Strings. String Processing. Periodicity. Strings. Typical applications:

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Oscar Nelson
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1 String Processing Typicl pplictions: pttern mtching/recognition moleculr biology, comprtive genomics, informtion retrievl dt/text mining dt/text compression, coding, encryption string processing in lrge dtbses 25/02/2011 Applied Algorithmics - week3 1 Strings A string is sequence of symbols drwn from some well defined set cll the lphbet. Exmples of lphbets include: ASCII code, Unicode binry lphbet {0,1} System of DNA bse-pirs {A,C,G,T} Ltin, Greek, Chinese lphbet Exmples of strings Jv/C/ADA progrms, HTML/XML documents, DNA sequences, imge/video/udio files 25/02/2011 Applied Algorithmics - week3 2 Strings Bsic definitions: Let A be n lphbet. We sy tht A + contins ll nonempty strings bsed on symbols from A, nd A * =A + {ε}, { where ε is n empty string. Let w be string of length n. We sy tht w=w[0..n-1]. Any initil frgment w[0..i] is clled prefix of w. Any finl frgment w[j..n-1] is clled suffix of w. Any frgment of form w[i..j] is clled substring of w. Periodicity We sy tht w=w[0..n-1] hs period p iff w[i]=w[i+p], for ll 0 i n-p-1, see below w b p w b p p+1 p+2 b For exmple string w=bbbb hs period 3 n-1 n-p-1 n-1 25/02/2011 Applied Algorithmics - week3 3 25/02/2011 Applied Algorithmics - week3 4

2 Periodicity Lemm But w=bbbb hs lso periods 6, 9 nd 11 Lemm: p is the shortest period in w=w[0..n-1] iff w[0..n-p-1] is the longest prefix of w which is lso suffix w[p..n-1], see figure on previous slide Periodicty Lemm: If string w=w[0..n-1] hs periods p nd q such tht p+q n then w hs lso period gcd(p,q), where gcd(,) stnds for the gretest common divisor of two integers. 25/02/2011 Applied Algorithmics - week3 5 Periodicity Lemm Proof: The min observtion is bsed on the fct tht if string w hs two periods p q then it lso hs period p-q. p p-q The thesis of the periodicity lemm follows from the observtion (Euclid s Algorithm) tht for ny positive integers >b, gcd(,b)=gcd(b,-b). 25/02/2011 Applied Algorithmics - week3 6 q String pttern mtching Input: given two strings: P=P[0..m-1] clled the pttern nd T=T[0..n-1] clled the text. Tsk: is to find ll occurrences of P in T using s smll number s possible text symbol comprisons, where n occurrence of P t position i in T is defined s P[j]=T[i+j] for ll 0 j m-1, see exmple below T P b i b 0 i+m-1 25/02/2011 Applied Algorithmics - week3 7 m-1 Brute-Force Algorithm The brute-force lgorithm tests nively (vi consecutive symbols comprison) whether pttern P occurs t ny permissible position 0 i n-m-1 in text T. The test t ech position cn cost s much s m, for exmple when T=.. nd P= possible scenrio in imges, unlikely in nturl lnguges, codes Thus the time complexity of brute force lgorithms is bounded by (n-m) m=o(n m) 25/02/2011 Applied Algorithmics - week3 8

3 Brute-force lgorithm - code Algorithm Brute-Force-First-Mtch(T,P): integer; for i 0 to n-m-1 do j 0; while (j<m) nd (T[i+j]=P[j]) do j j +1; if (j=m) then return (i); return (-1); 25/02/2011 Applied Algorithmics - week3 9 More efficient pttern mtching Cn we perform pttern mtching in time O(m+n) in ny, even the worst cse, scenrio? The nswer is yes, nd the solution is bsed on proper use of periodicity of strings. But wht is the cuse of high complexity nywy? It must be multiple comprisons of text symbols. Cn we do something bout it? Indeed, we cn, t lest on most of the occsions. 25/02/2011 Applied Algorithmics - week3 10 Principle of Knuth-Morris-Prtt KMP Algorithm In Brute-force solution, when the lgorithm moves from position i to i+1 it forgets ll text symbols tht hve been recognized previously KMP lgorithm similrly to Brute-force solution serches consecutive text positions storing t ny time the longest currently recognized prefix π of P But when the mismtch between P nd T is found KMP moves by the length of the smllest period of π remembering ll recognized text symbols 25/02/2011 Applied Algorithmics - week3 11 Principle of Knuth-Morris-Prtt KMP Algorithm shortest period of π clled shift s no occurrence of P in rnge (i..i+s-1) prefix π 0 i s i+s longest prefix/suffix of π? pttern P pttern P mismtch b text T If shorter thn s shift ws fesible s would not be the shortest period of π. b 25/02/2011 Applied Algorithmics - week3 12

4 KMP Filure Function The KMP lgorithm works in two stges: pttern preprocessing nd ctul text serch. During pttern preprocessing we: compute the longest proper prefix/suffix of ech prefix P[0..i] nd store its length in n rry F[1..m] t position i+1. Vector F clled the KMP filure function. During the text serch we: trverse consecutive text positions looking for pttern occurrences nd voiding redundnt positive tests with help of the filure function F[1 m]. 25/02/2011 Applied Algorithmics - week3 13 KMP filure function - exmple Let P[0..5] = bb Then the KMP filure function looks s follows F[0] is not defined F[1] = 0 (string hs no proper prefix/suffix) F[2] = 0 (string b hs no proper prefix/suffix) F[3] = 1 (the longest prefix/suffix in b is ) F[4] = 1 (the longest prefix/suffix in b is ) F[5] = 2 (the longest prefix/suffix in bb is b) F[6] = 3 (the longest prefix/suffix in bb is b) 25/02/2011 Applied Algorithmics - week3 14 KMP lgorithm - text serch Algorithm KMP-First-Mtch(T,P): integer; i j 0; while (j<m) nd (T[i+j]=P[j]) { //test next text symbol// j j +1; if (j=m) { then return (i); // return the first occurrence of P // else if (j>0) { then { i i +(j-f[j]); j F[j]; } // shift bsed on F // else i+1; // shift bsed on empty prefix // } } if (i>n-m) return (-1); // end of the text, no pttern occurrences // } 25/02/2011 Applied Algorithmics - week3 15 KMP text serch complexity New symbol is mtched New symbol cuses mismtch left end moves b right end moves Since either left end of the recognized pttern prefix or its right end lwys move the time complexity (number of symbol comprisons) is bounded by 2n. 25/02/2011 Applied Algorithmics - week3 16

5 KMP lgorithm - preprocessing Algorithm Brute-Force-KMP-Mtch(P): integer; F[1] 0; i 1; j F[1]; while (i m-1) do if (P[j]=P[i]) then F[i+1] j+1; j F[i+1]; i i+1; else if (j=0) then F[i+1] 0; j F[i+1]; i i+1; else j F[j]; 25/02/2011 Applied Algorithmics - week3 17 KMP complexity Using similr rgument to the one used in the text serch one cn prove tht the preprocessing requires t most 2 m comprisons. Theorem: The totl time (number of comprisons) complexity of KMP pttern mtching lgorithm is bounded by 2 m+2 n =O(m+n) nd the extr spce required for filure function is of size O(m). We show lter tht one cn obtin similr time bounds hving only O(1) spce. 25/02/2011 Applied Algorithmics - week3 18 Other string mtching lgorithms Boyer-Moore (BM) lgorithm symbols in pttern P re tested ginst the text symbols from right to left, i.e., the lgorithm is bsed on suffix recognition this pproch llows to perform text serch in time c n, for constnt c<1 on verge (in rndom nd nturl texts), but the method works in time O(n m) in the worst cse. It is possible to improve the worst time complexity of BM lgorithm O(n) if we keep in the memory informtion bout the lst recognized suffix of the pttern Other string mtching lgorithms Boyer-Moore lgorithm long shift pttern P text T 25/02/2011 Applied Algorithmics - week3 19 smll number of comprisons 25/02/2011 Applied Algorithmics - week3 20

6 Other string mtching lgorithms Krp-Rbin lgorithm is bsed on the use of reltively simple hsh function f() ech symbol in the lphbet A hs unique integer score s(), e.g., ll symbols cn be enumerted from 1 to A or using nother (ASCII, Unicode) encoding the score is extendble from symbols to strings with the help of hsh function f(), s.t., for,b A nd strings s, s 1 = s, nd s 2 =s b the score f(s) is esily computble from f(s 1 ) nd s(), s well s the score f(s 2 ) is esily computble from f(s) nd s(b) 25/02/2011 Applied Algorithmics - week3 21 Other string mtching lgorithms Krp-Rbin lgorithm the lgorithm computes initilly the score f(p) in the serch stge it compres the score of consecutive text substrings f(t[i..i+m-1]), for ll i 1,..,n-m-1 for every position i, s.t., f(t[i..i+m-1])=f(p) we test the pproprite text nd pttern symbols nively The lgorithm works in time O(n) on verge (in rndom nd nturl texts) but in time O(n m) in the worst cse 25/02/2011 Applied Algorithmics - week3 22 Other string mtching lgorithms Krp-Rbin lgorithm shift of size 1 is f(p)=f(sb)? pttern P is f(p)=f(s)? text T s = P[i+1..i+m-1] b Other string mtching lgorithms There exists n lgorithm tht uses O(n log(m)/m) symbol comprisons in rndom texts fter O(m) time preprocessing; this is the best result possible in this model. There exists text serch lgorithm bsed on n+o(n/m) symbol comprisons in the worst cse fter O(m 2 ) time preprocessing; this is the best result possible in this model For extr informtion on string mtching see: 25/02/2011 Applied Algorithmics - week /02/2011 Applied Algorithmics - week3 24

One Minute To Learn Programming: Finite Automata

One Minute To Learn Programming: Finite Automata Gret Theoreticl Ides In Computer Science Steven Rudich CS 15-251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge