Applied Databases. Sebastian Maneth. Lecture 16 Suffix Array, Burrows-Wheeler Transform. University of Edinburgh - March 16th, 2017
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1 Applied Dtbses Lecture 16 Suffix Arry, Burrows-Wheeler Trsform Sebsti Meth Uiversity of Ediburgh - Mrch 16th, 2017
2 Outlie 2 1. Suffix Arry 2. Burrows-Wheeler Trsform
3 Outlie 3 1. Suffix Arry 2. Burrows-Wheeler Trsform Lecture 17: XPth Lecture 18: XSLT Lecture 19: Recp I Lecture 20: Recp I Lecture 21: guest lecture NULLs cosidered hrmful (April-3) (to be cofirmed) Lecture 22: o lecture! (April-6)
4 1. Suffix Arry 4
5 Suffix Arry Costructio 5 p
6 Serch 6
7 Serch 7 M1 M2
8 Serch 8 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]...]
9 History [wikipede] The LCP rry ws itroduced i 1993, by Udi Mber d Gee Myers logside the suffix rry i order to improve the ruig time of their strig serch lgorithm. Gee Myers lter becme the vice presidet of Iformtics Reserch t Celer Geomics, d Udi Mber the vice presidet of egieerig t Google. 9
10 10
11 Suffix Arrys 11 much more spce efficiet th Suffix Tree used i prctise (suffix tree more used i theory) Suffix Arry Costructio, without Suffix Trees? [ Lier Work Suffix Arry Costructio, Kärkkäie, Sders, Burkhrdt, Jourl of the ACM, 2006 ] See lso: [ A txoomy of suffix rry costructio lgorithms, Puglisi, Smyth, Turpi, ACM Computig Surveys 39, 2007 ] liked from course web pge
12 2. Burrows-Wheeler Trsform 12 T = b$ b$ $b $b $b $b $b $b geerte ll cyclic shifts of T
13 2. Burrows-Wheeler Trsform 13 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b sort them lexicogrphiclly $ < < b < c <...
14 2. Burrows-Wheeler Trsform 14 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the first colum? sort them lexicogrphiclly $ < < b < c <...
15 2. Burrows-Wheeler Trsform 15 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the first colum? sorted #occ of ech letter: oe time $ three times oe time b two times sort them lexicogrphiclly $ < < b < c <...
16 2. Burrows-Wheeler Trsform 16 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the first colum? sorted #occ of ech letter: oe time $ three times oe time b two times sort them lexicogrphiclly C you retrieve the origil text T, give oly the first colum? $ < < b < c <...
17 2. Burrows-Wheeler Trsform 17 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the first colum? sorted #occ of ech letter: oe time $ three times oe time b two times sort them lexicogrphiclly C you retrieve the origil text T, give oly the first colum? Of course ot!! $ < < b < c <...
18 2. Burrows-Wheeler Trsform 18 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the first colum? sorted #occ of ech letter: oe time $ three times oe time b two times sort them lexicogrphiclly $ < < b < c <... C you retrieve the origil text T, give oly the first colum? Note ech colum cotis the sme letters ($, 3*, b, 2*)
19 2. Burrows-Wheeler Trsform 19 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the secod colum? sortig with respect to cosiderig oe previous letter = sortig of ll two-letter substrigs! sort them lexicogrphiclly $ < < b < c <...
20 2. Burrows-Wheeler Trsform 20 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the secod colum? sortig with respect to cosiderig oe previous letter C you retrieve the origil T from the secod colum oly? sort them lexicogrphiclly $ < < b < c <...
21 2. Burrows-Wheeler Trsform 21 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the secod colum? sortig with respect to cosiderig oe previous letter C you retrieve the origil T from the secod colum oly? sort them lexicogrphiclly C you retrieve the first colum, give the secod oe? $ < < b < c <...
22 2. Burrows-Wheeler Trsform 22 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the third colum? sortig with respect to cosiderig two previous letters sort them lexicogrphiclly $ < < b < c <...
23 2. Burrows-Wheeler Trsform 23 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the lst colum? sortig with respect to cosiderig ll previous letters sort them lexicogrphiclly $ < < b < c <...
24 2. Burrows-Wheeler Trsform 24 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the lst colum? sortig with respect to cosiderig ll previous letters C you retrieve the origil text T, give oly the lst colum? sort them lexicogrphiclly $ < < b < c <...
25 2. Burrows-Wheeler Trsform 25 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Wht iformtio is cptured by the lst colum? 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 <...
26 2. Burrows-Wheeler Trsform 26 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Burrows-Wheeler Trsform L of text T sort them lexicogrphiclly $ < < b < c <...
27 2. Burrows-Wheeler Trsform 27 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Burrows-Wheeler Trsform L of text T Why is this useful? E.g., if T cotis my occ's of the, the BWT[T] cotis my cosecutive t letters. sort them lexicogrphiclly BWT[T] is esily compressible by simple ru-legth code this is the ide behid bzip2 $ < < b < c <...
28 2. Burrows-Wheeler Trsform 28 T = b$ b$ $b $b $b $b $b $b $b $b $b $b b$ $b $b Burrows-Wheeler Trsform L of text T Give lst colum L, how c we recostruct the origil text T? sort them lexicogrphiclly $ < < b < c <...
29 2. Burrows-Wheeler Trsform 29 Nively: b $
30 2. Burrows-Wheeler Trsform 30 Nively: b $ sort $ b first colum!
31 2. Burrows-Wheeler Trsform 31 Nively: b $ sort $ b first colum! $b $b $b $b b$ $b $b previous letter!
32 2. Burrows-Wheeler Trsform 32 Nively: b $ sort $ b first colum! $b $ $b $b $b b$ $b $b must be substrig of T previous letter!
33 2. Burrows-Wheeler Trsform 33 Nively: b $ sort $ b first colum! $b $ $b $b $b b$ $b $b must be substrig of T previous letter!
34 2. Burrows-Wheeler Trsform 34 Nively: b $ sort $ b first colum! $b $ $b $b $b b$ $b $b must be substrig of T previous letter!
35 2. Burrows-Wheeler Trsform 35 Nively: b $ sort $ b first colum! $b $ $b $b $b b b$ $b $b $b All 2-letter substrigs of T! previous letter!
36 2. Burrows-Wheeler Trsform 36 Nively: b $ sort $ b first colum! $ b $b sort $b $ b colums 1 d 2.
37 2. Burrows-Wheeler Trsform 37 Nively: b $ sort previous letter $ b first colum! $ b $b sort $b $ b $b $ b $b
38 38 2. Burrows-Wheeler Trsform Nively: b $ sort $ b third colum! $ b $b sort $b $ b previous letter $b $ b $b sort $b $b b $
39 39 2. Burrows-Wheeler Trsform Nively: b $ sort $ b third colum! $ b $b sort $b $ b previous letter $b $ b $b sort $b $b b $ Et ceter
40 2. Burrows-Wheeler Trsform 40 Nive method: very expesive! (my sortigs) rk b ( L, k ) = #occ of b i L, up to positio k (excludig k) L = b $ rk ( L, 4 ) = 2
41 2. Burrows-Wheeler Trsform 41 Nive method: very expesive! (my sortigs) rk b ( L, k ) = #occ of b i L, up to positio k (excludig k) L = b $ rk ( L, 4 ) = 2 rk ( L, 3 ) = 1
42 2. Burrows-Wheeler Trsform 42 Nive method: very expesive! (my sortigs) rk b ( L, k ) = #occ of b i L, up to positio k (excludig k) L = b $ rk ( L, 4 ) = 2 rk ( L, 3 ) = 1 O(log S ) time (fter lier time preprocessig of L) rk ( L, 7 ) = 2
43 2. Burrows-Wheeler Trsform rk b ( L, k ) = #occ of b i L, up to positio k $ b L = b $ C $ 2 5b 6 43 Lst-to-Frot Mppig LF(k)=C[L[k]] + rk L[k] ( L, k ) first lie strtig with the letter i the first colum $b $b $b $b b$ $b $b secod, comig from the top where is the secod from top, i the first colum?
44 2. Burrows-Wheeler Trsform rk b ( L, k ) = #occ of b i L, up to positio k $ b L = b $ C $ 2 5b 6 44 Lst-to-Frot Mppig LF(k)=C[L[k]] + rk L[k] ( L, k ) first lie strtig with the letter i the first colum $b $b $b $b b$ $b $b 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
45 Decodig 45 rk b ( L, k ) = #occ of b i L, up to positio k $ b L = b $ C Lst-to-Frot Mppig LF(k)=C[L[k]] + rk L[k] ( L, k ) first lie strtig with the letter i the first colum $b $b $b $b b$ $b $b previous letter! strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ]
46 Decodig 46 rk b ( L, k ) = #occ of b i L, up to positio k $ b L = b $ C Lst-to-Frot Mppig LF(k)=C[L[k]] + rk L[k] ( L, k ) first lie strtig with the letter i the first colum $b $b $b $b b$ $b $b strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2 L[2] = [ b$ ]
47 Decodig 47 rk b ( L, k ) = #occ of b i L, up to positio k $ b L = b $ C Lst-to-Frot Mppig LF(k)=C[L[k]] + rk L[k] ( L, k ) $b $b $b $b b$ $b $b strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2 L[2] = [ b$ ] LF(2) = 6 L(6) = [ $ ]
48 Decodig 48 rk b ( L, k ) = #occ of b i L, up to positio k $ b L = b $ C Lst-to-Frot Mppig LF(k)=C[L[k]] + rk L[k] ( L, k ) $b $b $b $b b$ $b $b strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2, L[2] = [ b$ ] LF(2) = 6, L[6] = [ $ ] LF(6) = 3, L[3] = [ $ ]
49 Decodig 49 rk b ( L, k ) = #occ of b i L, up to positio k $ b L = b $ C Lst-to-Frot Mppig LF(k)=C[L[k]] + rk L[k] ( L, k ) $b $b $b $b b$ $b $b strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2, L[2] = [ b$ ] LF(2) = 6, L[6] = [ $ ] LF(6) = 3, L[3] = [ $ ] LF(3) = 7, L[7] = [ $ ]
50 Decodig 50 rk b ( L, k ) = #occ of b i L, up to positio k $ b L = b $ C Lst-to-Frot Mppig LF(k)=C[L[k]] + rk L[k] ( L, k ) $b $b $b $b b$ $b $b strt with $ (positio 5) LF(5) = 1 L[1] = [curret decodig: $ ] LF(1) = 2, L[2] = [ b$ ] LF(2) = 6, L[6] = [ $ ] LF(6) = 3, L[3] = [ $ ] LF(3) = 7, L[7] = [ $ ] LF(7) = 4, L[4] = b [ b$ ]
51 2. Burrows-Wheeler Trsform 51 $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 ( t,18733) mi motivtio used i bzip2 compressor
52 2. Burrows-Wheeler Trsform 52 $b $b $b $b b$ $b $b Wht is specil bout the BTW? efficiet bckwrd serch! coutig #occ s of ptter P i O( P log S ) time!
53 Bckwrd Serch o BWT 53
54 Bckwrd Serch o BWT 54
55 Bckwrd Serch o BWT 55
56 BWT Costructio 56 How c we costruct BWT[T]??
57 BWT Costructio 57 use the suffix rry SA(T)! BWT[ k ] = T[ SA[ k ] 1 ] (ssumig T[0]=$)
58 BWT Costructio 58 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 $b
59 BWT Costructio 59 use the suffix rry SA(T)! BWT[ k ] = T[ SA[ k ] 1 ] (ssumig T[0]=$) expli why this equtio is correct! $b $b $b $b b$ $b $b
60 60 END Lecture 16
Applied Databases. Sebastian Maneth. Lecture 16 Suffix Array, Burrows-Wheeler Transform. University of Edinburgh - March 10th, 2016
Applied Dtbses Lecture 16 Suffix Arry, Burrows-Wheeler Trsform Sebsti Meth Uiversity of Ediburgh - Mrch 10th, 2016 2 Outlie 1. Suffix Arry 2. Burrows-Wheeler Trsform 3 Olie Strig-Mtchig how c we do Horspool
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