Algorithms for bioinformatics Part 2: Data structures
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1 Alorithms for bioinformtics Prt 2: Dt structures Greory Kucherov LIGM/CNRS Mrne-l-Vllée
2 Pln Clssicl indexes Suffix trees DAWG nd Position heps Suffix rrys Succinct (compressed) indexes Burrows-Wheeler trnsform nd FM-index
3 Suffix rry
4 Suffix rry: definition T=ctct ctct ctct tct tct ct ct t t t t
5 Suffix rry: definition T=ctct ct ctct t tct t ct ctct t tct t
6 Suffix rry: definition T=ctct ct ctct t tct t ct ctct t tct t SA Suffix rry (SA) is permuttion of the positions of T tken in the lexicorphicl order of correspondin suffixes T[SA[i]n]< lex T[SA[i+]n] [Mnber, Myers 90] [Gonnet, Bez-Ytes, Snider 92]
7 Strin mtchin with suffix rrys T=ctct ct ctct t tct t ct ctct t tct t SA c c For ny substrin, its occurrences form n intervl of SA Intervl [L P,R P ] for P cn be found by two binry serches Serch for L P : while l<r do m=(l+r)/2 if P< lex T[SA[m]n] then l m else r m the whole serch for P tkes time O(m lo(n)+occ)
8 Strin mtchin with suffix rrys T=ctct ct ctct t tct t ct ctct t tct t SA LCP LCP[i] = lenth of the lonest common prefix between suffixes strtin t SA[i] nd SA[i-] Note: the lonest common prefix between ny two suffixes SA[i] nd SA[j] cn be computed in constnt time fter liner-time preprocessin of LCP rry. (Use rne minimum queries) usin LCP rry, intervl serch time O(m lo(n)) cn be turned into O(m+lo(n))
9 Computin LCP from SA (nd T) compute Rnk[i] = SA - [i] compute LCP[i] for i=rnk[],,rnk[n] to compute LCP[Rnk[i]], look t LCP[Rnk[i-]] let q=rnk[i], p=rnk[i-] key observtion: if h=lcp[p]>0, then LCP[q] h- h- comprisons cn be sved! k i- i h j- j h bcbdbe be h- bcbdbe bebe be SA j- i- j k i p- p q- q j=sa[p-]+
10 Computin LCP from SA (nd T) compute Rnk[i] = SA - [i] compute LCP[i] for i=rnk[],,rnk[n] to compute LCP[Rnk[i]], look t LCP[Rnk[i-]] let q=rnk[i], p=rnk[i-] key observtion: if h=lcp[p]>0, then LCP[q] h- h- comprisons cn be sved! k i- i h j- j compre successively T[i+h], T[i+h+], with respectively T[k+h], T[k+h+], h bcbdbe be h- bcbdbe bebe be SA j- i- j k i p- p q- q j=sa[p-]+
11 Computin LCP from SA (nd T) compute Rnk[i] = SA - [i] compute LCP[i] for i=rnk[],,rnk[n] to compute LCP[Rnk[i]], look t LCP[Rnk[i-]] let q=rnk[i], p=rnk[i-] key observtion: if h=lcp[p]>0, then LCP[q] h- h- comprisons cn be sved! k i- i h j- j compre successively T[i+h], T[i+h+], with respectively T[k+h], T[k+h+], compute new h=lcp[q] iterte to i+ runnin time: O(n) h bcbdbe be h- bcbdbe bebe be SA j- i- j k i j=sa[p-]+ p- p q- q
12 Strin mtchin with suffix rrys P=ccc lcp l suf l = suf r = cccc SA l r ssume we know lcp l =LCP(suf l,p), lcp r =LCP(suf r,p) nd ssume lcp l <lcp r lcp r
13 Strin mtchin with suffix rrys P=ccc lcp l suf l = suf m = ccc suf r = cccc lcp r SA l m r ssume we know lcp l =LCP(suf l,p), lcp r =LCP(suf r,p) nd ssume lcp l <lcp r consider suf m nd consider LCP(suf m,suf r )
14 Strin mtchin with suffix rrys P=ccc lcp l suf l = suf m = ccc suf r = cccc lcp r SA l m r ssume we know lcp l =LCP(suf l,p), lcp r =LCP(suf r,p) nd ssume lcp l <lcp r consider suf m nd consider LCP(suf m,suf r ) Cse : LCP(suf m,suf r )<suf r Then P > lex suf m, LCP(suf m,p)=lcp(suf m,suf r ) nd we set l m for the next itertion
15 Strin mtchin with suffix rrys P=cc lcp l suf l = suf m = ccc suf r = cccc lcp r SA l m r ssume we know lcp l =LCP(suf l,p), lcp r =LCP(suf r,p) nd ssume lcp l <lcp r consider suf m nd consider LCP(suf m,suf r ) Cse 2: LCP(suf m,suf r )>suf r Then P < lex suf m, LCP(suf m,p)=lcp r nd we set r m for the next itertion
16 Strin mtchin with suffix rrys P=ccc lcp l suf l = suf m = ccc suf r = cccc lcp r SA l m r ssume we know lcp l =LCP(suf l,p), lcp r =LCP(suf r,p) nd ssume lcp l <lcp r consider suf m nd consider LCP(suf m,suf r ) Cse 3: LCP(suf m,suf r )=suf r Then we keep comprin chrs of P with those of suf m until P[j] suf m [j] which determines if P lex suf m or P> lex suf m, nd lso the vlue LCP(suf m,p)
17 Strin mtchin with suffix rrys P=ccc lcp l suf l = suf m = ccc suf r = cccc lcp r SA l m r ssume we know lcp l =LCP(suf l,p), lcp r =LCP(suf r,p) nd ssume lcp l <lcp r consider suf m nd consider LCP(suf m,suf r ) Cse 3: LCP(suf m,suf r )=suf r Then we keep comprin chrs of P with those of suf m until P[j] suf m [j] which determines if P lex suf m or P> lex suf m, nd lso the vlue LCP(suf m,p)
18 Strin mtchin with suffix rrys P=ccc suf l = suf m = ccc lcp m suf r = cccc lcp r SA l m r ssume we know lcp l =LCP(suf l,p), lcp r =LCP(suf r,p) nd ssume lcp l <lcp r consider suf m nd consider LCP(suf m,suf r ) Cse 3: LCP(suf m,suf r )=suf r Then we keep comprin chrs of P with those of suf m until P[j] suf m [j] which determines if P lex suf m or P> lex suf m, nd lso the vlue lcp m =LCP(suf m,p)
19 Strin mtchin with suffix rrys P=ccc suf l = suf m = ccc lcp m suf r = cccc lcp r SA l m r ssume we know lcp l =LCP(suf l,p), lcp r =LCP(suf r,p) nd ssume lcp l <lcp r consider suf m nd consider LCP(suf m,suf r ) Cse 3: LCP(suf m,suf r )=suf r Then we keep comprin chrs of P with those of suf m until P[j] suf m [j] which determines if P lex suf m or P> lex suf m, nd lso the vlue lcp m =LCP(suf m,p) Conclusion: t ech step we either do binry division, or move one chr forwrd in the pttern time O( P +lo(n))
20 Construction of suffix rry construction in O(n lo(n)) [Mnber, Myers 90] construction in O(n) [Kärkkäinen, Snders 03] [Ko, Aluru 03] [Kim et l 03] works on prcticl liner-time construction of suffix rry: [Non et l 09]
21 Suffix rry nd suffix tree T=ctct ct ctct t tct t ct ctct t tct t SA LCP t c t c t 7 t t c t 3 9 c t 6 t c t 2 t t 8 c t SA lef lbels left-to-riht provided tht children of ny node re ordered ccordin to the order of chrcters LCP strin lenth of lowest common ncestor (lc) 4 0
22 Enhnced suffix rrys 2 c t c t 5 t 7 t t 9 3 t c c 6 t t 2 c t t 8 t 0 4 t c T=ctct LCP LCP rry encodes the tree topoloy of ST: internl node intervl LCP[ij] s.t. min{lcp[i+j]}=q nd LCP[i]<q, LCP[j+]<q on LCP nd SA rrys, one cn simulte bottom-up nd top-down nvition in ST, s well s suffix links [Abouelhod et l 04]
23 Enhnced suffix rrys 2 c t c t 5 t 7 t t 9 3 t c c 6 t t 2 c t t 8 t 0 4 t c T=ctct LCP LCP rry encodes the tree topoloy of ST: internl node intervl LCP[ij] s.t. min{lcp[i+j]}=q nd LCP[i]<q, LCP[j+]<q on LCP nd SA rrys, one cn simulte bottom-up nd top-down nvition in ST, s well s suffix links [Abouelhod et l 04]
24 Suffix rrys: prcticl issues SA is populr dt structure tht provides spceefficient lterntive to ST Enhnced suffix rrys tke bits/chr, this results to 5-27Gb for humn enome Enhnced SAs re used in prcticl bioinformtics softwre (Vmtch,seemehl) Efficient SA construction is n ctive re of reserch (externl memory, prlleliztion, ) Compressed suffix rrys [Grossi,Vitter 00] (different structures surveyed in [Nvrro, Mkinen 07])
25 Burrows-Wheeler trnsform nd FM-index
26 Succinct nd compressed indexes succinct index tkes spce in bits proportionl to tht of the text itself previous indexes re not succinct s they tke O(n) computer words but O(n lo(n)) bits compressed index tkes spce in bits proportionl to tht of the compressed text self-index does not require storin the text
27 Burrows-Wheeler trnsform T=ctct ctct ctct ctct tctc tctc tctc ctct ctct ctct tctc tctc tctc
28 Burrows-Wheeler trnsform T=ctct ctct ctct ctct tctc tctc tctc ctct ctct ctct tctc tctc tctc SA
29 Burrows-Wheeler trnsform T=ctct ctct ctct ctct tctc tctc tctc ctct ctct ctct tctc tctc tctc
30 Burrows-Wheeler trnsform T=ctct T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t BWT hs been defined for the purpose of ctct compression, s BWT compresses better tct c thn the input text tct c BWT is reversible! tctc c tct c tct ctc t tctc t ctc t ctc
31 Burrows-Wheeler trnsform T=ctct T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct c tct ctc t tctc t ctc t ctc F L
32 Burrows-Wheeler trnsform T= T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct c tct ctc t tctc t ctc t ctc F L
33 Burrows-Wheeler trnsform T= T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct c tct ctc t tctc t ctc t ctc F L
34 Burrows-Wheeler trnsform T= T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct c tct ctc t tctc t ctc t ctc F L
35 Burrows-Wheeler trnsform T= T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct Obs 2: for identicl chrs, their reltive c tct order in F nd L is the sme j ctc t x tctc x i t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] F L Ex: LF[]=8+
36 Burrows-Wheeler trnsform T= t T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct Obs 2: for identicl chrs, their reltive c tct order in F nd L is the sme j ctc t x tctc x i t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] F L Ex: LF[]=8+
37 Burrows-Wheeler trnsform T= t T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct Obs 2: for identicl chrs, their reltive c tct order in F nd L is the sme j ctc t x tctc x i t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] F L Ex: LF[]=8+
38 Burrows-Wheeler trnsform T= t T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct Obs 2: for identicl chrs, their reltive c tct order in F nd L is the sme j ctc t x tctc x i t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] F L Ex: LF[]=8+
39 Burrows-Wheeler trnsform T= t T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct Obs 2: for identicl chrs, their reltive c tct order in F nd L is the sme j ctc t x tctc x i t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] F L Ex: LF[]=8+
40 Burrows-Wheeler trnsform T= t T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct Obs 2: for identicl chrs, their reltive c tct order in F nd L is the sme j ctc t x tctc x i t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] F L Ex: LF[]=8+
41 Burrows-Wheeler trnsform T= t T[SA[i]] BWT BWT[i]=T[SA[i]-] if SA[i], otherwise ctct ctc t Obs : the first column (F) is esy to ctct reconstruct, it cn be represented by n tct c rry C[x]= y<x occ(y,t) for ech letter x tct c Ex: C=[0,,6,8,0] tctc c tct Obs 2: for identicl chrs, their reltive c tct order in F nd L is the sme j ctc t x tctc x i t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] F L Ex: LF[]=8+
42 LF function T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] LF[i] yields the index (in SA) of the suffix immeditely precedin (in T) the i-th suffix (in SA). Formlly, LF[SA[i]]=SA[i-] F L
43 rnk function T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc LF[i]=C[BWT[i]]+rnk[BWT[i],i] F L
44 rnk function T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc F L rnk[bwt[i],i] LF[i]=C[BWT[i]]+rnk[BWT[i],i]
45 rnk function T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc F L rnk[bwt[i],i] LF[i]=C[BWT[i]]+rnk[BWT[i],i] how bout enerl queries rnk[,i] for ny letter nd ny position i?
46 rnk/select functions iven strin T, efficiently nswer queries rnk(,i) on the number of s in T[i] rnk function (on bit vectors) turns out to be fundmentl lorithmic block for buildin succinct dt structures [Jcobson 89] rnk cn be supported in time O() usin o(n) dditionl bits of memory complementry function select(,j): output the position of the j-th occurrence of in T. select cn lso be supported in O() time
47 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc P=tc F L
48 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc P=tc F L
49 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc P=tc F L
50 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc e f P=tc [e,f] : current intervl x : letter e:= C[x]+rnk[x,e]+ f:= C[x]+rnk[x,f] F L
51 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc e f P=tc [e,f] : current intervl x : letter e:= C[x]+rnk[x,e]+ f:= C[x]+rnk[x,f] F L
52 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc e f P=tc [e,f] : current intervl x : letter e:= C[x]+rnk[x,e]+ f:= C[x]+rnk[x,f] F L
53 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc e f P=tc [e,f] : current intervl x : letter e:= C[x]+rnk[x,e]+ f:= C[x]+rnk[x,f] F L
54 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc e f P=tc [e,f] : current intervl x : letter e:= C[x]+rnk[x,e]+ f:= C[x]+rnk[x,f] F L
55 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc e f P=tc [e,f] : current intervl x : letter e:= C[x]+rnk[x,e]+ f:= C[x]+rnk[x,f] F L
56 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc P=tc [e,f] : current intervl x : letter e:= C[x]+rnk[x,e]+ f:= C[x]+rnk[x,f] F L
57 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc P=tc Wht position is it?? F L
58 Strin mtchin with BWT T=ctct T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc P=tc It is position 4! F L
59 FM-index T=ctct SA T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc F L Solution: store only frction of vlues of SA. Storin one vlue over lo(n) leds to O(n lo(n)/lo(n))=o(n) bits
60 FM-index T=ctct SA T[SA[i]] BWT ctct ctc t ctct tct c tct c tctc c tct c tct ctc t tctc t ctc t ctc F L Solution: store only frction of vlues of SA. Storin one vlue over lo(n) leds to O(n lo(n)/lo(n))=o(n) bits Serch time becomes O( P lo(n)) [Ferrin, Mnzini 00] FM-index includes: BWT selection of SA vlues uxiliry structures: rry C, rnk, position mrkin
61 FM-index: prcticl issues FM-index cn be implemented usin ~3 bits/chr (!!) FM-index is now widely used in prcticl bioinformtics softwre: BWA, bowtie, SOAP2 (mppin), CGA (ssembly) other succinct dt structure exist (includin compct suffix rry) nd continue to pper construction my require much more spce thn the resultin structure externl memory lorithms re importnt lorithms specilized to multi-core or GPU processor rchitectures dynmic indexes
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