CPE702 Algorithm Analysis and Design Week 11 String Processing

Size: px

Start display at page:

Download "CPE702 Algorithm Analysis and Design Week 11 String Processing"

Molly Wade
5 years ago
Views:

1 CPE702 Agorithm Anaysis and Dsign Wk 11 String Procssing Prut Boonma Dpartmnt of Computr Enginring Facuty of Enginring, Chiang Mai Univrsity Basd on Sids by M.T. Goodrich and R. Tamassia

2 2 In this wk String Procssing Strings Pattrn Matching Tri

3 Strings A string is a squnc of charactrs Examps of strings: Java program HTML documnt DNA squnc Digitizd imag An aphabt Σ is th st of possib charactrs for a famiy of strings Examp of aphabts: ASCII

3 3 Strings A string is a squnc of charactrs Examps of strings: Java program HTML documnt DNA squnc Digitizd imag An aphabt Σ is th st of possib charactrs for a famiy of strings Examp of aphabts: ASCII Unicod {0, 1} {A, C, G, T} Lt P b a string of siz m A substring P[i.. j] of P is th subsqunc of P consisting of th charactrs with ranks btwn i and j A prfix of P is a substring of th typ P[0.. j] A suffix of P is a substring of th typ P[i..m 1] Givn strings T (txt) and P (pattrn), th pattrn matching probm consists of finding a substring of T qua to P Appications: Txt ditors Sarch ngins Bioogica rsarch

4 4 Brut-Forc Pattrn Matching Th brut-forc pattrn matching agorithm compars th pattrn P with th txt T for ach possib shift of P rativ to T, unti ithr a match is found, or a pacmnts of th pattrn hav bn trid Brut-forc pattrn matching runs in tim O(nm) Examp of worst cas: T = aaa ah P = aaah may occur in imags and DNA squncs uniky in Engish txt Agorithm BrutForcMatch(T, P) Input txt T of siz n and pattrn P of siz m Output starting indx of a substring of T qua to P or 1 if no such substring xists for i 0 to n m { tst shift i of th pattrn } j 0 whi j < m T[i + j] = P[j] j j + 1 if j = m rturn i {match at i} s brak whi oop {mismatch} rturn -1 {no match anywhr}

5 5 Boyr-Moor Huristics Th Boyr-Moor s pattrn matching agorithm is basd on two huristics Looking-gass huristic: Compar P with a subsqunc of T moving backwards Charactr-jump huristic: Whn a mismatch occurs at T[i] = c If P contains c, shift P to aign th ast occurrnc of c in P with T[i] Es, shift P to aign P[0] with T[i + 1] Examp T a p a t t r n m a t c h i n g a g o r i t h m P 1 r i t h m 3 r i t h m 5 r i t h m r i t h m 2 r i t h m 4 r i t h m 6 r i t h m

6 6 Last-Occurrnc Function Boyr-Moor s agorithm prprocsss th pattrn P and th aphabt Σ to buid th ast-occurrnc function L mapping Σ to intgrs, whr L(c) is dfind as th argst indx i such that P[i] = c or 1 if no such indx xists Examp: Σ = {a, b, c, d} P = abacab c a b c d L(c) Th ast-occurrnc function can b rprsntd by an array indxd by th numric cods of th charactrs Th ast-occurrnc function can b computd in tim O(m + s), whr m is th siz of P and s is th siz of Σ

7 7 Th Boyr-Moor Agorithm Agorithm BoyrMoorMatch(T, P, Σ) L astoccurncfunction(p, Σ ) i m 1 j m 1 rpat if T[i] = P[j] if j = 0 rturn i { match at i } s i i 1 j j 1 s { charactr-jump } L[T[i]] i i + m min(j, 1 + ) j m 1 unti i > n 1 rturn 1 { no match } Cas 1: j a i b a j m j.... b a j Cas 2: 1 + j a a.. i b. j m (1 + ). a.. b. 1 +

8 8 Examp a b a c a a b a d c a b a c a b a a b b 1 a b a c a b a b a c a b 5 a b a c a b a b a c a b 8 a b a c a b 6 a b a c a b

9 9 Anaysis Boyr-Moor s agorithm runs in tim O(nm + s) Examp of worst cas: T = aaa a P = baaa Th worst cas may occur in imags and DNA squncs but is uniky in Engish txt Boyr-Moor s agorithm is significanty fastr than th brut-forc agorithm on Engish txt a a a a a a a a a b a a a a a 12 b a a a a a b a a a a a b a a a a a

10 10 Th KMP Agorithm Knuth-Morris-Pratt s agorithm compars th pattrn to th txt in ft-toright, but shifts th pattrn mor intignty than th brut-forc agorithm. Whn a mismatch occurs, what is th most w can shift th pattrn so as to avoid rdundant comparisons? Answr: th argst prfix of P[0..j] that is a suffix of P[1..j].. a b a a b x..... a b a a b a No nd to rpat ths comparisons j a b a a b a Rsum comparing hr

11 11 KMP Faiur Function Knuth-Morris-Pratt s agorithm prprocsss th pattrn to find matchs of prfixs of th pattrn with th pattrn itsf Th faiur function F(j) is dfind as th siz of th argst prfix of P[0..j] that is aso a suffix of P[1..j] Knuth-Morris-Pratt s agorithm modifis th brut-forc agorithm so that if a mismatch occurs at P[j] T[i] w st j F(j j P[j] a b a a b a F(j) a b a a b x..... a b a a b a F(j 1) j a b a a b a

12 12 Th KMP Agorithm Th faiur function can b rprsntd by an array and can b computd in O(m) tim At ach itration of th whi-oop, ithr i incrass by on, or th shift amount i j incrass by at ast on (obsrv that F(j Agorithm KMPMatch(T, P) F faiurfunction(p) i 0 j 0 whi i < n if T[i] = P[j] if j = m 1 rturn i j { match } s i i + 1 j j + 1 s if j > 0 j F[j 1] s i i + 1 rturn 1 { no match }

13 Computing th Faiur Function 13 Th faiur function can b rprsntd by an array and can b computd in O(m) tim Th construction is simiar to th KMP agorithm itsf At ach itration of th whi-oop, ithr i incrass by on, or th shift amount i j incrass by at ast on (obsrv that F(j 1) < j) Hnc, thr ar no mor Agorithm faiurfunction(p) F[0] 0 i 1 j 0 whi i < m if P[i] = P[j] {w hav matchd j + 1 chars} F[i] j + 1 i i + 1 j j + 1 s if j > 0 thn {us faiur function to shift P} j F[j 1] s F[i] 0 { no match } i i + 1

14 14 Examp a b a c a a b a c c a b a c a b a a b b a b a c a b 7 a b a c a b j P[j] a b a c a b F(j) a b a c a b 13 a b a c a b a b a c a b

15 15 Prprocssing Strings Prprocssing th pattrn spds up pattrn matching quris Aftr prprocssing th pattrn, KMP s agorithm prforms pattrn matching in tim proportiona to th txt siz If th txt is arg, immutab and sarchd for oftn (.g., works by Shakspar), w may want to prprocss th txt instad of th pattrn A tri is a compact data structur for rprsnting a st of strings, such as a th words in a txt A tris supports pattrn matching quris in tim proportiona to th pattrn siz

16 16 Standard Tris Th standard tri for a st of strings S is an ordrd tr such that: Each nod but th root is abd with a charactr Th chidrn of a nod ar aphabticay ordrd Th paths from th xtrna nods to th root yid th strings of S Examp: standard tri for th st of strings S = { bar, b, bid, bu, buy, s, stock, stop } b s i u t a d y o r c p k

17 17 Anaysis of Standard Tris A standard tri uss O(n) spac and supports sarchs, insrtions and dtions in tim O(dm), whr: n tota siz of th strings in S m siz of th string paramtr of th opration d siz of th aphabt b s i u t a d y o r c p k

18 18 Word Matching with a Tri W insrt th words of th txt into a tri Each af stors th occurrncs of th associatd word in th txt s a b a r? s s t o c k! s a b u? b u y s t o c k! b i d s t o c k! b i d s t o c k! h a r t h b? s t o p! b h s i u t a d y a o r , r 69 0, c k p 84 17, 40, 51, 62

19 19 Comprssd Tris A comprssd tri has intrna nods of dgr at ast two It is obtaind from standard tri by comprssing chains of rdundant nods ar b id u y s ck to p b s i u t a d y o r c p k

20 20 Compact Rprsntation Compact rprsntation of a comprssd tri for an array of strings: Stors at th nods rangs of indics instad of substrings Uss O(s) spac, whr s is th numbr of strings in th array Srvs as an auxiiary indx structur S[0] = S[1] = S[2] = S[3] = s b a r s s t o c k S[4] = S[5] = S[6] = S[7] = b u b u y b i d S[8] = S[9] = h a r b s t o p 1, 0, 0 7, 0, 3 0, 0, 0 1, 1, 1 6, 1, 2 4, 1, 1 0, 1, 1 3, 1, 2 1, 2, 3 8, 2, 3 4, 2, 3 5, 2, 2 0, 2, 2 2, 2, 3 3, 3, 4 9, 3, 3

21 21 Suffix Tri Th suffix tri of a string X is th comprssd tri of a th suffixs of X m i n i z m i i mi nimiz z miz nimiz z nimiz z

22 22 Anaysis of Suffix Tris Compact rprsntation of th suffix tri for a string X of siz n from an aphabt of siz d Uss O(n) spac Supports arbitrary pattrn matching quris in X in O(dm) tim, whr m is th siz of th pattrn Can b constructd in O(n) tim m i n i z m i , 7 1, 1 0, 1 2, 7 6, 7 4, 7 2, 7 6, 7 2, 7 6, 7

23 23 Homwork Brut-forc matching for string with prfix, suffix and infix. Pattrn can b h* which can match any words startd with h For xamp, if T = This is monday, P = mon*, thn matchd substring is monday *day which can match any words ndd with day For xamp, if T = This is monday, P = *day, thn matchd substring is monday h*o which can match any substring startd with h and ndd with o For xamp, if T = This is monday, P = T*s, thn matchd substring is This Input: two strings, th first is txt (T) and th scond is pattrn (P) Output: th ocation of th first matchd substring in T. Nu if no match. Du dat: Spt 9, 2010

Indexed Search Tree (Trie)

Indexed Search Tree (Trie) Indxd Sarch Tr (Tri) Fawzi Emad Chau-Wn Tsng Dpartmnt of Computr Scinc Univrsity of Maryand, Cog Park Indxd Sarch Tr (Tri) Spcia cas of tr Appicab whn Ky C can b dcomposd into a squnc of subkys C 1, C