Applied Databases. Sebastian Maneth. Lecture 16 Suffix Array, Burrows-Wheeler Transform. University of Edinburgh - March 10th, 2016
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1 Applied Dtbses Lecture 16 Suffix Arry, Burrows-Wheeler Trsform Sebsti Meth Uiversity of Ediburgh - Mrch 10th, 2016
2 2 Outlie 1. Suffix Arry 2. Burrows-Wheeler Trsform
3 3 Olie Strig-Mtchig how c we do Horspool o Uicode Texts?
4 4 Suffix Tree T = bbb$ dd ed mrker $ oe-to-oe correspodece of leves to suffixes tree with m+1 leves hs <= 2m+1 odes! Lemm Size of suffix tree for T$ is lier i = T, i.e., i O(). serch time still O( P ), s for suffix trie! perfect dt structure for our tsk!
5 5 Suffix Tree Costructio Good ews: Suffix tree c be costructed i lier time! Complex costructio lgorithms Weier 1973 McCreight 1976 Lier time oly for costt-size lphbets! Otherwise, O( log ) Ukkoe 1995 Frch 1997 Lier time for y iteger lphbet, drw from polyomil rge gi big brekthrough
6 6 Suffix Tree My pplictios E.g., ifiite-widow Lempel-Ziv like compressio b b b bb b b b (1,4) (1,3) (9,4) (1,2) b (positio, legth) M. C. Escher (1948)
7 Questios is the size of this tree relly i O()? i terms of #odes/edges: OK how bout the sizes of lbels??
8 Questios is the size of this tree relly i O()? i terms of #odes/edges: OK how bout the sizes of lbels?? ech requires log() bits!?
9 Yes, but: log() is smll, e.g., 64 bits c be cosidered costt! ech requires log() bits!?
10 Lesso to ler: log() i terms of ru-time fctor, c be ftl i terms of spce-fctor, it is fie! how log will text be? 2^64??? 4 / 8 Bytes ech is eough!
11 For y text with fewer chrcters th #toms i the uiverse, the lbel size for the suffix tree is costt of x bits.. x Bits ech is eough!
12 5*4 Bytes = 20 Bytes for edge poiters lbel size is ot issue but, size of edge-poiters? imgie ech edge requires 32-bit poiter!!
13 13 Actul Spce of Suffix Trees Spce for edge-poiters is problemtic: ctul spce of suffix tree, c. 20 T o commodity hrdwre, texts of more th 1GB re ot doble
14 how to void the huge spce eeded for edges?
15 15 1. Suffix Arry
16 16 Suffix Arry Costructio p
17 17 1. Suffix Arry red leves from left-to-right! SA(T) = [ 12,11,8,5,2,10,9,7,4,6,3 ]
18 18 1. Suffix Arry red leves from left-to-right! SA(T) = [ 12,11,8,5,2,10,9,7,4,6,3 ] Theorem The suffix rry of T c be costructed i time O( T ).
19 19 Serch
20 20 Serch
21 21 Serch
22 22 Serch
23 23 Serch
24 24 Serch
25 25 Serch
26 26 Serch
27 27 Serch
28 28 Serch
29 29 Serch
30 30 Serch
31 31 Serch O( P + log T ) i prctise, usig simple trick O( P + log T ) gurteed, usig LCP-rry LCP(k,j) = logest commo prefix of T[SA[k]...] d T[SA[j]...]
32 32 1. Suffix Arrys much more spce efficiet th Suffix Tree used i prtise (suffix tree more used i theory) Suffix Arry Costructio, without Suffix Trees? [ Lier Work Suffix Arry Costructio, J. Kärkkäie, Sders, Burkhrdt, Jourl of the ACM, 2006 ] See lso: [ A txoomy of suffix rry costructio lgorithms, S. J. Puglisi, W. F. Smyth, A. Turpi, ACM Computig Surveys 39, 2007 ]
33 33 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b geerte ll cyclic shifts of T
34 34 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <...
35 35 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the first colum? $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <...
36 36 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the first colum? $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <... sorted #occ of ech letter: oe time $ three times oe time b two times
37 37 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the first colum? $b $b $b $b b$ $b $b sorted #occ of ech letter: oe time $ three times oe time b two times C you retrieve the origil text T, give oly the first colum? sort them lexicogrphiclly $ < < b < c <...
38 38 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the first colum? $b $b $b $b b$ $b $b sorted #occ of ech letter: oe time $ three times oe time b two times C you retrieve the origil text T, give oly the first colum? sort them lexicogrphiclly $ < < b < c <... Of course ot!!
39 39 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the first colum? $b $b $b $b b$ $b $b sorted #occ of ech letter: oe time $ three times oe time b two times C you retrieve the origil text T, give oly the first colum? sort them lexicogrphiclly $ < < b < c <... Note ech colum cotis the sme letters ($, 3*, b, 2*)
40 40 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the secod colum? $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <... sortig with respect to cosiderig oe previous letter
41 41 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the secod colum? $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <... sortig with respect to cosiderig oe previous letter C you retrieve the origil T from the secod colum oly?
42 42 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the secod colum? $b $b $b $b b$ $b $b sortig with respect to cosiderig oe previous letter C you retrieve the origil T from the secod colum oly? C you retrieve the first colum, give the secod oe? sort them lexicogrphiclly $ < < b < c <...
43 43 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the third colum? $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <... sortig with respect to cosiderig two previous letters
44 44 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the lst colum? $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <... sortig with respect to cosiderig ll previous letters
45 45 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the lst colum? $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <... sortig with respect to cosiderig ll previous letters C you retrieve the origil text T, give oly the lst colum?
46 46 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b Wht iformtio is cptured by the lst colum? $b $b $b $b b$ $b $b sortig with respect to cosiderig ll previous letters C you retrieve the origil text T, give oly the lst colum? YES, you c!! sort them lexicogrphiclly $ < < b < c <...
47 47 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <... Burrows-Wheeler Trsform L of text T
48 48 2. Burrows-Wheeler Trsform T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <... Burrows-Wheeler Trsform L of text T Give lst colum L, how c we recostruct the origil text T?
49 49 2. Burrows-Wheeler Trsform Nively: b $
50 50 2. Burrows-Wheeler Trsform Nively: b $ sort $ b first colum!
51 51 2. Burrows-Wheeler Trsform Nively: b $ sort $b $b $b $b b$ $b $b previous letter! $ b first colum!
52 52 2. Burrows-Wheeler Trsform Nively: b $ $ b sort $b $b $b $b b$ $b $b $ previous letter! first colum! must be substrig of T
53 53 2. Burrows-Wheeler Trsform Nively: b $ $ b sort $b $b $b $b b$ $b $b $ previous letter! first colum! must be substrig of T
54 54 2. Burrows-Wheeler Trsform Nively: b $ $ b sort $b $b $b $b b$ $b $b $ previous letter! first colum! must be substrig of T
55 55 2. Burrows-Wheeler Trsform Nively: b $ $ b sort $b $b $b $b b$ $b $b $ b $b previous letter! first colum! All 2-letter substrigs of T!
56 56 2. Burrows-Wheeler Trsform Nively: b $ $ b $b sort sort $ b $b $ b first colum! colums 1 d 2.
57 57 2. Burrows-Wheeler Trsform Nively: b $ sort $ b first colum! previous letter $ b $b sort $b $ b $b $ b $b
58 58 2. Burrows-Wheeler Trsform Nively: b $ sort $ b third colum! previous letter $ b $b sort $b $ b $b $ b $b sort $b $b b $
59 59 2. Burrows-Wheeler Trsform Nively: b $ sort $ b third colum! previous letter $ b $b sort $b $ b $b $ b $b sort $b $b b $ Et ceter
60 60 2. Burrows-Wheeler Trsform Nive method: very expesive! (my sortigs) rkb( L, k ) = #occ of b i L, up to positio k (excludig k) L = b $ rk( L, 4 ) = 2
61 61 2. Burrows-Wheeler Trsform Nive method: very expesive! (my sortigs) rkb( L, k ) = #occ of b i L, up to positio k (excludig k) L = b $ rk( L, 4 ) = 2 rk( L, 3 ) = 1
62 62 2. Burrows-Wheeler Trsform Nive method: very expesive! (my sortigs) rkb( L, k ) = #occ of b i L, up to positio k (excludig k) L = b $ rk( L, 4 ) = 2 rk( L, 3 ) = 1 rk( L, 7 ) = 2 O(log S ) time (fter lier time preprocessig of L)
63 2. Burrows-Wheeler Trsform rkb( L, k ) = #occ of b i L, up to positio k L = b $ $ b m C Lst-to-Frot Mppig LF(k)=C[L[k]] + rkl[k]( L, k ) $b $b $b $b b$ $b $b 1$ 2 5b 6 first lie strtig with the letter secod, comig from the top where is the secod from top, i the first colum? 63
64 2. Burrows-Wheeler Trsform rkb( L, k ) = #occ of b i L, up to positio k L = b $ $ b m C Lst-to-Frot Mppig LF(k)=C[L[k]] + rkl[k]( L, k ) $b $b $b $b b$ $b $b 1$ 2 5b 6 first lie strtig with the letter secod, comig from the top where is the secod from top, i the first colum? LF(6) = C[L[6]] + rk( L, 6 ) = C[ ] + 1 = = 3 64
65 65 Decodig rkb( L, k ) = #occ of b i L, up to positio k L = b $ $ b m C Lst-to-Frot Mppig LF(k)=C[L[k]] + rkl[k]( L, k ) previous letter! $b $b $b $b b$ $b $b first lie strtig with the letter strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ]
66 66 Decodig rkb( L, k ) = #occ of b i L, up to positio k L = b $ $ b m C Lst-to-Frot Mppig LF(k)=C[L[k]] + rkl[k]( L, k ) $b $b $b $b b$ $b $b first lie strtig with the letter strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2 L[2] = [ b$ ]
67 67 Decodig rkb( L, k ) = #occ of b i L, up to positio k L = b $ $ b m C Lst-to-Frot Mppig LF(k)=C[L[k]] + rkl[k]( L, k ) $b $b $b $b b$ $b $b first lie strtig with the letter strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2 L[2] = [ b$ ] LF(2) = 6 L(6) = [ $ ]
68 68 Decodig rkb( L, k ) = #occ of b i L, up to positio k L = b $ $ b m C Lst-to-Frot Mppig LF(k)=C[L[k]] + rkl[k]( L, k ) $b $b $b $b b$ $b $b first lie strtig with the letter strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2, L[2] = LF(2) = 6, L(6) = LF(6) = 3, L(3) = [ b$ ] [ $ ] [ $ ]
69 69 Decodig rkb( L, k ) = #occ of b i L, up to positio k L = b $ $ b m C Lst-to-Frot Mppig LF(k)=C[L[k]] + rkl[k]( L, k ) $b $b $b $b b$ $b $b first lie strtig with the letter strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2, L[2] = LF(2) = 6, L(6) = LF(6) = 3, L(3) = LF(3) = 7, L(7) = [ b$ ] [ $ ] [ $ ] [ $ ]
70 70 Decodig rkb( L, k ) = #occ of b i L, up to positio k L = b $ $ b m C Lst-to-Frot Mppig LF(k)=C[L[k]] + rkl[k]( L, k ) $b $b $b $b b$ $b $b first lie strtig with the letter strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2, L[2] = LF(2) = 6, L(6) = LF(6) = 3, L(3) = LF(3) = 7, L(7) = LF(7) = 4, L(4) = b [ b$ ] [ $ ] [ $ ] [ $ ] [ b$ ]
71 71 2. Burrows-Wheeler Trsform $b $b $b $b b$ $b $b Wht is specil bout the BTW? hs my repetig chrcters (WHY?) c be ru-legth compressed! Imgie the word the ppers my times i text. he...t he...t he...t he...t he...t he...t he...t mi motivtio used i bzip2 compressor ( t,18733)
72 72 2. Burrows-Wheeler Trsform $b $b $b $b b$ $b $b Wht is specil bout the BTW? efficiet bckwrd serch! coutig #occ s of ptter P i O( P log S ) time!
73 73 Bckwrd Serch o BWT
74 74 Bckwrd Serch o BWT
75 75 Bckwrd Serch o BWT
76 76 BWT Costructio use the suffix rry SA(T)! BWT[ k ] = T[ SA[ k ] 1 ] (ssumig T[0]=$)
77 77 BWT Costructio use the suffix rry SA(T)! BWT[ k ] = T[ SA[ k ] 1 ] (ssumig T[0]=$) e.g SA[b$] = [7,6,4,2,1,5,3] T[6] T[5] T[3] T[1] T[0] T[4] T[2] b $ $b $b $b $b b$ $b $b
78 78 BWT Costructio use the suffix rry SA(T)! BWT[ k ] = T[ SA[ k ] 1 ] (ssumig T[0]=$) expli why this equtio is correct! $b $b $b $b b$ $b $b
79 79 END Lecture 16
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