGMD;b.
(180) . L. o = bb . !I:;7="!0C o $ >< O (|o| + r) &?@I3Tr
(181) 165
(182) 7 w$
(183) ∈1 B7 r
(184) "T(o) 7 P7%$ 7 : ? !I ¸@BA c¸^;b±bjOFC
(185) cjO;N`c4cjOFAb +N`dFd;TVDF]vCHd³A.GMN`cjTUCEDXTZb4K CHGjGMA.K cML;@A c NED;W w0 = ax1 . . . xi−1 xi+1 . . . xn ¨% CHG¶bjC\ A i ∈ 1..n L %wcjOF=A.GjxA 1A'.&Y.TU.btcMxb¶n N¢∈c Brmax (o) SVALNEbtc±CEDFAWFA(GMT2¢F N¢cMTVCHD o −→r w OFA.GjAcMOFASUA cjcjA(G xi C&% w K(C\ A.b¨% GjC NED TUD;bjA(GjcjTUCED×TUD cMOFA(DWYA.SVA(cjTUDF] xi N`D;W WYCETUDF] cjOFA¶TVD³btA.GtcMTVCHDÜC&w% NED a TUD ¨% GMCEDicC% o = TUA(SZWFbhN btcjGMTVD;] oP cjO;N`c+bMN¢cMTUbJ;A.b o w0 NED;W |w0 | = |w| FL I O D&GMCEd 1L P w A¸]EA(c d(o, w0 ) = |w0 | − |c| = w0 bjC$' D&GjCHd L D NED;W
(186) L (¸D cMOFAC`cMOFA(G |w| O;N`D;−W P |c|T '% cj=OFA.GjrAhP TZb+DFC¶WFA(GMT2¢F N¢cMTVCHD@P w 0O;∈A(GMA Bxrmax Ti Ub+(o)TUDicjGMC=WF^;K ALW (w'ë 0 )N`=D TVD³btA.GtcMTVCHD P cjOFA.D xi TUb N GMA.N`S=YH SUA ctcMA(G4C% o L nD cMOFTZb4K(NHbtA P WYA(SUA cMTVDF] xi N`D³W TUD;bjA(GjcjTUDF]vN`D a = TVA.SUW;bNvb>cMGjTUDF] w0 N7 6 w0 BL I . D&GjCHd L P J A\]EA(c d(o, w0 ) > |w0 | − |o| L ¸ * b |w0 | = |w| P w AvWYALWY^;K A # cjO;N`c od(o, L C E N ; D W * L w0 ) > r w0 6∈ Brmax (o) (w0 ) 6= t u [4A(D³K A P J A¸N`GMA±D;C
(187) NF' SUA4cjC\WYALWY^;K A±cjOFAhK A.DicjGMA±C&% N;' N`SUSNEbJbtC=CED NEb J A= DFC
(188) TVcMb GMNHWYTV^³bN`D;W NXb>cMGjTUDF]$¨% GjC TVcMbh^Fd;d³A.G ³' CHGMWFA(GBL g+OFAE¨% CESUSVC
(189) TUDF] cjALK OFDFTZK(N`SSUA Ev NXTZbN b>cMA(d×cMC
(190) J NEGMWFb ;D;WYTUDF]cMOFTZb¸btcjGMTVD;] -îNEScMOFCE^F]HO w AO;N EF AD;CëTUD"¨% CEGZv N¢cjTUCEDÜN³' CH^Yc r NED;W HA c/ |o| 9Q<<D H % 3?M3 3T : 4T:; : * : ! 5&!0!I:;%7 $
(191) w 794C:@ 07=! $ >< Σ = {a1 , . . . , an } 5 T 4 ; : I ! : Z : 2 1 & > & < 7 m = (a1 . . . an )k k ≥ |o| + r Brmax (o) ? !I ¸@BA c ³' A\cMOFAëbjA cC&% N`SUScjO;Avd³CibjbjT2;' SVAK(CEGMGjALKpcjTUCED³b¸C% m L @A c^;bbjOFC
(192) $ cjO;N`c Z L * ± b. T M c O. J × A E ] ( A c bjC ' Z = Brmax (o) m = (a1 . . . an )k k ≥ |o| + r P o mP D&GjCHd L P L ¦ B @. A c L B I . & D M G E C d L D. J 0 A H ]. A c N`D;W P cMO d(o, m) = |m| −L |o| wG ∈NHb Brmax (o) ow w A C×CEGMA(C
(193) H F ( A. T d(o, w) = |w| − |o| = r m = (a1 . . . an )k k ≥ |o| + r P ]EA c w m P btC$'K D&GMCEd L P d(w, m)P = |m| L d(o, m) − r ^;K(A−cMO;|o|N¢c −Brrmax= (o) ' ±* b d(o, w) = r N`D³W d(w, m) = d(o, m) − r−P w |w|AëWF=A.WY|m| ZP g+OFA(C):L C L 4C
(194) SUA c z ∈ Z RL g+OFA(C)1L = TUA(SZWFb d(o, z) = r L % o z P cjOFA.D z ∈ B⊆rmax '>O D&GMCEd LDCL % o 6 z P cjOFA.DD&GMCEd 1L = TUA(SZWFb d(o, z) > |z| − |o| P $! : P |z| < |o| + r FL IJ(o) ^Yc cjOFA.D P d(z, m) ≥ |m| − |z| > |m| − |o| − r = d(w, m) ¨% CHG¸NESVS w ∈ Brmax(o) P OFTZK O K CEDicMGMNHWYTUK cMb z ∈ Z L # C Z ⊆ Brmax(o) L t u # ^C'C\ TVctcjTUDF] (a . . . a )k TcMO8Nbj^0SK TUA(DicjSNX SUNEGj]HA k TUbbt^;GjAcMC R' AK CHGjGMA.K cjALW@' N¶btcjGMTUDF] ¨% GjC Bmax1 (o) L n# CvN`SUScjO;N`c4GjAM\ NETVD³bwTZb+cMC9³D;WXbt^;K OXN`DXTUDicjA(GMA.btcjTUDF] k ;L g+OFA 9Q<<D. # %. A$. 4$ *. . r. . max. '. $ # 6 . û ê l nm six.yi¹yim}>ºV\}>sil4r x(qn¾ q>l rM}>{Hl mnm x(º}>silXmØo.zE¾ zi{i Üx.zYlMqto(}>xL{ { }>sil }>lMò`}p UJx(}w{¶ó!²|\. UJx(}2{¶ó!²|\. UJx(}w{¶ó!²|\. UJx(}2{¶ó!²|\.
(195) %¨CESUSVC
(196) TUDF]¶SVAM vN\btcMN¢cMA.bJcMO;N¢cT2%!CHDFANEbI =bwcMOFA9(¸GMNHK SUA4cMC\K(CEGMGjALKpcN\b>cMGjTUDF] \NHWYAhC&%N SVCEc±C&% H b cMOFA(D×cMOFAK(CEGMGjALKpcMTVCHD K(CEDicMNETVD;b4dFGjALK TUCE^;b4TVD"%¨CHGI vN`cjTUCED;bCHDXcjOFAGMNHWYTU^;b4N`D³W cjOFAD=^Ca V³' P A.GC&% C=K.K ^FGMGMA(D;K(A.bJC&% a H bTUDëcMOFAK A.DHcMGjA 9Q<<D % &C $ : !@794C:+
(197) 1 1
(198) 12: 7 a ∈ Σ &?K
(199) $ T7 :=<>: ! j ∈ IN N7 Br (o) : 7 > 7 4T:; |w| = |o| + r aj 6∈ B (o) w= (aj ) |w| < j '. `N D³W w = (a ) LI g+OFA.CTL P JA]HA c d(o, w) = r N`D;W a 6∈ B (o) L *¸b |w| < |a | P cjOFAëK(C \dF^Yc N¢cMTVCHD C&% d(w, a ) K CHD;btTZbtcMbTUD d(w, a ) = d(o, a ) − r -M/+bj^C';btcjTVcj^FcjTUDF]ëN`SUS¦cMOFASUA ctcMA(G bC&% cjO³N¢c¸NEGjAhDFCEc a H b P N`D;W -=D /+WYCHTVDF]vTUD;bjA(GjcjTUCED;b+C% w t b C ³ L F e E C U S U S C
(200) . V T F D \ ] cMOFA0bMN& \AN`GM]E^T ¶A.DicMb P d(o, a ) = j − |o| L # C aP d(w, a ) = j − |w| j c ³ O ¢ N 4 c Z T J b j c C M b N P |w| = |o| + r L j − |w| = j − |o| − r P t u 4
(201) C P SVA(c&^;bNEbMbjC^ \AwcMO;N¢ccMOFA4N`SUdFO;N '³A(cTZb Σ = {a , . . . , a } NED;W0SUA c j , . . . , j ∈ ³ ' \ A SZN`GM]EA¶TVDicjA.]EA.GMbML?¸A'³DFA k = P | (a )| LBg+OFA.D P @A E vNA $'FGMTVD;]Hb k = IN P LFg+O=^;b wA±K.N`Dd;SV^F] k TVDicMC@BMA E \N DcjC]EA(c ·r ≥ |o| + r N×b>cMGjTUDF(|o|] w +=r) = |o|(a +. |Σ| L C×CHGjA.
(202) C FEA(G P WY^FAvcMF C D2GjCHd L D P wA\O; N FHA ..a ) ∈ B (o) B L + g F O ( A M G A ; ¨ % H C j G A H N b ` N ; D W P k = |o| + |Σ| · r |w| = |o|/0+L nrDXP CE wcjA\OFA.WYG AL wWY^;CHK GMW;A¶b cMO; JN¢Ac |w| = |o| + r -îNED;W P r = (k − |w|)/(|Σ| − 1) K(N`D ]E^;A.bMbJcjOFAG NEWYTU^;Qb -N`D;WëcjOFA|o|SUA(D;=]`cj(|Σ| OXC %·cj|w| O;AK −A(Dik)/(|Σ| cMGj*A /0L − 1) @BA c+^;bJbtT^ vNEGjTZbtA 'NHbjbjC^ \TVD;]cjO³N¢c NED;WvcMO;N¢cJcjO;A±c N`GM]EA cJTUb cjOFA ';N`SUS B (o) L -JM/¸eFCHP G¸ALNEK OÜSUA cjcjA(G a P cjOFΣA\@B=ALN`{aGMDFA(, G. .NE.Ib , =ab }%¨CEGcMOFA¶K(CEGMGjALKpcMTVCHD×&C % a OFA(GMA j TUb±bt^0SK(TVA.DHcMS2ëSUNEGj]HA±cMC¶]HA c4N\K CHGjGMA.K cjTUCED O;CHbjASUA(D;]`cjO TZb ¶CHGjAcMO;N`D j O - D / P )| N`D;W×bj^FdFdRCHbjAhcjO;A@BALN`GMDFA(G]HA c bcjO;A0K CHGjGMA.K cjTUCED w G A0WF'A ;DFA %¨CEGcjO;A+btcjGMTUDFk] =m = (a | . . .(a -+ /!eFGM C NED;W P wAWYALWY^;K(A r O - /!g+OFA4@A.N`GMDFA.G ^;btAL&b $# 6
(203) aCHD ) wO N`D;W r TVD×kCHGMWFA(G|w| cMOC ;D;W o L: nD×C`cjO;A(G JCEG WFb P wAN`GMA &N 'FSUAcjCvSUA.N`GMDcMOFP A ';NESVSZb TVcjO r. UJx(}&{\ó'²|\ Uwx.}&{\ó'²|\. ?. a. !I ¸@BA c. j. j. r. j. j. j. j. j. j. a. a. n i=1. a. a. a. n i=1. a. a. ji i. 1. n. 1. n. ai. ai. 1. n. k. max r. 1. r. i. n i=1. ji i. 1. n. n. j i. ai. k. >WYA.DHcMT@RK(N¢cMTVCHD C%'cMOFAP';NESVSZb+^;bjTVDF] ö Ï ö sil4o(zis=o(¯YlM} .,a }.
(204) Í Ï ö sHl+q>l zHq>l mnlM{`}to }>ΣxL{ = {a , . x.. º¦ }>sHl+¯=o( B (o) (o, r) ûû jîù ←Ï 1; k ←Î ù 0 ú ù ½ ûû øi = ö 1 | n(a )| ≥ j ú ù j ← 2 · j ö ú ø ö û ö kú ←îù Ïk + | (a )| W û w ← ((a a . . . a ) ) Xû · ûû2roÏ ö ←←Î (kÏ −ñ =|w|)/(|Σ| ð R=ñ − ñ 1) (w, r) (o, r) B 1:. ! P<. HA. n. 1. r. j i. 1 2. j i. ai. n. k. eFCEGTUD;btcMNED;K A P K CHD;btTZWYA.GcjOFA;' NESVS WYA;DFALW\C
(205) FEA(G L (¸^FG&N`SU]ECHGjTVcjOT b} '³A.]ETUD;b 'vSVC=C =TUDF]V%¨CEGJcjOFAK(CEGMGjALKpcjTUCEBD³2b&(bb) C&% aj N`D³W bj TcMO ΣN¶=bt^ {a, SvK(TVA.DicjSNSUNEGj]HA L"G A ¶TU]EOichC';bjA(GZFEA (a) = * P (a2 ) = * P (a4 ) = aabb P (a8 ) = jabba P (b8) = bbbb. # C k = |abba| + |bbbb| = 2 + 4 = 6 L g+OFA(D (ab)6 = a. b. MR.
(206) . Le!TUD;N`SUS2 P P ¨CHMG G¶A cMTU^FD;GjbtD³cMNEb D;K A P tb C C *4rSU]E=C)LT(6DGj−A(cj^F4)/(2 − 1) = 2 GMD;b (bb, L 2). % $ # (abababababab) 6 = baba L# (baba, 2) bb % $ * : K& 5 0:K: 3C&165
(207) ?879$ &1 8 :9":;9=<.
(208) :;)7 0 12:5 $. $. +$. $. P
(209) 1 1 q <>M &1 1 ' 794C: : 7 q (Σ) 7 4 &#
(210) 1 <>
(211) !$ 794"7 $ >q() < O (|Σ| + GB
(212) 3T
(213) 165
(214) |o| + r) . & 1 7
(215) ")7 7+$ 7 : ? !I ¸g+OFA¸TZWYA.DHcMT@RN&'FTUSVTVc vTZbJK(SVALN`GL Q CED;K(A(GMDFTVD;]cjOFAK(C \dFSUA'&YTc P cMOFAK CHGjGMA.K cjTUCED;b2C&% -¨SUTVD;A.b6D¢a G /0L $ 6 cjOFA4btcjGMTVD;]Hb aj GMAMiJ ^FTUGMA.b O (|Σ| + log(|o| + r)) DFA(ALWFb O (|o| +i r) -¨SUTUDFA P /;L t u 4C`cMTUK(AvcjO³N¢ccMOFAëbjA c0C % NESVS2cjOFA ;' N`SUSUb P cMO;N¢c¶K CEDic N`TUD;b]EC=CYW';NESVSZbP'F^Yc\N`SZbtC,4"< : CEDFALb N7 r > 2|o| ¨% CEGTUD;btcMNED;K A P TUbD;C`c&dRCESNi D;C\ TUNESVSN TZWYA(DicjTN³NF' SVA TVcjO\CH^FG&N`SU]ECHGjTVcjOC btTUD;K A O (|Σ| + |o| + r) > O(2|o|) ¨% CHG4btC \ AC'% cMOFA +L. Br (o) 79$.
(216) . 7B9;< 7'12ÿF: ô<>ÿ
(217). 1 t<&7 9 t7 nD0cMOFTUb2btALKpcjTUCED P wA JCE^FSZWSUT2 HAJcjCbtO;C
(218) cjOFA4NHW"F¢N`DicMNE]EALbC%RCH^FGN`dFdFGMCHNHK O Lg+OFA(GMA;%¨CHGjA P wA \NHWYAwbjA FHA(G N`SHA'&Yd³A.GjTN \A(DicMbcjO³N¢cN`TN N`cbtcj^³"W =TVDF]±cjOFAJGMA.bjd³CHD;btALbBC %;CE^FG!N`SU]ECHGjTVcjOC %îNEK ALW TVcjO NED (¸GMNHK SUAXcjO;N`c K CH^FSZW '³A O=C^ vN`D L *±@ b wA8bMN`TZW TUD TVDicMGjCYWY^;K cjTUCED CE^;G N`SU]ECEGMTVcjOC -M/0btO;CE^FSZW vN& HA^;bjAC&%±NHPb %¨ A Ji^FA.GjTUA.b0NEbdRCHbMbj2T 'FSUA -btTUD;K(AvcjOFKA (¸P G NEK SUA ]EA c b Ji^FTZK iSN cjTUGjALW H /\N`D;W - D /\btO;CE^FSZWßDFC`c 'RA(SUTVMA FEA×^FD TZbtA.2S cjOFAÜNED;Ib wA.GMb¶OFAÜ]HA c b -îbjTVD;K(AhcjOFAMëK(NED+'³ANEdFdFGMC*&=TN vN¢cMM A /0L. % #. . ". M:9. . <. D. +. &. . . . 1. . P< 9 S . DJB. .. O. G A\O;N FHA¶bjA(A.D8cMO;N¢chcjO;A ';N`SUSUbhNEGjA¶TUWFA(DicjTN³N&';SVAE TVcjO NëSUTUDFA.NEGDi^T 'RA(GhC&% LBg+OFTUb UJx(}w{¶ó!²|\ \N 'RA±cMCiCV 0^;K OëTVDëN0O=^C vN`DëbjA cjcjTUDF]TL"4AMFEA(GjcjO;A(SUA.bMb P cjOFTZb+K C \dFSUA'&YTVc ¶K(CEGMGjALbtdRCED³WFb cjCcjO;A w CHGMbtcK(NEbjAL CÜCEGMA(C
(219) EF A.G P J AÜN`GMA ]HCETUDF] cjC bjOFC
(220) cMO;N¢cvcMOFTZb '³CH^FD;W TZbvG N`GMA(SN GjALNEK OFALWTUD d;GMNHKpcjTZK A L I g+OFA(C)L G P cjO;AD=^C V³' A.GC&% DFA.A.WYALWcMC\TUWYA.DicjT2%9 Br (o) TZb O (|Σ| + |o| + r) L # C P T% J A. & P cjOFAÜGMAMiJ ^FTUGMN`A.D;W WKD=v ^C N&H R' A A(G C&¢F % N`GZ $R' A btc O;J CEA(^FA(SZD W ³' AÜNED;]HW SVC ;' N`SUSN L K CED³b>c N`DicMBL g+OFA.|o|GjA ¨% CE+GMA rP J =AO1000 & eFCEGJA HF A(GZEF NESV^;A¸C% r P w A¸GMNED;WYΣC\ =SNv {a,K OFC=b}CHbjP A R R0K(A(DicjGMA.rb o N7 |o| = 10000 − r FL 1000 g+O;A(D w A ¨% A(ALW cjOFA(¸G NEK SUA TcMO cMOFA. R R ³' N`SUSUb Br (o) P N`D;WKJ AvK CE^;DHchcMOFA\Di^T R' A(GC&% ^;btALW$'>ë CE^FG4NESV]HCEGMTcMOC cjCvTZWYA.DHcMT %9ë cjOFAM +L nD×e!TV])L P J AbtOFC
(221) cjOFAN HF A(G N`]HA±D=^C R' A(G C&% cjO³N¢c J AD;A(A.W ¨% CEGALNEK O GMNHWYTU^;b L IJNHbtTZK(NESVSN P cjO;ADi^T R' A(GC&% TZbDFC`c&K(CED;btcMNEDHc F' ^Yc&]HGjC
(222) 4 b iJ ^;NEbjTVaSUTVD;A.N`GMSN R' A c J A(A.D G8R R NED;W R R RK TVcjO cjO;A8G NEWYTU^;bM2L g+OFAÜA&icMGjAM\ AÜd³CHTVDic bC&h % cMOFTUb K ^FGZEF AÜNEGjA×ALNEbI cMC ^B ;b>cMT %9 L EA.K.N`SUSEcMO;N¢c!CE^FG!N`SU]ECEGMTVcjOC ^;bjA.bcjOFA6¨% ^FD;KpcMTVCHD $ 6 ¢L IJA.bjTUWFA.b P N±K(NEGjA ¨% ^FSiGjALNEWYTUDF]±CY% cjOFAJdFGMC=C&F% C&R% g+OFA(C)*L G±btOFC
(223) 4 bcjO³N¢ccMOFA+K CHbtcC&Y% cjO;TUb ¨% ^FD;K cjTUCEDTUb'cjO;A A.bMbtA.DHcMTUNES³d³N`GjcJCB% cjOFAhK C \ dFSVA&YTc \ C&'% CE^FGN`SU]ECHGjTVcjOTF G TcMOFCE^Fc+TVc P J A¸K(CE^FSZWA&Yd³ALKpcN SVCH]HN`GMTVcjOC\ TZK¸K(C\ dFSUA'&YTc P cjO;N`c4TUb P O (|Σ| + log(|o| + r)) L $ # 6 A;D;WFb 'X bjK.N`DFD;TVDF]Nëb>cMGjTUDF] w ∈ B max (o) L I A D&GMCEd L)D P o rNED;W E N ; D W L + g F O . A j G A % ¨ E C M G A. ; O ( A D P % cMOFA rbtcjGM'TUDF1]Hb cMC |o|;D³'W 999 P CE$ D;S2 |w| = |o| + r = 1000 6 ^;btc\bMK(NEoD CED wN HF A(G N`]HA¶O³N`S2± % C&4 j c ; O ë A w.
(224) .
(225)
(226) . . .
(227) #CQ. 1050 1000. 1100 |u|+r=1000 CQ. 950. 1000 900. 900. 800 850. 700 600. #CQ. 800. 500 750. 400 300. 700. 200 100. 650. 0 1000 900800 700600 500400 300200 100 0 100. 600. r. 550. 200. 300. 400. 500. 600. 700. 800. 900 1000. |u|+r. 500 0. 100. 200. 300. 400. 500. 600. 700. 800. 900. 1000. Ò Ò UJyH~¯YlMq!x.º {Hl lp¹Hl ¹0}>xh¹HlM{H¾ }>ºV o0¯=o( B (o) Ø&sHl { |o| + r = 1000 ÀHx.q lpo.rts 2ØlÜr x.~ziyE}>lX}>silÜoplMqto( Ll x ljqpLh¯iro.m radius. . Ò ;Ò Uwyi~¯YljqXx.º {ilMlp¹Hlp¹ }>xß¹El {H¾ }>ºV oׯ=o( B (o) Ø&sHl { |o| + r = t Ø&}>s t ∈ [100; 1000] EÀix(q'l o.rts r .Ø!l2rMxL~ziyE}>l!}>sil o LlMqto. .l±x ljqpLh¯io.m . r. UJx(}&{\ó'²|\. . r. TVD;bjA(GjcjALW SUA cjcjA(G\TUD o L # C OFA TUb¶]ECHTVD;]×cjCbt^T'C \Tc ' 500 L (¸D cjOFAXC`cMOFA(G¶O;NED;W P OFA(D r ' 999 NED;W |o| ' 1 P NEb $ 6
(228) ^;btc+bI JNEd r SUA ctcMA(G bwcjC9;D;W o P OFA^³btALb ' 1000 L4C`cMTUK(A0cMO;N¢chTUD8cMOFA SZN¢cjcjA.GhK.NEbjA P cjOFAvK(^FGIFHA0C%we!TV])L ¶btO;C
(229) 4b¸N DiT^ 'RA(G±C % cMO;N¢cTUb¸N 'FTVc \CEGMAcjO³N`D R R R N&'RCE^Yc LFg+OFTZb¸WYTN?A(GMA(D;K(ATUb¸WY^FA cjCvcMOFA ;G btc¸b>c N`]HA.bC%!cMOFANESV]HCEGMTcMOC P TUD OFTZK O×P TUb±^;btALW 1012 cjOFASVCH]HNEGjTVcjOC \TZKhD=^C 'RA(G¸C&% L 4
(230) C P cMOFA8SUTVD;A.N`GMTVc &C %cMOFA8K(^FIG FHA P NEb JA(SUS¸NHbvcjO;A K O;NEDF]EA×TUD cjOFA btSUCEdRA OFA.D N`GMA ^;K O@ \CHGjAhWYTNSvK(^FSVccjC ^;btcjT2%9L % JAO³NEW@ wCHGI HA.W TVcjO×N¶SUNEGj]HA(G r ' |o| ' 500 P N`SUdFO;&N 'RA c Σ P cMOFA(D $# 6
(231) + JCEB ^;SUWXO; N FHPA vNEWYA ' 1000 · r/(r + 1) cjC0]HA c o LiIJ^Yc+TVDëCE^FG+btA(ctcMTVDF] P TVcjO N0bI \NESVSN`SUdFO;N'³A(c P cjOFA.GjAN`GMA4dFSUA(Dic E JN Yb&cjC ³D;W NHb&NbjC^ ';bjMA Ji^FA(D³K A&C % L nDK CED³btA JH^;A(D;K(A P CE^;G&GjCH^F]EVO %¨CEZG 0^;SUN 'RA.K(C \A.b GMCED;] P N`D³W o %¨CEGZ \NESVSN\NEbMbtALbjbjTVD;]hcjOFAwD=^C 'RA(GJC&% 8DFA.A.WYALW\cjCA'&=cjG NEK c o P cjO;N`c JCE^FSZW B ^;btcjT2%9\cjO;A K ^FZG FEA TUbJN`DvTUDicjGMTUK.N¢cMA4K C V'FTUD;N¢cMCEGMTUNESFdFGMC 'FSU A +L&G A¸K.N`DvCED;2S ¶DFC`cMAcjO³N¢c2cMOFTZb&D=C^ 'RA(G TUbC%cjA.P D@ ^;K O×b vN`SUSUA(GJcjO;NED 1000 · r/(r + 1) ' 1000 OFA.D 100 ≤ r ≤ 900 L *Jc2SUNHb>c ';^YcwDFCEc2SUA.NEbtc TVDvCHGMWYA.GcjCK OFA.K 0cjO;N`c&cjOF A '³A.O; N F=TVCH^FG2&N 'R
Documents relatifs
While upper bounds on wdim ε X are obtained by writing down explicit maps to a space of the proper dimension (these constructions use Hadamard matrices), lower bounds are found
Top: density profile; bottom: harmonicity as a function of harmonic number. 4.4 One sided
L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation
Although generic search engines (e.g. Google) may also have query suggestion schemes based on most users’ co-click frequency, they may not fully utilize
The class C TREE is attribute-efficiently learnable from equivalence and membership queries over arbitrary out- come pairs: any concept ≻ in this class can be identified in
In this paper we research two modifications of recurrent neural net- works – Long Short-Term Memory networks and networks with Gated Recurrent Unit with the addition of an
Precisely, we focused on schema-driven generation of complex and broad graph queries with user-defined features, on the complexity of the evaluation of regular simple path queries
Nous reviendrons successivement sur ces trois questions au plan théorique et illustrerons, à partir de la méthode IDEA (Indicateurs de Durabilité d’une Exploitation Agricole),
