Suffix Trees ind Suffix Arriys

Transcription

1 Suffix Tr id Suffix Arriy

2 Outli fr Tdiy Suffix Trf A impl diti tructur fr trig irchig. Suffix Tr A pwrful, id fxibl diti tructur fr trig ilgrithm. Suffix Array A cmpict iltritiv t uffix tr. Applfcatf f Suffix Tr ad Array Thr ir miy!

3 Rviw frm Lit Tim

4 Tri A trf i i tr thit tr i cllcti f trig vr m ilphibt Σ. Eich d crrpd t i prfx f m trig i th t. Tri ir mtim cilld prfix tr, ic ich d i i tri crrpd t i prfx f f th wrd i th tri. u t b t a b d g d g t

5 Ah-Criick Strig Mitchig Th Ah-Crafck trfg matchfg algrfthm i i ilgrithm fr fdig ill ccurrc f i t f trig P₁,, Pₖ iid i trig T. Rutim i O(), O(m + z), whr m = T, = P₁ + + Pₖ, id z i th umbr f mitch. Grit fr th ci whr th pittr ir fxd id th txt t irch chig.

6 Gmic Ditibi Miy trig ilgrithm th diy ir dvlpd fr r ud xtivly i cmputitiil gmic. Typicilly, w hiv i hug ditibi with miy vry lirg trig (gm) thit w'll prprc t pd up futur priti. Cmm prblm: giv i fxd trig T t irch id chigig pittr P₁,, Pₖ, fd ill mitch f th pittr i T. Qutf: Ci w itid prprc T t mik it iy t irch fr viriibl pittr?

7 Suffix Tri

8 Subtrig, Prfx, id Suffix Uful Fact : Giv i tri trig i t f trig S₁, S₂,, Sₖ, it' pibl t dtrmi, i tim O( Q ), whthr i qury trig Q i i prfx f iy Sᵢ. Uful Fact : A trig P i i ubtrig f a r t i trig T if id ly if T i i prfx f m uffix f P. a r Spcifcilly, writ T = αpω; th T i i prfx f th uffix Pω f T. a r

9 T Subtrig, Prfx, id Suffix Uful Fact : Giv i tri trig i t f trig S₁, S₂,, Sₖ, it' pibl t dtrmi, i tim O( Q ), whthr i qury trig Q i i prfx f iy Sᵢ. Uful Fact : A trig P i i ubtrig f i trig T if id ly if P i i prfx f m uffix f T. Spcifcilly, writ T = αpω; th P i i prfx f th uffix Pω f T. α P ω

10 Suffix Tri A uffix trf f T i i tri f ill th uffix f T. Giv iy pittr trig P, w ci chck i tim O( P ) whthr P i i ubtrig f T by ig whthr P i i prfx i T' uffix tri. (Thi chck whthr P i i prfx f m uffix f T.)

11 Suffix Tri A uffix trf f T i i tri f ill th uffix f T. Mr grilly, giv iy mpty pittr P₁,, Pₖ f ttil lgth, w ci dtct hw miy f th pittr ir ubtrig f T i tim O(). (Fidig ill mitch i i bit trickir; mr thit litr.)

12 A Typicil Trifrm Appd m w chirictr Σ t th d f T, th ctruct th tri fr T. Th w chirictr lxicgriphicilly prcd ill thr chirictr. Thi i uuilly cilld th tfl; thik f it lik th Thrylid vri f i ull trmiitr. Lif d crrpd t uffix. Itril d crrpd t prfx f th uffix.

13 Ctructig Suffix Tri Oc w build i igl uffix tri fr trig T, w ci fficitly dtct whthr pittr mitch i tim O(). Qutf: Hw lg d it tik t ctruct i uffix tri? Prblm: Thr' i Ω(m ) lwr bud th wrt-ci cmplxity f ay ilgrithm fr buildig uffix tri.

14 A Dgrit Ci a b a b b a b b b b b a m b m b b b b b b b

15 A Dgrit Ci a b a b b a b b b b b a m b m b b b b b b b Thr Thr ir ir Θ(m) Θ(m) cpi cpi f f d d chiid chiid tgthr tgthr i i b m m.. Spic Spic uig: uig: Θ(m Θ(m ). ).

16 A Dgrit Ci a b a b b a b b b b b a m b m b b b b b b b Thr Thr ir ir Θ(m) Θ(m) cpi cpi f f d d chiid chiid tgthr tgthr i i b m m.. Spic Spic uig: uig: Ω(m Ω(m ). ).

17 Crrctig th Prblm Bciu uffix tri miy hiv Ω(m ) d, ill uffix tri ilgrithm mut ru i tim Ω(m ) i th wrt-ci. Ci w rduc th umbr f d i th tri?

18 Pitricii Tri A illy d i i tri i i d thit hi xictly child. A Patrfcfa trf (r radfix trf) i i tri whr ill illy d ir mrgd with thir pirt.

19 Pitricii Tri A illy d i i tri i i d thit hi xictly child. A Patrfcfa trf (r radfix trf) i i tri whr ill illy d ir mrgd with thir pirt. 567

20 Suffix Tr A uffix tr fr i trig T i i Pitricii tri f T whr ich lif i libld with th idx whr th crrpdig uffix tirt i T. (Nt thit uffix tr ir t th im i uffix tri. T th bt f my kwldg, uffix tri ir t ud iywhr.) 567

21 Suffix Tr A uffix tr fr i trig T i i Pitricii tri f T whr ich lif i libld with th idx whr th crrpdig uffix tirt i T. (Nt thit uffix tr ir t th im i uffix tri. T th bt f my kwldg, uffix tri ir t ud iywhr.) 567

22 Suffix Tr A uffix tr fr i trig T i i Pitricii tri f T whr ich lif i libld with th idx whr th 7 crrpdig uffix tirt i T. (Nt thit uffix tr ir t th im i uffix tri. T th bt f my kwldg, uffix tri ir t ud iywhr.)

23 Prprti f Suffix Tr If T = m, th uffix tr hi xictly m + lif d. Fr iy T ε, ill itril d i th uffix tr hiv it lit tw childr. Numbr f d i i uffix tr i Θ(m)

24 Suffix Tr Rprtiti Suffix tr miy hiv Θ(m) d, but th libl th dg ci hiv iz ω(). Thi mi thit i iïv rprtiti f i uffix tr miy tik ω(m) pic. Uful fact: Eich dg i i uffix tr i libld with i ccutiv rig f chirictr frm w. Trfck: Rprt ich dg libld with i trig α i i piir f itgr [tirt, d] rprtig whr i th trig α ippir.

25 Suffix Tr Rprtiti

26 Suffix Tr Rprtiti

27 Suffix Tr Rprtiti tart d child

28 Buildig Suffix Tr Clafm: It pibl t build i uffix tr fr i trig f lgth m i tim Θ(m). Th algrithm ar t trivial! W'll dicu f thm xt tim.

29 Applfcatf: Strig Sirch

30 Strig Mitchig Supp w prprc i trig T by buildig i uffix tr fr it. Giv iy pittr trig P f lgth, w ci dtrmi, i tim O(), whthr i i ubtrig f P by lkig it up i th uffix tr

31 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

32 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch. Obrvatf : : Evry Evry ccurrc f f P i i T i i i prfx prfx f f m m uffix uffix f f T. T

33 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch. Obrvatf : : Evry Evry uffix uffix f f T T bgiig with with m m pittr pittr P ippir ippir i i th th ubtr ubtr fud fud by by irchig irchig fr fr P. P

34 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

35 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

36 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

37 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

38 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

39 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

40 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

41 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

42 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

43 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

44 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

45 Strig Mitchig Clafm: Aftr pdig O(m) tim prprcig T, ci fd all mitch f i trig P i tim O( + z), whr z i th umbr f mitch

46 Fidig All Mitch T fd ill mitch f trig P, tirt by irchig th tr fr P. If th irch fill ff th tr, rprt mitch. Othrwi, lt v b th d it which th irch tp, r th dpit f th dg whr it tp if it d i th middl f i dg. D i DFS id rprt th umbr f ill th liv fud i thi ubtr. Th idic rprtd thi wiy giv bick ill piti it which P ccur.

47 Fidig All Mitch T fd ill mitch f trig P, tirt by irchig th tr fr P. If th irch fill ff th tr, rprt mitch. Othrwi, lt v b th d it which th irch tp, r th dpit f th dg whr it tp if it d i th middl f i dg. D i DFS id rprt th umbr f ill th liv fud i thi ubtr. Th idic rprtd thi wiy giv bick ill piti it which P ccur.

48 Fidig All Mitch T fd ill mitch f trig P, tirt by irchig th tr fr P. If th irch fill ff th tr, rprt mitch. Othrwi, lt v b th d it which th irch tp, r th dpit f th dg whr it tp if it d i th middl f i dg. D i DFS id rprt th umbr f ill th liv fud i thi ubtr. Th idic rprtd thi wiy giv bick ill piti it which P ccur. Hw Hw fit fit i i thi thi tp? tp?

49 Clafm: Th DFS t fd ill liv i th ubtr crrpdig t prfx P tik tim O(z), whr z i th umbr f mitch. Prf: If th DFS rprt z mitch, it mut hiv viitd z diffrt lif d. Sic ich itril d f i uffix tr hi it lit tw childr, th ttil umbr f itril d viitd durig th DFS i it mt z. Durig th DFS, w d't d t ictuilly mitch th chirictr th dg. W jut fllw th dg, which tik tim O(). Thrfr, th DFS viit it mt O(z) d id dg id pd O() tim pr d r dg, th ttil rutim i O(z).

50 Rvr Ah-Criick Giv pittr P₁, Pₖ f ttil lgth, uffix tr ci fd ill mitch f th pittr i tim O(m + + z). Build th tr i tim O(m), th irch fr ill mitch f ich Pᵢ; ttil tim icr ill irch i O( + z). Act i i rvr Ah-Criick: Ah-Criick trig mitchig ru i tim O(), O(m+z) Suffix tr trig mitchig ru i tim O(m), O(+z)

51 Athr Applfcatf: Lgt Rpitd Subtrig

52 Lgt Rpitd Subtrig Cidr th fllwig prblm: Giv i trig T, fd th lgt ubtrig w f T thit ippir i it lit tw diffrt piti. Sm ximpl: I m, th lgt rpitd ubtrig i. I baaa, th lgt rpitd ubtrig i aa. (Th ubtrig ci vrlip.) Appliciti t cmputitiil bilgy: mr thi hilf f th humi gm i frmd frm rpitd DNA quc!

53 Lgt Rpitd Subtrig

54 Lgt Rpitd Subtrig Obrvatf : : If If w i i i rpitd rpitd ubtrig ubtrig f f T, T, it it mut mut b b i prfx prfx f f it it lit lit tw tw diffrt diffrt uffix. uffix. 567

55 Lgt Rpitd Subtrig Obrvatf : : If If w i i i rpitd rpitd ubtrig ubtrig f f T, T, it it mut mut crrpd t t i prfx prfx f f i pith pith t t i i itril itril d. d. 567

56 Lgt Rpitd Subtrig Obrvatf Obrvatf : : If If w i i i lgt lgt rpitd rpitd ubtrig, ubtrig, it it crrpd crrpd t t i full full pith pith t t i i itril itril d. d. 567

57 Lgt Rpitd Subtrig Obrvatf Obrvatf : : If If w i i i lgt lgt rpitd rpitd ubtrig, ubtrig, it it crrpd crrpd t t i full full pith pith t t i i itril itril d. d. 567

58 Lgt Rpitd Subtrig Fr ich d v i i uffix tr, lt (v) b th trig thit it crrpd t. Th trfg dpth f i d v i dfd i (v), th lgth f th trig v crrpd t. Th lgt rpitd ubtrig i T ci b fud by fdig th itril d i T with th miximum trig dpth.

59 Lgt Rpitd Subtrig Hr' i O(m)-tim ilgrithm fr lvig th lgt rpitd ubtrig prblm: Build th uffix tr fr T i tim O(m). Ru i DFS vr th uffix tr, trickig th trig dpth i yu g, t fd th itril d f miximum trig dpth. Rcvr th trig thit d crrpd t. Gd ixrcf: Hw might yu fd th lgt ubtrig f T thit rpit it lit k tim?

60 Challg Prblm: Slv thi prblm i liir tim withut uig uffix tr (r uffix irriy).

61 Tim-Out fr Aucmt!

62 Prblm St Prblm St luti will b up th cur wbit litr tdiy. W ll try t gt it gridd id rturd i i pibl. Prblm St i du Tudiy it :PM. Stp by ffic hur with quti! Ak quti Piizzi!

63

64 Bick t CS66!

65 Grilizd Suffix Tr

66 Suffix Tr fr Multipl Strig Suffix tr tr ifrmiti ibut i igl trig id xprt i hug imut f tructuril ifrmiti ibut thit trig. Hwvr, miy ippliciti rquir ifrmiti ibut th tructur f multipl diffrt trig.

67 Grilizd Suffix Tr A gralfzd uffix tr fr T₁,, Tₖ i i Pitricii tri f ill uffix f T₁₁,, Tₖₖ. Eich Tᵢ hi i uiqu d mirkr. Liv ir tiggd with i j, miig ith uffix f trig T j ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₂ ₁ ₂ f ₂ ₁ 567₁ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₂ ₁ ₁ f f ₂ ₁ ₁ ₂ ff₂ 567₂ ₁ 6₁ ₂ 5₂ ₁ ₁

68 Grilizd Suffix Tr Clafm: A grilizd uffix tr fr trig T₁,, Tₖ f ttil lgth m ci b ctructd i tim Θ(m). U i tw-phi ilgrithm: Ctruct i uffix tr fr th igl trig T₁₁T₂₂ Tₖₖ i tim Θ(m). Thi will d up with m ivilid uffix. D i DFS vr th uffix tr id pru th ivilid uffix. Ru i tim O(m) if implmtd itlligtly.

69 Appliciti f Grilizd Suffix Tr

70 Lgt Cmm Subtrig Cidr th fllwig prblm: Giv tw trig T₁ id T₂, fd th lgt trig w thit i i ubtrig f bth T₁ id T₂. Ci lv i tim O( T₁ T₂ ) uig dyimic prgrimmig. Ci w d bttr?

71 Lgt Cmm Subtrig ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₁ 567₁ ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₂ ₁ ₁ ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567₂ ₁

72 Lgt Cmm Subtrig ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₁ 567 ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₂ ₁ ₁ ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567₂ ₁

73 Lgt Cmm Subtrig ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₁ 567 ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₂ ₁ ₁ ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567₂ ₁

74 Lgt Cmm Subtrig ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₁ 567 ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₂ ₁ ₁ ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567₂ ₁

75 Lgt Cmm Subtrig ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₁ 567 ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₂ ₁ ₁ ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567₂ ₁

76 Lgt Cmm Subtrig ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₁ 567 ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₂ ₁ ₁ ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567 ₁

77 Lgt Cmm Subtrig ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₁ 567 ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₂ ₁ ₁ ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567 ₁

78 Lgt Cmm Subtrig ₁ ₂ ₁ ₁ ₂ 7₂ 7₁ 6₂ ₁ ₁ 567 ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₂ ₁ ₁ ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567 ₁

79 Lgt Cmm Subtrig ₁ ₁ ₂ 7₂ ₁ 7₁ ₂ 6₂ ₁ ₁ 567 ₁ ₂ ₂ f ₂ f ₂ ₂ ₂ ₁ ₁ ₁ ₂ 5₁ ₂ ₁ ₁ ₁ Obrvatf: Obrvatf: Ay Ay cmm cmm ubtrig ubtrig f f T₁ T₁ id id T₂ T₂ will will b b i prfx prfx f f i uffix uffix f f T₁ T₁ id id i prfx prfx f f i uffix uffix f f T₂. T₂. ₁ f f ₂ ₂ ₁ 6₁ ₂ 5₂ ₁ ff₂ 567 ₁

80 Lgt Cmm Subtrig Build i grilizd uffix tr fr T₁ id T₂ i tim O(m). Atit ich itril d i th tr with whthr thit d hi it lit lif d frm ich f T₁ id T₂. Tik tim O(m) uig DFS. Ru i DFS vr th tr t fd th mirkd d with th hight trig dpth. Tik tim O(m) uig DFS Ovrill tim: O(m).

81 Sufix Tr: Th Catch

82 Spac Uag Sufix tr ar mmrry hrg. Suppr Σ = {A, C, G, T, }. Each itral rd d 5 machi wrrd: frr ach charactr, wrrd frr th tart/d idx ad a child pritr. tart d child A C T G Thi i till O(m), but it' a hug hidd crtat!

83 Ca w gt th fxibility rf a ufix tr withrut th mmrry crt?

84 Y kida!

85 Sufix Array A ufix array frr a trig T i a array rf th ufix rf T, trrd i rrtd rrdr. By crvtir, prcd all rthr charactr

86 Sufix Array A ufix array frr a trig T i a array rf th ufix rf T, trrd i rrtd rrdr. By crvtir, prcd all rthr charactr

87 Rprtig Sufix Array Sufix array ar typically rprtd implicitly by jut trrig th idic rf th ufix i rrtd rrdr rathr tha th ufix thmlv. Spac rquird: Θ(m). Mrr prcily, pac frr T, plu r xtra wrrd frr ach charactr

88 Rprtig Sufix Array Sufix array ar typically rprtd implicitly by jut trrig th idic rf th ufix i rrtd rrdr rathr tha th ufix thmlv. Spac rquird: Θ(m). Mrr prcily, pac frr T, plu r xtra wrrd frr ach charactr

89 Sarchig a Sufix Array Rcall: P i a ubtrig rf T if it' a prfx rf a ufix rf T. All match rf P i T hav a crmmr prfx, r thy'll b trrd crcutivly. Ca fd all match rf P i T by drig a biary arch rvr th ufix array

90 Aalyzig th Rutim Th biary arch will rquir O(lrg m) prrb itr th ufix array. Each crmparir tak tim O(): hav tr crmpar P agait th currt ufix. Tim frr biary archig: O( lrg m). Tim tr rprrt all match aftr that prit: O(z). Trtal tim: O( lg m + z).

91 Why th Slrwdrw?

92 A Lr rf Structur May algrrithm r ufix tr ivrlv lrrkig frr itral rd with variru prrprti: Lrgt rpatd ubtrig: itral rd with largt trig dpth. Lrgt crmmr ubtrig: itral rd with largt trig dpth that ha a child frrm ach trig. Bcau ufix array dr rt trr th tr tructur, w lr acc tr thi ifrrmatir.

93 Sufix Tr ad Sufix Array

94 Sufix Tr ad Sufix Array

95 Sufix Tr ad Sufix Array

96 Sufix Tr ad Sufix Array

97 Sufix Tr ad Sufix Array

98 Th lrgt crmmr prfx rf a rag rf trig i a ufix array crrrprd tr th lrwt crmmr actrr rf thr ufix i th ufix tr.

99 Lrgt Crmmr Prfx Giv twr trig x ad y, th lgt cmm prfx rr (LCP) rf x ad y i th lrgt prfx rf x that i alr a prfx rf y. Th LCP rf x ad y i drtd lcp(x, y). Fu fact: Thr i a O(m)-tim algrrithm frr crmputig LCP ifrrmatir r a ufix array. Lt' hrw it wrrk.

100 Pairwi LCP Fact: Thr i a algrrithm (du tr Kaai t al.) that crtruct, i tim O(m), a array rf th LCP rf adjact ufix array tri. Th algrrithm i't that crmplx, but th crrrct argumt i a bit rtrivial

101 Pairwi LCP Claim: Thi ifrrmatir i rugh frr u tr fgur rut th lrgt crmmr prfx rf a rag rf lmt i th ufix array

102 Pairwi LCP Claim: Thi ifrrmatir i rugh frr u tr fgur rut th lrgt crmmr prfx rf a rag rf lmt i th ufix array

103 Pairwi LCP Claim: Thi ifrrmatir i rugh frr u tr fgur rut th lrgt crmmr prfx rf a rag rf lmt i th ufix array. Hy, Hy, lk! lk! It It a a rag rag miimum miimum qury qury prblm! prblm! 7 5 6

104 Crmputig LCP Tr prprrc a ufix array tr upprrt O() LCP quri: U Kaai' O(m)-tim algrrithm tr build th LCP array. Build a RMQ tructur rvr that array i tim O(m) uig Fichr-Hu. U th prcrmputd RMQ tructur tr awr LCP quri rvr rag. Rquir O(m) prprrcig tim ad rly O() qury tim.

105 Sarchig a Sufix Array Rcall: Ca arch a ufix array rf T frr all match rf a pattr P i tim O( lrg m + z). If w'v dr O(m) prprrcig tr build th LCP ifrrmatir, w ca pd thi up.

106 Sarchig a Sufix Array

107 Sarchig a Sufix Array

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112 Sarchig a Sufix Array

113 Sarchig a Sufix Array

114 Sarchig a Sufix Array

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117 Sarchig a Sufix Array

118 Sarchig a Sufix Array

119 Sarchig a Sufix Array

120 Sarchig a Sufix Array

121 Sarchig a Sufix Array

122 Sarchig a Sufix Array

123 Sarchig a Sufix Array

124 Sarchig a Sufix Array

125 Sarchig a Sufix Array Ituitivly, imulat drig a biary arch rf th lav rf a ufix tr, rmmbrig th dpt ubtr yru'v matchd r far. At ach prit, if th biary arch prrb a laf rutid rf th currt ubtr, kip it ad crtiu th biary arch i th dirctir rf th currt ubtr. Tr implmt thi r a actual ufix array, w u LCP ifrrmatir tr implicitly kp track rf whr th brud r th currt ubtr ar.

126 Sarchig a Sufix Array Claim: Th algrrithm w jut ktchd ru i tim O( + lrg m + z). Prf Sktch: Th O(lrg m) trm crm frrm th biary arch rvr th lav rf th ufix tr. Th O() trm crrrprd tr dcdig dpr itr th ufix tr r charactr at a tim. Fially, w hav tr pd O(z) tim rprrtig match.

127 Applicatir: Lgt Cmm Exti

128 Lrgt Crmmr Extir Giv twr trig T₁ ad T₂ ad tart pritir i ad j, th lgt cmm xti rf T₁ ad T₂, tartig at pritir i ad j, i th lgth rf th lrgt trig w that appar at pritir i i T₁ ad pritir j i T₂. W'll drt thi valu by LCET₁, T₂ (i, j). Typically, T₁ ad T₂ ar fxd ad multipl (i, j) quri ar pcifd. f f

129 Lrgt Crmmr Extir Giv twr trig T₁ ad T₂ ad tart pritir i ad j, th lgt cmm xti rf T₁ ad T₂, tartig at pritir i ad j, i th lgth rf th lrgt trig w that appar at pritir i i T₁ ad pritir j i T₂. W'll drt thi valu by LCET₁, T₂ (i, j). Typically, T₁ ad T₂ ar fxd ad multipl (i, j) quri ar pcifd. f f

130 Lrgt Crmmr Extir Giv twr trig T₁ ad T₂ ad tart pritir i ad j, th lgt cmm xti rf T₁ ad T₂, tartig at pritir i ad j, i th lgth rf th lrgt trig w that appar at pritir i i T₁ ad pritir j i T₂. W'll drt thi valu by LCET₁, T₂ (i, j). Typically, T₁ ad T₂ ar fxd ad multipl (i, j) quri ar pcifd. f f

131 Lrgt Crmmr Extir Giv twr trig T₁ ad T₂ ad tart pritir i ad j, th lgt cmm xti rf T₁ ad T₂, tartig at pritir i ad j, i th lgth rf th lrgt trig w that appar at pritir i i T₁ ad pritir j i T₂. W'll drt thi valu by LCET₁, T₂ (i, j). Typically, T₁ ad T₂ ar fxd ad multipl (i, j) quri ar pcifd. f f

132 Lrgt Crmmr Extir Giv twr trig T₁ ad T₂ ad tart pritir i ad j, th lgt cmm xti rf T₁ ad T₂, tartig at pritir i ad j, i th lgth rf th lrgt trig w that appar at pritir i i T₁ ad pritir j i T₂. W'll drt thi valu by LCET₁, T₂ (i, j). Typically, T₁ ad T₂ ar fxd ad multipl (i, j) quri ar pcifd. f f

133 Lrgt Crmmr Extir Giv twr trig T₁ ad T₂ ad tart pritir i ad j, th lgt cmm xti rf T₁ ad T₂, tartig at pritir i ad j, i th lgth rf th lrgt trig w that appar at pritir i i T₁ ad pritir j i T₂. W'll drt thi valu by LCET₁, T₂ (i, j). Typically, T₁ ad T₂ ar fxd ad multipl (i, j) quri ar pcifd. f f

134 Lrgt Crmmr Extir Obrvati: LCET₁, T₂ (i, j) i th lgth rf th lrgt crmmr prfx rf th ufix rf T₁ ad T₂ tartig at pritir i ad j. f f

135 Lrgt Crmmr Extir Obrvati: LCET₁, T₂ (i, j) i th lgth rf th lrgt crmmr prfx rf th ufix rf T₁ ad T₂ tartig at pritir i ad j.

136 Lrgt Crmmr Extir Obrvati: LCET₁, T₂ (i, j) i th lgth rf th lrgt crmmr prfx rf th ufix rf T₁ ad T₂ tartig at pritir i ad j.

137 Lrgt Crmmr Extir Obrvati: LCET₁, T₂ (i, j) i th lgth rf th lrgt crmmr prfx rf th ufix rf T₁ ad T₂ tartig at pritir i ad j.

138 Sufix Array ad LCE Claim: Thr i a O(m), O() data tructur frr LCE. Prprrcig: Crtruct a gralizd ufix array frr T₁ ad T₂ augmtd with LCP ifrrmatir. (Jut build a ufix array frr T₁₁T₂₂.) Th build a tabl mappig ach idx i th trig tr it idx i th ufix array. Qury: Dr a RMQ rvr th LCP array at th apprrpriat idic. ₁ 5₂ 7₁ ₂ ₁ ₂ ₁ 5₁ ₂ ₁ ₁ 6₁ ₂ ₁ ₂ ₁ ₂ ₁ ₂ ₁ ₂ ₁ ₁ ₂ ₁ ₁ ₁ ₂ ₁ t₂ ₂ t₂

139 Sufix Array ad LCE Claim: Thr i a O(m), O() data tructur frr LCE. Prprrcig: Crtruct a gralizd ufix array frr T₁ ad T₂ augmtd with LCP ifrrmatir. (Jut build a ufix array frr T₁₁T₂₂.) Th build a tabl mappig ach idx i th trig tr it idx i th ufix array. Qury: Dr a RMQ rvr th LCP array at th apprrpriat idic. ₁ 5₂ 7₁ ₂ ₁ ₂ ₁ 5₁ ₂ ₁ ₁ 6₁ ₂ ₁ ₂ ₁ ₂ ₁ ₂ ₁ ₂ ₁ ₁ ₂ ₁ ₁ ₁ ₂ ₁ t₂ ₂ t₂

140 Sufix Array ad LCE Claim: Thr i a O(m), O() data tructur frr LCE. Prprrcig: Crtruct a gralizd ufix array frr T₁ ad T₂ augmtd with LCP ifrrmatir. (Jut build a ufix array frr T₁₁T₂₂.) Th build a tabl mappig ach idx i th trig tr it idx i th ufix array. Qury: Dr a RMQ rvr th LCP array at th apprrpriat idic. ₁ 5₂ 7₁ ₂ ₁ ₂ ₁ 5₁ ₂ ₁ ₁ 6₁ ₂ ₁ ₂ ₁ ₂ ₁ ₂ ₁ ₂ ₁ ₁ ₂ ₁ ₁ ₁ ₂ ₁ t₂ ₂ t₂

141 Sufix Array ad LCE Claim: Thr i a O(m), O() data tructur frr LCE. Prprrcig: Crtruct a gralizd ufix array frr T₁ ad T₂ augmtd with LCP ifrrmatir. (Jut build a ufix array frr T₁₁T₂₂.) Th build a tabl mappig ach idx i th trig tr it idx i th ufix array. Qury: Dr a RMQ rvr th LCP array at th apprrpriat idic. ₁ 5₂ 7₁ ₂ ₁ ₂ ₁ 5₁ ₂ ₁ ₁ 6₁ ₂ ₁ ₂ ₁ ₂ ₁ ₂ ₁ ₂ ₁ ₁ ₂ ₁ ₁ ₁ ₂ ₁ t₂ ₂ t₂

142 A Applicati: Lrgt Palidrrmic Subtrig

143 Palidrrm A palidrm i a trig that' th am frrward ad backward. A palidrmic ubtrig rf a trig T i a ubtrig rf T that' a palidrrm. Surpriigly, rf grat imprrtac i crmputatiral birlrgy. A C U G C A G U

144 Palidrrm A palidrm i a trig that' th am frrward ad backward. A palidrmic ubtrig rf a trig T i a ubtrig rf T that' a palidrrm. Surpriigly, rf grat imprrtac i crmputatiral birlrgy. A C U G C A G U

145 Palidrrm A palidrm i a trig that' th am frrward ad backward. A palidrmic ubtrig rf a trig T i a ubtrig rf T that' a palidrrm. Surpriigly, rf grat imprrtac i crmputatiral birlrgy. A C U G U G A C

146 Palidrrm A palidrm i a trig that' th am frrward ad backward. A palidrmic ubtrig rf a trig T i a ubtrig rf T that' a palidrrm. Surpriigly, rf grat imprrtac i crmputatiral birlrgy. A C U G U G A C

147 Lrgt Palidrrmic Subtrig Th lgt palidrmic ubtrig prrblm i th frllrwig: Giv a trig T, fd th lrgt ubtrig rf T that i a palidrrm. Hrw might w rlv thi prrblm?

148 A Iitial Ida Tr dal with th iu rf trig grig frrward ad backward, tart rf by frrmig T ad T R, th rvr rf T. Iitial Ida: Fid th lrgt crmmr ubtrig rf T ad T R. Ufrrtuatly, thi dr't wrrk: T = abcdba T R = abdcba Lrgt crmmr ubtrig: ab / ba Lrgt palidrrmic ubtrig: a / b / c / d

149 Palidrrm Ctr ad Radii Frr rw, lt' frcu r v-lgth palidrrm. A v-lgth palidrrm ubtrig ww R rf a trig T ha a ctr ad radiu: Ctr: Th prt btw th duplicatd ctr charactr. Radiu: Th lgth rf th trig grig rut i ach dirctir. Ida: Frr ach ctr, fd th largt crrrprdig radiu.

150 Palidrrm Ctr ad Radii a b b a c c a b c c b

151 Palidrrm Ctr ad Radii a b b a c c a b c c b

152 Palidrrm Ctr ad Radii a b b a c c a b c c b

153 Palidrrm Ctr ad Radii a b b a c c a b c c b

154 Palidrrm Ctr ad Radii w a b b a c c a b c c b

155 Palidrrm Ctr ad Radii w a b b a c c a b c c b w R b c c b a c c a b b a

156 Palidrrm Ctr ad Radii w a b b a c c a b c c b w R b c c b a c c a b b a

157 Palidrrm Ctr ad Radii w a b b a c c a b c c b w R b c c b a c c a b b a

158 Palidrrm Ctr ad Radii w a b b a c c a b c c b w R b c c b a c c a b b a

159 Palidrrm Ctr ad Radii w a b b a c c a b c c b w R b c c b a c c a b b a

160 Palidrrm Ctr ad Radii w a b b a c c a b c c b w R b c c b a c c a b b a

161 Palidrrm Ctr ad Radii w a b b a c c a b c c b w R b c c b a c c a b b a

162 Palidrrm Ctr ad Radii w a b b a c c a b c c b w R b c c b a c c a b b a

163 A Algrrithm I tim O(m), crtruct T R. Prprrc T ad T R i tim O(m) tr upprrt LCE quri. Frr ach prt btw twr charactr i T, fd th lrgt palidrrm ctrd at that lrcatir by xcutig LCE quri r th crrrprdig lrcatir i T ad T R. Each qury tak tim O() if it jut rprrt th lgth. Trtal tim: O(m). Rprrt th lrgt trig frud thi way. Trtal tim: O(m).

164 Nxt Tim Ctructig Sufix Tr Hrw r arth dr yru build ufix tr i tim O(m)? Ctructig Sufix Array Start by buildig ufix array i tim O(m)... Ctructig LCP Array ad addig i LCP array i tim O(m).