Outline for Today. A space-efficient data structure for substring searching. Converting from suffix arrays to suffix trees.
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- Hugo Marsh
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1 Suffix Array
2 Outli for Today Rviw from Lat Tim Suffix Array A urpriigly hlpful auxiliary tructur. Cotructig Suffix Tr A pac-fficit data tructur for ubtrig archig. LCP Array Quick rviw of uffix tr. Covrtig from uffix array to uffix tr. Cotructig Suffix Array A xtrmly clvr algorithm for buildig uffix array.
3 Rviw from Lat Tim
4 Suffix Tr A uffix tr for a trig T i a Patricia tri of T 8 whr ach laf i labld with th 7 idx whr th corrpodig uffix tart i T. 4 o 5 0 o 6 o
5 Suffix Tr If T = m, th uffix tr ha xactly m + laf od. For ay T ε, all itral od i th uffix tr hav at lat two childr. Numbr of od i a uffix tr i Θ(m) o 5 0 o 6 o
6 Spac Uag Suffix tr ar mmory hog. Suppo Σ = {A, C, G, T, }. Each itral od d 5 machi word: for ach charactr, w d thr word for th tart/d idx of th labl ad for a child poitr. Thi i till O(m), but it' a hug hidd cotat.
7 Suffix Array
8 Suffix Array A uffix array for a trig T i a array of th uffix of T, tord i ortd ordr. By covtio, prcd all othr charactr o o
9 Rprtig Suffix Array Suffix array ar typically rprtd implicitly by jut torig th idic of th uffix i ortd ordr rathr tha th uffix thmlv. Spac rquird: Θ(m). Mor prcily, pac for T, plu o xtra word for ach charactr o
10 Sarchig a Suffix Array Rcall: P i a ubtrig of T iff it' a prfix of a uffix of T. All match of P i T hav a commo prfix, o thy'll b tord cocutivly. Ca fid all match of P i T by doig a biary arch ovr th uffix array o o
11 Aalyzig th Rutim Th biary arch will rquir O(log m) prob ito th uffix array. Each compario tak tim O(): hav to compar P agait th currt uffix. Tim for biary archig: O( log m). Tim to rport all match aftr that poit: O(z). Total tim: O( log m + z).
12 Why th Slowdow?
13 A Lo of Structur May algorithm o uffix tr ivolv lookig for itral od with variou proprti: Logt rpatd ubtrig: itral od with largt trig dpth. Logt commo xtio: lowt commo actor of two od. Bcau uffix array do ot tor th tr tructur, w lo acc to thi iformatio.
14 Suffix Tr ad Suffix Array o 5 0 o 6 o o o Nifty NiftyFact: Fact:Th Thuffix uffixarray arrayca ca b bcotructd cotructdfrom froma aordrd ordrd DFS DFSovr ovraauffix uffixtr! tr!
15 Suffix Tr ad Suffix Array o 5 0 o 6 o o o
16 Suffix Tr ad Suffix Array o 5 0 o 6 o o o Nifty NiftyFact: Fact:Adjact Adjacttrig trigwith with aacommo commoprfix prfixcorrpod corrpodto to ubtr ubtri ith thuffix uffixtr. tr.
17 Logt Commo Prfix Giv two trig x ad y, th logt commo prfix or (LCP) of x ad y i th logt prfix of x that i alo a prfix of y. Th LCP of x ad y i dotd lcp(x, y). LCP iformatio i fudamtally importat for uffix array. With it, w ca implicitly rcovr much of th tructur prt i uffix tr.
18 Suffix Tr ad Suffix Array o 5 0 o 6 o o o Nifty NiftyFact: Fact:Th Thlowt lowtcommo commo actor actorof ofuffix uffixxxad adyyha ha trig triglabl lablgiv givby bylcp(x, lcp(x,y). y).
19 Computig LCP Iformatio Claim: Thr i a O(m)-tim algorithm for computig LCP iformatio o a uffix array. Lt' how it work.
20 Pairwi LCP Fact: Thr i a algorithm (du to Kaai t al.) that cotruct, i tim O(m), a array of th LCP of adjact uffix array tri. Th algorithm i't that complx, but th corrct argumt i a bit otrivial o o
21 Pairwi LCP Som otatio: SA[i] i th ith uffix i th uffix array. H[i] i th valu of lcp(sa[i], SA[i + ]) Claim: Claim:For Foray ay00<<i i<<j j<<m: m: lcp(sa[i], lcp(sa[i],sa[j]) SA[j])==RMQ RMQHH(i, (i,j j ) ) o o
22 Computig LCP To prproc a uffix array to upport O() LCP quri: U Kaai' O(m)-tim algorithm to build th LCP array. Build a RMQ tructur ovr that array i tim O(m) uig Fichr-Hu. U th prcomputd RMQ tructur to awr LCP quri ovr rag. Rquir O(m) prprocig tim ad oly O() qury tim.
23 Sarchig a Suffix Array Rcall: Ca arch a uffix array of T for all match of a pattr P i tim O( log m + z). If w'v do O(m) prprocig to build th LCP iformatio, w ca pd thi up.
24 Sarchig a Suffix Array Ituitivly, imulat doig a biary arch of th lav of a uffix tr, rmmbrig th dpt ubtr you'v matchd o far. At ach poit, if th biary arch prob a laf outid of th currt ubtr, kip it ad cotiu th biary arch i th dirctio of th currt ubtr. To implmt thi o a actual uffix array, w u LCP iformatio to implicitly kp track of whr th boud o th currt ubtr ar.
25 Sarchig a Suffix Array Claim: Th algorithm w jut ktchd ru i tim O( + log m + z). Proof ida: Th O(log m) trm com from th biary arch ovr th lav of th uffix tr. Th O() trm corrpod to dcdig dpr ito th uffix tr o charactr at a tim. Fially, w hav to pd O(z) tim rportig match.
26 Logt Commo Extio
27 Aothr Applicatio: LCE Rcall: Th logt commo xtio of two trig T₁ ad T₂ at poitio i ad j, dotd LCET₁, T₂ (i, j), i th lgth of th logt ubtrig of T₁ ad of T₂ that bgi at poitio i i T₁ ad poitio j i T₂. a p p d p p a l Uig gralizd uffix tr ad LCA, w hav a O(m), O() -tim olutio to LCE. Claim: Thr' a much air olutio uig LCP.
28 Suffix Array ad LCE Rcall: LCET₁, T₂(i, j) i th lgth of th logt commo prfix of th uffix of T₁ tartig at poitio i ad th uffix of T₂ tartig at poitio j. Suppo w cotruct a gralizd uffix array for T₁ ad T₂ augmtd with LCP iformatio. W ca th u LCP to awr LCE quri i tim O(). W'll d a tabl mappig uffix to thir idic i th tabl to do thi, but that' ot that hard to t up ₁ ₂ ₁ ₂ ₁ ₂ o₁ ₁ ₂ ₁ o₁ ₁ ₂ ₁ t₂ o₂ t₂
29 Uig LCP: Cotructig Suffix Tr
30 Cotructig Suffix Tr Lat tim, I claimd it wa poibl to cotruct uffix tr i tim O(m). W'll do thi by howig th followig: A uffix array for T ca b built i tim O(m). A LCP array for T ca b built i tim O(m). Chck Kaai' papr for dtail. A uffix tr ca b built from a uffix array ad LCP array i tim O(m).
31 From Suffix Array to Suffix Tr
32 Uig LCP o 5 0 o 6 o o o Claim: Claim:Ay Ay0' 0'i ith thuffix uffixarray array rprt rprtdmarcatio dmarcatiopoit poit btw btwubtr ubtrof ofth throot rootod. od.
33 Uig LCP o 5 o 6 Th 0proprty Tham amproprty hold holdfor forth th ubarray, xcpt o ubarray, xcpt uig ubarray uigth th ubarray mi miitad itadof of o 5 3 o 6 3
34 a aaaba aaabbabaaaba aaba aabaaabbabaaaba aabbabaaaba aba abaaaba abaaabbabaaaba abbabaaaba ba baaaba baaabbabaaaba bababaaaba bbabaaaba
35 Thi Thii iaalightly lightly modifid modifid Cartia Cartiatr! tr! a aaaba aaabbabaaaba aaba aabaaabbabaaaba aabbabaaaba aba abaaaba abaaabbabaaaba abbabaaaba ba baaaba baaabbabaaaba bababaaaba bbabaaaba
36 a aaaba aaabbabaaaba aaba aabaaabbabaaaba aabbabaaaba aba abaaaba abaaabbabaaaba abbabaaaba ba baaaba baaabbabaaaba bababaaaba bbabaaaba
37 A Liar-Tim Algorithm Cotruct a Cartia tr from th LCP array, fuig togthr od with th am valu if o bcom a part of th othr. Ru a DFS ovr th tr ad add miig childr i th ordr i which thy appar i th uffix array. Aig labl to th dg bad o th LCP valu. Total tim: O(m).
38 Tim-Out For Aoucmt!
39 Problm St Two Problm St Two go out today. It' du xt Tuday (April 9th) at th tart of cla. Play aroud with tri, Aho-Coraick, uffix tr, ad uffix array! Problm St O ha b gradd. Grad ar availabl o GradScop. Solutio ar availabl i hardcopy i lctur. Thy'll b i th filig cabit i th Gat B wig (ar Kith' offic) if you wr't abl to pick thm up. Lua mad om xcllt graph howig th actual prformac of th RMQ data tructur i practic, icludig chart for how commo rror brak th rutim boud. Highly rcommdd!
40 Offic Hour Locatio Look lik w'r o logr allowd to hold offic hour i th Huag Bamt. W'v movd our Moday / Tuday offic hour ito Gat B6. Kith' offic hour will till b i Gat 78.
41 WiCS Caual CS Dir Staford WiCS i holdig th firt of thir biquartrly CS Caual Dir xt Moday, April 8 from 6:30PM 7:30PM at th WCC. Highly rcommdd! Your prpctiv at thi poit i your CS carr would b rally valuabl to popl who ar jut tartig out.
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43 HackOvrflow HackOvrflow i thi Saturday, April 6, from 0:00AM 0:00PM i th Huag Bamt. It' a grat hackatho for firt-timr. Highly rcommdd!
44 DivrityBa: Itrtd? DivrityBa i a joit ffort by SOLE, SBSE, AISES, ad FLIP with a focu o computr cic. Thy'r lookig for popl to tak o ladrhip poitio. Thi i a phomal orgaizatio ad it would b a grat plac to mak a hug impact. Itrtd? Apply hr:
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46 Back to CS66!
47 Th Hard Part: Buildig Suffix Array
48 A Naïv Algorithm Hr' a impl algorithm for buildig a uffix array: Cotruct all th uffix of th trig i tim Θ(m ). Sort tho uffix uig haport or mrgort. Mak O(m log m) compario, but ach compario tak O(m) tim. Tim rquird: O(m log m). Total tim: O(m log m). Ca w do bttr?
49 Radix Sort Radix ort i a fat ortig algorithm for trig ad itgr. It' a powrful primitiv for buildig othr algorithm ad data tructur ad com up all th tim i job itrviw. I ca you hav't it bfor (it' oly itrmitttly taught i CS6), lt' tart with a quick radix ort rviw.
50 Aalyzig Radix Sort Suppo thr ar t total trig with maximum lgth k, draw from alphabt Σ. Tim to t up iitial buckt: Θ( Σ ). Tim to ditribut trig lmt ach roud: O(t). Tim to collct trig ach roud: O(t + Σ ). Numbr of roud: O(k) Rutim: O(k(t + Σ )).
51 Spdig Up with Radix Sort What happ if w u radix ort itad of haport i our origial uffix array algorithm? Numbr of trig: Θ(m). Strig lgth: Θ(m). Numbr of charactr: Σ. Rutim i thrfor Θ(m + m Σ ) Aumig Σ = O(m), th rutim i Θ(m ), a log factor fatr tha bfor. Ca w do bttr?
52 Radix Sort Uful obrvatio: it' poibl to ort t trig i tim O(t) if th trig all hav a cotat lgth, ad th alphabt iz i O(t). W'r goig to u thi obrvatio i a littl bit, but mak a ot of it for ow.
53 Th DC3 Algorithm
54 DC3 O of th implt ad fatt algorithm for buildig uffix array i calld DC3 (Diffrc Covr, iz 3). It' a matrpic of a algorithm it' clvr, brilliat, ad ot that hard to cod up. It' alo quit uacd ad tricky. W'r goig to pd th rt of today workig through th dtail. You'll th play aroud with it o th problm t.
55 Som Aumptio Aum th iitial iput alphabt coit of a t of itgr 0,,,, Σ -. If thi i't th ca, w ca alway ort th lttr ad rplac ach with it rak. Aumig that Σ = O(), thi do't affct th rutim.
56 Som Trmiology Dfi Tₖ to b th poitio i T who idic ar qual to k mod 3. T₀ i th t of poitio that ar multipl of thr. T₁ i th t of poitio that follow th poitio i T₀. T₂ i th t of poitio that follow th poitio i T₁. m o o o o m o m
57 DC3, Ituitivly At a high-lvl, DC3 work a follow: Rcurivly gt th ortd ordr of all uffix i T₁ ad T₂. Uig thi iformatio, fficitly ort th uffix i T₀. Mrg th two lit of ortd uffix (th uffix i T₀ ad th uffix i T₁/T₂) togthr to form th full uffix array. Th dtail ar bautiful, but tricky.
58 DC3, Ituitivly At a high-lvl, DC3 work a follow: Rcurivly gt th ortd ordr of all uffix i T₁ ad T₂. Uig thi iformatio, fficitly ort th uffix i T₀. Mrg th two lit of ortd uffix (th uffix i T₀ ad th uffix i T₁/T₂) togthr to form th full uffix array. Th dtail ar bautiful, but tricky.
59 Th Firt Stp Our objctiv i to gt th rlativ rakig of th uffix at poitio T₁ ad T₂. High-lvl ida: Cotruct a w trig bad o uffix tartig at poitio i T₁ ad T₂. Comput th uffix array of that trig, rcurivly. U th rultig uffix array to dduc th ordrig of th uffix from T₁ ad T₂.
60 Embiggig Our Strig Form two w trig from T by droppig off th firt charactr ad firt two charactr ad paddig with xtra markr. Th, cocatat tho trig togthr. m o o o o m o m o o o o m o m o o o m o m
61 Embiggig Our Strig Form two w trig from T by droppig off th firt charactr ad firt two charactr ad paddig with xtra markr. Th, cocatat tho trig togthr. o o o o m o m o o o m o m
62 Um, Why? Claim: Th rlativ ordr of th uffix i th firt half of th trig tartig at poitio i T₁ ad th uffix i th cod half of th trig at poitio i T₂ i th am a th rlativ ordr of tho uffix i T. Ituitio: Strig withi th am half ar rlativly ordrd. Strig acro th two halv ar protctd by th dmarkr. o o o o m o m o o o m o m
63 So, Um... w jut doubld th iz of our iput trig. You'r ot uppod to do that i a divid-ad-coqur algorithm. o o o o m o m o o o m o m
64 Playig with Block Ky Iight: Brak th rultig trig apart ito block of iz thr. Thik about what happ if w compar two uffix tartig at th bgiig of a block: Sic th uffix ar ditict, thr' a mimatch at om poit. All block prior to that poit mut b th am. Th diffrig block of thr i th tibrakr. o o o o m o m o o o m o m
65 Th Rcuriv Stp Th Trick: Trat ach block of thr charactr a it ow charactr. Dtrmi th rlativ ordrig of tho charactr by a O(m)-tim radix ort. Rplac ach block of thr charactr with th rak of it mtacharactr. Rcurivly comput th uffix array of th rultig trig. 0 o o m o o o o om m o m o o o o mo o m o o o o m o m o o o m o m
66 Th Rcuriv Stp Th Trick: Trat ach block of thr charactr a it ow charactr. Dtrmi th rlativ ordrig of tho charactr by a O(m)-tim radix ort. Rplac ach block of thr charactr with th rak of it mtacharactr. Rcurivly comput th uffix array of th rultig trig o o o o m o m o o o m o m
67 Th Rcuriv Stp Oc w hav thi uffix array, w ca u it to gt th uffix from T₁ ad T₂ ito ortd ordr o o o o m o m o o o m o m
68 Th Rcuriv Stp Oc w hav thi uffix array, w ca u it to gt th uffix from T₁ ad T₂ ito ortd ordr m o o o o m o m o o o o m o m o o o m o m
69 Th Rcuriv Stp Oc w hav thi uffix array, w ca u it to gt th uffix from T₁ ad T₂ ito ortd ordr m o o o o m o m o o o o m o m o o o m o m
70 Rakig T₁ ad T₂ W pd a total of O(m) work i thi tp doublig th array, groupig it ito block of iz 3, radix ortig it, ad covrtig th rult of th call ito maigful data. W alo mak a rcuriv call o a array of iz m / 3. Total work: O(m), plu a rcuriv call o a array of iz m / 3.
71 DC3, Ituitivly At a high-lvl, DC3 work a follow: Rcurivly gt th ortd ordr of all uffix i T₁ ad T₂. Uig thi iformatio, fficitly ort th uffix i T₀. Mrg th two lit of ortd uffix (th uffix i T₀ ad th uffix i T₁/T₂) togthr to form th full uffix array. Th dtail ar bautiful, but tricky.
72 A Bautiful Iight Claim: If w kow th rlativ ordrig of uffix at poitio T₁ ad T₂, w ca dtrmi th rlativ ordr of uffix i poitio T₀. Ida: U a modifid radix ort! m o o o o m o m m o o o o m o m A B C D E
73 A Bautiful Iight Claim: If w kow th rlativ ordrig of uffix at poitio T₁ ad T₂, w ca dtrmi th rlativ ordr of uffix i poitio T₀. Ida: U a modifid radix ort! m m o o o o m o m A B C D E
74 A Bautiful Iight Claim: If w kow th rlativ ordrig of uffix at poitio T₁ ad T₂, w ca dtrmi th rlativ ordr of uffix i poitio T₀. Ida: U a modifid radix ort! o o o m o m m m o o o o m o m A B C D E
75 A Bautiful Iight Claim: If w kow th rlativ ordrig of uffix at poitio T₁ ad T₂, w ca dtrmi th rlativ ordr of uffix i poitio T₀. Ida: U a modifid radix ort! 8 m m o o o o m o m A B C D E
76 A Bautiful Iight Claim: If w kow th rlativ ordrig of uffix at poitio T₁ ad T₂, w ca dtrmi th rlativ ordr of uffix i poitio T₀. Ida: U a modifid radix ort! m o o o o m o m A B C D E m m m 7 9 8
77 Sortig T₀ To ort T₀, w do th followig: For ach poitio i T₀, form a pair of th lttr at that poitio ad th idx of th uffix right aftr it (which i i T₁). Th pair ar ffctivly trig draw from a alphabt of iz Σ + m. Radix ort thm i tim O(m).
78 DC3, Ituitivly At a high-lvl, DC3 work a follow: Rcurivly gt th ortd ordr of all uffix i T₁ ad T₂. Uig thi iformatio, fficitly ort th uffix i T₀. Mrg th two lit of ortd uffix (th uffix i T₀ ad th uffix i T₁/T₂) togthr to form th full uffix array. Th dtail ar bautiful, but tricky.
79 Mrgig th Lit At thi poit, w hav two ortd lit: A ortd lit of all th uffix i T₀. A ortd lit of all th uffix i T₁ ad T₂. W alo kow th rlativ ordr of ay two uffix i T₁ ad T₂. How ca w mrg th lit togthr?
80 Th Mrgig A 8 B 5 4 D E 9 0 C o m o m o m o m m o o o o m o m A B C D E
81 Th Mrgig A 8 B 5 4 D E 9 0 C Ky Ky ida: ida: W W kow kow th th rlativ rlativ ordrig ordrig of of th th uffix uffix at at poitio poitio that that ar ar cogrut cogrut to to or or mod mod o m o m m o o o o m o m A B C D E
82 Th Mrgig A 8 B 5 4 D E 9 0 C m o o o o m o m A B C D E
83 Th Mrgig 3 8 B 5 4 D I I thi thi ca ca it it do't do't mattr, mattr, but but what what would would happ happ if if th th firt firt lttr lttr wr wr th th am? am? W W do't do't kow kow th th rlativ rlativ ordrig ordrig of of th th uffix. uffix. E 9 0 C 6 7 A m o o o o m o m A B C D E
84 Th Mrgig 3 8 B 5 4 D Th Th ca ca b b rakd rakd rgardl of of whthr whthr th th firt firt two two charactr ar ar th th am. am. E 9 0 C 6 7 A o m o o o o m o m A B C D E
85 Th Mrgig E 9 0 C 6 7 A 8 B 5 4 D 3 B 7 E C A D 0 m o o o o m o m A B C D E
86 Th Mrgig Comparig ay two uffix rquir at mot O() work bcau w ca u th xitig rakig of th uffix i T₁ ad T₂ to trucat log uffix. Thr ar a total of m uffix to mrg. Total rutim: O(m).
87 Th Ovrall Rutim Th rcuriv tp to ort T₁ ad T₂ tak tim Θ(m) plu th cot of a rcuriv call o a iput of iz m / 3. Uig T₁ ad T₂ to ort T₀ tak tim Θ(m). Mrgig T₀, T₁, ad T₂ tak tim Θ(m). Rcurrc rlatio: R(m) = R(m / 3) + O(m) Via th Matr Thorm, w that th ovrall rutim i Θ(m).
88 Th Ovrall Algorithm Although thi algorithm ha a lot of tricky dtail, it' actually ot that tough to cod it up. Th origial papr giv a two-pag C++ implmtatio of th tir algorithm. Ad bcau w'r Dct Huma Big, w'r ot goig to ak you to writ it up o your ow.
89 Qutio to Podr Thi algorithm i xtrmly clvr ad ha lot of itrlockig movig part. Why i th umbr 3 o igificat? Why did w hav to doubl th lgth of th trig bfor groupig ito block? You'll xplor om of th qutio i th problm t.
90 How Did Ayo Ivt Thi? Thi algorithm ca m totally magical ad cofuig th firt tim you it. A with mot algorithm, thi o wa bad o a lot of prior work. I 997, Marti Farach publihd a algorithm (ow calld Farach' algorithm) for dirctly buildig a uffix tr i tim O(m). It ivolvd may of th am tchiqu (jut ort uffix at om pcific poitio, u that to fill i th miig uffix, th mrg th rult), but ha a lot mor dtail bcau it work dirctly o uffix tr rathr tha array. Th algorithm itlf i a bit tricky but i totally bautiful. It would mak for a rally fu fial projct!
91 Mor to Explor Thr ar a umbr of othr data tructur i th family of uffix tr ad uffix array. Th uffix automato or DAWG i a miimal-tat DFA for all th uffix of a trig T. It alway ha iz O( T ), ad thi i ot obviou! A factor oracl i a rlaxd automato that match all th ubtrig of om trig T, plu poibly om puriou match. Th Burrow-Whlr traform i a tchiqu rlatd to uffix array that wa origially dvlopd for data comprio. Ay of th would b mak for grat fial projct topic.
92 Summary Suffix tr ar a compact, flxibl, powrful tructur for awrig qutio o trig. Suffix array giv a pac-fficit altrativ to uffix tr that hav a light tim tradoff. LCP array lik uffix tr ad uffix array ad ca b built i tim O(m). Suffix array ca b cotructd i tim O(m). Suffix tr ca b cotructd i tim O(m) from a uffix array ad LCP array.
93 Nxt Tim Balacd Tr B-tr, -3-4 tr, ad rd/black tr. Whr th hck did rd/black tr com from? Thr' a amazig awr to thi qutio. Trut m.
Outline for Today. A simple data structure for string searching. A compact, powerful, and flexible data structure for string algorithms.
Suffix Tr Outli fr Tday Rviw frm Lat Tim A quick rfrhr tri. Suffix Tri A impl data tructur fr trig archig. Suffix Tr A cmpact, pwrful, ad flxibl data tructur fr trig algrithm. Gralizd Suffix Tr A v mr
(1) Then we could wave our hands over this and it would become:
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