Module 9: String Matching
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- Lorraine Cole
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1 Module 9: Strig Mtchig CS 240 Dt Structures d Dt Mgemet T. Biedl K. Lctot M. Sepehri S. Wild Bsed o lecture otes y my previous cs240 istructors Dvid R. Cherito School of Computer Sciece, Uiversity of Wterloo Witer 2018 Refereces: Goodrich & Tmssi 9.1 versio :16 Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
2 Outlie 1 Strig Mtchig Itroductio Strig Mtchig with Fiite Automt Kuth-Morris-Prtt lgorithm Boyer-Moore Algorithm Ri-Krp Algorithm Suffix Trees Coclusio Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer 2018
3 Outlie 1 Strig Mtchig Itroductio Strig Mtchig with Fiite Automt Kuth-Morris-Prtt lgorithm Boyer-Moore Algorithm Ri-Krp Algorithm Suffix Trees Coclusio Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer 2018
4 Ptter Mtchig Defiitio [1] Serch for strig (ptter) i lrge ody of text T [0.. 1] The text (or hystck) eig serched withi P[0..m 1] The ptter (or eedle) eig serched for Strigs over lphet Σ Retur the first i such tht P[j] = T [i + j] for 0 j m 1 This is the first occurrece of P i T If P does ot occur i T, retur FAIL Applictios: Iformtio Retrievl (text editors, serch egies) Bioiformtics Dt Miig Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
5 Ptter Mtchig Defiitio [2] Exmple: T = Where is he? P 1 = he P 2 = who Defiitios: Sustrig T [i..j] 0 i j < : strig of legth j i + 1 which cosists of chrcters T [i],... T [j] i order A prefix of T : sustrig T [0..i] of T for some 0 i < A suffix of T : sustrig T [i.. 1] of T for some 0 i 1 Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
6 Geerl Ide of Algorithms Ptter mtchig lgorithms cosist of guesses d checks: A guess is positio i such tht P might strt t T [i]. Vlid guesses (iitilly) re 0 i m. A check of guess is sigle positio j with 0 j < m where we compre T [i + j] to P[j]. We must perform m checks of sigle correct guess, ut my mke (my) fewer checks of icorrect guess. We will digrm sigle ru of y ptter mtchig lgorithm y mtrix of checks, where ech row represets sigle guess. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
7 Brute-force Algorithm Ide: Check every possile guess. BruteforcePM(T [0.. 1], P[0..m 1]) T : Strig of legth (text), P: Strig of legth m (ptter) 1. for i 0 to m do 2. for j 0 to m 1 do 3. if T [i + j]!= P[j] the 4. rek 5. if j = m the 6. retur i 7. retur FAIL Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
8 Brute-Force Exmple Exmple: T =, P = Wht is the worst possile iput? P = m 1, T = Worst cse performce Θ(( m + 1)m) m /2 Θ(m) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
9 How to improve? More sophisticted lgorithms Do extr preprocessig o the ptter P Determiistic fiite utomt (DFA), KMP Boyer-Moore Ri-Krp We elimite guesses sed o completed mtches d mismtches. Do extr preprocessig o the text T Suffix-trees We crete dt structure to fid mtches esily. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
10 Outlie 1 Strig Mtchig Itroductio Strig Mtchig with Fiite Automt Kuth-Morris-Prtt lgorithm Boyer-Moore Algorithm Ri-Krp Algorithm Suffix Trees Coclusio Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer 2018
11 Strig Mtchig with Fiite Automt Exmple: Automto for the ptter P = c Σ Σ c You should e fmilir with: fiite utomto, DFA, NFA, covertig NFA to DFA trsitio fuctio δ, sttes Q, cceptig sttes F Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
12 Strig Mtchig with Fiite Automt Exmple: Automto for the ptter P = c Σ Σ c You should e fmilir with: fiite utomto, DFA, NFA, covertig NFA to DFA trsitio fuctio δ, sttes Q, cceptig sttes F The ove fiite utomtio is NFA Stte q expresses we hve see P[0..q 1] NFA ccepts T if d oly if T cotis c But evlutig NFAs is very slow. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
13 Strig mtchig with DFA C show: There exists equivlet smll DFA.,c Σ c c,c c,c,c Esy to test whether P is i T. But how do we fid the rcs? Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
14 Strig mtchig with DFA Code There is o eed to uild explicit DFA with odes d rcs. All tht is eeded is the trsitio-fuctio δ(q, c) = stte to go to c q δ(q, c) C the test for ptter i O() time y prsig text with DFA. DFA-PM(T [0.. 1], P[0..m 1]) T : Strig of legth (text), P: Strig of legth m (ptter) 1. δ trsitio fuctio for DFA 2. q 0 // curret stte 3. for i 0 to 1 do 4. q δ(q, T [i]) 5. if q = m 6. retur Ptter occurs with shift i (m 1) 7. retur FAIL Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
15 Strig mtchig with DFA Exmple Exmple: T = c, P = c,c Σ c c,c c,c,c T : c P: c to stte 4 () () () () c to stte 4 () () () () c q: (fter redig this chrcter) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
16 Strig mtchig with DFA Trsitios Let P the ptter to serch, of legth m. The stte of the utomto is i {0,..., m} Stte q represets the text tht hs ee prsed so fr eds with P[0..q 1], (d there is o loger prefix of P tht it eds with) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
17 Strig mtchig with DFA Trsitios Let P the ptter to serch, of legth m. The stte of the utomto is i {0,..., m} Stte q represets the text tht hs ee prsed so fr eds with P[0..q 1], (d there is o loger prefix of P tht it eds with) If the ext chrcter is c d does ot exted ptter: The text tht hs ee prsed so fr eds with P[0...q 1] c This is ot prefix of P Wt: The logest prefix of P tht is suffix of P[0..q 1] c The ew stte is the legth of tht prefix Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
18 Strig mtchig with DFA Trsitios Let P the ptter to serch, of legth m. The stte of the utomto is i {0,..., m} Stte q represets the text tht hs ee prsed so fr eds with P[0..q 1], (d there is o loger prefix of P tht it eds with) If the ext chrcter is c d does ot exted ptter: The text tht hs ee prsed so fr eds with P[0...q 1] c This is ot prefix of P Wt: The logest prefix of P tht is suffix of P[0..q 1] c The ew stte is the legth of tht prefix Exmple: q c P[0..q 1] c Prefixes of P logest δ(q, c) 5 Λ,,,,,, Λ,,,,,,... 1 Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
19 Strig mtchig with DFA Rutime Mtchig time o text strig of legth is Θ() Preprocessig c clerly e doe i O(m 3 Σ ) time. There exists lgorithm to do preprocessig i O(m Σ ). Altogether, ptter mtchig with DFA hs ru-time O(m Σ + ). This is fst ( Σ is costt) ut we c e fster! Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
20 Outlie 1 Strig Mtchig Itroductio Strig Mtchig with Fiite Automt Kuth-Morris-Prtt lgorithm Boyer-Moore Algorithm Ri-Krp Algorithm Suffix Trees Coclusio Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer 2018
21 Kuth-Morris-Prtt Motivtio Σ c Σ Use ew type of trsitio ( filure ): Use this trsitio oly if o other fits. Does ot cosume chrcter. With these rules, computtios of the utomto re determiistic. (But it is formlly ot vlid DFA.) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
22 Kuth-Morris-Prtt Motivtio Σ c Σ Use ew type of trsitio ( filure ): Use this trsitio oly if o other fits. Does ot cosume chrcter. With these rules, computtios of the utomto re determiistic. (But it is formlly ot vlid DFA.) C store filure-fuctio i rry F [0..m 1] The filure rc from stte j leds to F [j 1] Give the filure-rry, we c esily test whether P is i T : Automto ccepts T if d oly if T cotis c Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
23 Kuth-Morris-Prtt Algorithm KMP(T, P), to retur the first mtch T : Strig of legth (text), P: Strig of legth m (ptter) 1. F filurearry(p) 2. i 0 // curret chrcter of T to prse 3. j 0 // curret stte tht we re i 4. while i < do 5. if P[j] = T [i] the 6. if j = m 1 the 7. retur i m + 1 //mtch 8. else 9. i i j j else // i. e. P[j] T [i] 12. if j > 0 the 13. j F [j 1] 14. else 15. i i retur FAIL // o mtch Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
24 Strig mtchig with KMP Exmple Exmple: T = c, P = c Σ c Σ T : c P : to stte 3 () () () to stte 3 () () () c q: , 4 5 3, (fter redig this chrcter) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
25 Strig mtchig with KMP Filure-fuctio Let P the ptter to serch, of legth m. The stte of the utomto is i {0,..., m} Stte j represets the text tht hs ee prsed so fr eds with P[0..j 1], (d there is o loger prefix of P tht it eds with) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
26 Strig mtchig with KMP Filure-fuctio Let P the ptter to serch, of legth m. The stte of the utomto is i {0,..., m} Stte j represets the text tht hs ee prsed so fr eds with P[0..j 1], (d there is o loger prefix of P tht it eds with) If the ext chrcter does ot exted ptter: The text tht hs ee prsed so fr eds with P[0...j 1] (Recll tht trsitio does ot cosume chrcter) Wt: The logest prefix of P tht is suffix of P[0..j 1] (ut exclude P[0..j 1] itself, sice tht leds to mismtch) The ew stte is the legth of tht prefix Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
27 Strig mtchig with KMP Filure-fuctio Let P the ptter to serch, of legth m. The stte of the utomto is i {0,..., m} Stte j represets the text tht hs ee prsed so fr eds with P[0..j 1], (d there is o loger prefix of P tht it eds with) If the ext chrcter does ot exted ptter: The text tht hs ee prsed so fr eds with P[0...j 1] (Recll tht trsitio does ot cosume chrcter) Wt: The logest prefix of P tht is suffix of P[0..j 1] (ut exclude P[0..j 1] itself, sice tht leds to mismtch) The ew stte is the legth of tht prefix Oe-setece descriptio: F [j] is the legth of the logest prefix of P[0..j] tht is suffix of P[1..j]. The filure-rc leds from stte j to stte F [j 1] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
28 KMP Filure Arry Exmple F [j] is the legth of the logest prefix of P[0..j] tht is suffix of P[1..j]. Cosider P = c j P[1..j] Prefixes of P logest F [j] 0 Λ Λ,,,,,,... Λ 0 1 Λ,,,,,,... Λ 0 2 Λ,,,,,, Λ,,,,,, Λ,,,,,, c Λ,,,,,,... Λ 0 6 c Λ,,,,,,... 1 This c clerly e computed i O(m 3 ) time, ut we c do etter! Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
29 Computig the Filure Arry filurearry(p) P: Strig of legth m (ptter) 1. F [0] 0 2. i 1 3. j 0 4. while i < m do 5. if P[i] = P[j] the 6. j j F [i] j 8. i i else if j > 0 the 10. j F [j 1] 11. else 12. F [i] i i + 1 Correctess-ide: F [ ] is defied vi ptter mtchig of P[1..i] i P. So KMP uses itself! Alredy-uilt prts of F [ ] re used to expd it. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
30 KMP filure fuctio fst computtio P = c Σ Σ c Prse P[1..m 1] = c while ddig filure-rcs: Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
31 KMP filure fuctio fst computtio P = c Σ Σ c Prse P[1..m 1] = c while ddig filure-rcs: i 1 P[i] P[j] j 0 F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
32 KMP filure fuctio fst computtio P = c Σ Σ c Prse P[1..m 1] = c while ddig filure-rcs: i 1 1 P[i] P[j] j 0 F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
33 KMP filure fuctio fst computtio P = c Σ Σ c Prse P[1..m 1] = c while ddig filure-rcs: i P[i] P[j] j 0 0 F [i] 0 Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
34 KMP filure fuctio fst computtio P = c Σ Σ c Prse P[1..m 1] = c while ddig filure-rcs: i P[i] P[j] j 0 0 F [i] 0 Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
35 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] P[j] j F [i] 0 1 Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
36 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] P[j] j F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
37 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] P[j] j F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
38 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] c P[j] j F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
39 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] c P[j] j F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
40 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] c c P[j] j F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
41 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] c c P[j] j F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
42 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] c c c P[j] j c 0 F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
43 KMP filure fuctio fst computtio P = c Σ Σ c 6 7 Prse P[1..m 1] = c while ddig filure-rcs: i P[i] c c c P[j] j c 0 1 F [i] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
44 KMP Rutime filurearry At ech itertio of the while loop, t lest oe of the followig hppes: 1 i icreses y oe, or 2 the idex i j icreses y t lest oe (F [j 1] < j) There re o more th 2m itertios of the while loop Ruig time: Θ(m) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
45 KMP Rutime filurearry At ech itertio of the while loop, t lest oe of the followig hppes: 1 i icreses y oe, or 2 the idex i j icreses y t lest oe (F [j 1] < j) KMP There re o more th 2m itertios of the while loop Ruig time: Θ(m) filurearry c e computed i Θ(m) time At ech itertio of the while loop, t lest oe of the followig hppes: 1 i icreses y oe, or 2 the idex i j icreses y t lest oe (F [j 1] < j) There re o more th 2 itertios of the while loop Ruig time KMP ltogether: Θ( + m) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
46 Outlie 1 Strig Mtchig Itroductio Strig Mtchig with Fiite Automt Kuth-Morris-Prtt lgorithm Boyer-Moore Algorithm Ri-Krp Algorithm Suffix Trees Coclusio Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer 2018
47 Boyer-Moore Algorithm Overview Bsed o three key ides: Reverse-order serchig: Compre P with susequece of T movig ckwrds Bd chrcter jumps: Whe mismtch occurs t T [i] = c, use loctio of c i P (if y) to elimite guesses. Good suffix jumps: Whe mismtch occurs, the use recetly see suffix of P to elimite guesses. Similr to filure rry i KMP. This gives two possile shifts (loctios of ext guess to try). Use the oe tht moves forwrd more. I prctice lrge prts of T will ot e looked t. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
48 Boyer-Moore Algorithm BoyerMoore(T,P) 1. L lst occurrece rry computed from P 2. S good suffix rry computed from P 3. k 0 // curret guess 4. while k m 5. i k + m 1, j m 1 6. while j 0 d T [i] = P[j] the 7. i i 1 8. j j 1 9. if j = 1 retur k 10. else 11. k k + mx(1, j L[T [i]], m 1 S[j]) 12. retur FAIL L d S will e explied elow. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
49 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o P = m o o r e T = o y e r m o o r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
50 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o P = m o o r e T = o y e r m o o r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
51 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o o P = m o o r e T = o y e r m o o r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
52 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o d o P = m o o r e T = o y e r m o o r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
53 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o l d o P = m o o r e T = o y e r m o o r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
54 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o l d o P = m o o r e T = o y e r m o o r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
55 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o l d o 6 chrcters ot looked t P = m o o r e T = o y e r m o o r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
56 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o l d o 6 chrcters ot looked t P = m o o r e T = o y e r m o o r e e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
57 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o l d o 6 chrcters ot looked t P = m o o r e T = o y e r m o o r e e (r) e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
58 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o l d o 6 chrcters ot looked t P = m o o r e T = o y e r m o o r e e (r) e (m) e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
59 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o l d o 6 chrcters ot looked t P = m o o r e T = o y e r m o o r e e (r) e (m) r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
60 Bd chrcter exmples P = l d o T = w h e r e i s w l d o o o l d o 6 chrcters ot looked t P = m o o r e T = o y e r m o o r e 4 chrcters ot looked t e (r) e (m) o o r e Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
61 Lst-Occurrece Fuctio Preprocess the ptter P d the lphet Σ Build the lst-occurrece fuctio L mppig Σ to itegers L(c) is defied s the lrgest idex i such tht P[i] = c or 1 if o such idex exists Exmple: P=moore c m o r e ll others L(c) T = o y e r m o o r e e k = 0, i = j = 4, T [i] = r, L[T [i]] = 3 k k = 1 (or igger) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
62 Lst-Occurrece Fuctio Preprocess the ptter P d the lphet Σ Build the lst-occurrece fuctio L mppig Σ to itegers L(c) is defied s the lrgest idex i such tht P[i] = c or 1 if o such idex exists Exmple: P=moore c m o r e ll others L(c) T = o y e r m o o r e e k = 0, i = j = 4, T [i] = r, L[T [i]] = 3 k k = 1 (or igger) (r) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
63 Good suffix exmples P = sells shells s h e i l s e l l s s h e l l s P = odetofood i l i k e f o o d f r o m m e x i c o Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
64 Good suffix exmples P = sells shells s h e i l s e l l s s h e l l s h e l l s P = odetofood i l i k e f o o d f r o m m e x i c o Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
65 Good suffix exmples P = sells shells s h e i l s e l l s s h e l l s h e l l s (e) (l) (l) (s) P = odetofood i l i k e f o o d f r o m m e x i c o o f o o d Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
66 Good suffix exmples P = sells shells s h e i l s e l l s s h e l l s h e l l s (e) (l) (l) (s) P = odetofood i l i k e f o o d f r o m m e x i c o o f o o d Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
67 Good suffix exmples P = sells shells s h e i l s e l l s s h e l l s h e l l s (e) (l) (l) (s) P = odetofood i l i k e f o o d f r o m m e x i c o o f o o d (o) (d) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
68 Good suffix exmples P = sells shells s h e i l s e l l s s h e l l s h e l l s (e) (l) (l) (s) P = odetofood i l i k e f o o d f r o m m e x i c o o f o o d (o) (d) Crucil igrediet: logest suffix of P[i+1..m 1] tht occurs i P. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
69 Good Suffix rry Precompute suffix skip rry S of size m: For 0 j < m, S[j] stores shift if serch filed t T [i] P[j] At this poit, hd T [i+1..k+m 1] = P[j+1..m 1] Also kow T [i] P[j] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
70 Good Suffix rry Precompute suffix skip rry S of size m: For 0 j < m, S[j] stores shift if serch filed t T [i] P[j] At this poit, hd T [i+1..k+m 1] = P[j+1..m 1] Also kow T [i] P[j] S[j] is the mximum idex l such tht OR P[j+1... m 1] is suffix of P[0... l] d P[j] P[l m+j+1] h e l l s (e) (l) (l) (s) P[0... l] is suffix of P[j+1,..., m 1] o f o o d (o) (d) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
71 Good Suffix rry Precompute suffix skip rry S of size m: For 0 j < m, S[j] stores shift if serch filed t T [i] P[j] At this poit, hd T [i+1..k+m 1] = P[j+1..m 1] Also kow T [i] P[j] S[j] is the mximum idex l such tht OR P[j+1... m 1] is suffix of P[0... l] d P[j] P[l m+j+1] h e l l s (e) (l) (l) (s) P[0... l] is suffix of P[j+1,..., m 1] o f o o d (o) (d) Computle (similr to KMP filure fuctio) i Θ(m) time. You do ot eed to kow how to fid the vlues S[j], ut you must e le to fid the ext guess o exmples. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
72 Boyer-Moore lgorithm coclusio Worst-cse ruig time O( + m + Σ ) if P is ot i T. This complexity is difficult to prove. Worst-cse ruig time O(m) if we wt to report ll occurreces (KMP c do i O( + m)) O typicl Eglish text the lgorithm proes pproximtely 25% of the chrcters i T Fster th KMP i prctice o Eglish text. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
73 Outlie 1 Strig Mtchig Itroductio Strig Mtchig with Fiite Automt Kuth-Morris-Prtt lgorithm Boyer-Moore Algorithm Ri-Krp Algorithm Suffix Trees Coclusio Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer 2018
74 Ri-Krp Figerprit Algorithm Ide Ide: use hshig Compute hsh fuctio for ech text positio No explicit hsh tle: just compre with ptter hsh If mtch of the hsh vlue of the ptter d text positio foud, the compres the ptter with the sustrig y ive pproch Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
75 Ri-Krp Figerprit Algorithm Ide Ide: use hshig Compute hsh fuctio for ech text positio No explicit hsh tle: just compre with ptter hsh If mtch of the hsh vlue of the ptter d text positio foud, the compres the ptter with the sustrig y ive pproch Exmple: P: , T : Hsh fuctio: h(x) = x mod 97 gives h(p) = 95. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
76 Ri-Krp Figerprit Algorithm Ide Ide: use hshig Compute hsh fuctio for ech text positio No explicit hsh tle: just compre with ptter hsh If mtch of the hsh vlue of the ptter d text positio foud, the compres the ptter with the sustrig y ive pproch Exmple: P: , T : Hsh fuctio: h(x) = x mod 97 gives h(p) = hsh-vlue 84 hsh-vlue 94 hsh-vlue 76 hsh-vlue 18 hsh-vlue 95 Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
77 Ri-Krp Figerprit Algorithm First Attempt Ri-Krp-Simple(T, P) 1. M suitle prime umer 2. h P compute-hsh-vlue(p[0..m 1], M) 3. for k 0 to m 4. h T compute-hsh-vlue(t [k..k+m 1], M) 5. if h T = h P the 6. if T [k..k+m 1] = P 7. retur foud t shift k 8. retur ot foud Never misses mtch sice h(k 1 ) h(k 2 ) implies k 1 k 2 h(t [k..k+m 1]) depeds o m chrcters, so ive computtio tkes Θ(m) time per shift Ruig time is Θ(m) for serch miss (how c we improve this?) Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
78 Ri-Krp Figerprit Algorithm Fst Rehsh The iitil hshes re clled figerprits. Crucil isight: We c updte these figerprits i costt time. Use previous hsh to compute ext hsh O(1) time per hsh, except first oe Exmple: Pre-compute: mod 97 = 9 Previous hsh: mod 97 = 76 Next hsh: mod 97 =?? Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
79 Ri-Krp Figerprit Algorithm Fst Rehsh The iitil hshes re clled figerprits. Crucil isight: We c updte these figerprits i costt time. Use previous hsh to compute ext hsh O(1) time per hsh, except first oe Exmple: Pre-compute: mod 97 = 9 Previous hsh: mod 97 = 76 Next hsh: mod 97 =?? Oservtio: mod 97= (41592 ( )) 10+6 mod97 = (76 (4 9 )) 10+6 mod97 = 406 mod97 = 18 Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
80 Ri-Krp Figerprit Algorithm Coclusio Ri-Krp-Fst(T, P) 1. M suitle prime umer 2. h P compute-hsh-vlue(p[0..m 1], M) 3. h T compute-hsh-vlue(t [0..m 1], M) 4. s 10 m 1 mod M 5. for k 0 to m 6. if h T = h P the 7. if T [k..k+m 1] = P 8. retur foud t shift k 9. if k < m the 10. h T ((h T T [k] s) 10 + T [k+m]) mod M 11. retur ot foud Choose tle size M t rdom to e huge prime Expected ruig time is O(m + ) Θ(m) worst-cse, ut this is (uelievly) ulikely Exteds to 2d ptters d other geerliztios Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
81 Outlie 1 Strig Mtchig Itroductio Strig Mtchig with Fiite Automt Kuth-Morris-Prtt lgorithm Boyer-Moore Algorithm Ri-Krp Algorithm Suffix Trees Coclusio Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer 2018
82 Tries of Suffixes d Suffix Trees Wht if we wt to serch for my ptters P withi the sme fixed text T? Ide: Preprocess the text T rther th the ptter P Oservtio: P is sustrig of T if d oly if P is prefix of some suffix of T. So wt to store ll suffixes of T i trie. To sve spce: Use compressed trie. Store suffixes implicitly vi idices ito T. This is clled suffix tree. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
83 Trie of suffixes: Exmple T = hs suffixes {,,,,,,,,, Λ} Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
84 Tries of suffixes Store suffixes vi idices: T = T[9..9] T[5..9] T[8..9] T[7..9] T[6..9] T[3..9] T[4..9] T[1..9] T[0..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
85 Suffix tree Suffix tree: Compressed trie of suffixes T = 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
86 Buildig Suffix Trees Text T hs chrcters d + 1 suffixes We c uild the suffix tree y isertig ech suffix of T ito compressed trie. This tkes time Θ( 2 ). There is wy to uild suffix tree of T i Θ() time. This is quite complicted d eyod the scope of the course. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
87 Suffix Trees: Strig Mtchig Assume we hve suffix tree of text T. To serch for ptter P of legth m: We ssume tht P does ot hve the fil. P is the prefix of some suffix of T. I the ucompressed trie, serchig for P would e esy: P exists i T if d oly serch for P reches ode i the trie. I the suffix tree, serch for P util oe of the follow occurs: 1 If serch fils due to o such child the P is ot i T 2 If we rech ed of P, sy t ode v, the jump to lef l i sutree of v. (We presume tht suffix trees stores such shortcuts.) 3 Else we rech lef l = v while chrcters of P left. Either wy, left idex t l gives the shift tht we should check. This tkes O( P ) time. Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
88 Ptter Mtchig i Suffix Tree: Exmple T = P = 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 o such child T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
89 Ptter Mtchig i Suffix Tree: Exmple T = P = 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 o such child T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
90 Ptter Mtchig i Suffix Tree: Exmple T = P = 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
91 Ptter Mtchig i Suffix Tree: Exmple T = P = 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
92 Ptter Mtchig i Suffix Tree: Exmple T = P = 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
93 Ptter Mtchig i Suffix Tree: Exmple T = P = 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
94 Ptter Mtchig i Suffix Tree: Exmple T = P = rir 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
95 Ptter Mtchig i Suffix Tree: Exmple T = P = rir 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
96 Ptter Mtchig i Suffix Tree: Exmple T = P = rir 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
97 Ptter Mtchig i Suffix Tree: Exmple T = P = rir 0 T[9..9] 1 3 T[5..9] 2 T[6..9] T[0..9] T[7..9] 3 T[3..9] T[1..9] 1 T[8..9] 2 T[4..9] T[2..9] Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
98 Outlie 1 Strig Mtchig Itroductio Strig Mtchig with Fiite Automt Kuth-Morris-Prtt lgorithm Boyer-Moore Algorithm Ri-Krp Algorithm Suffix Trees Coclusio Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer 2018
99 Strig Mtchig Coclusio Brute- Force DFA KMP BM RK Preproc. O(m Σ ) O(m) O(m + Σ ) O(m) Serch O() O( + m) O(m) O() O() time (ofte etter) (expected) Extr spce Suffix trees O( 2 ) ( O()) O(m) O(m Σ ) O(m) O(m + Σ ) O(1) O() Biedl, Lctot, Sepehri, Wild (SCS, UW) CS240 Module 9 Witer / 42
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