Text Algorithms (6EAP) Lecture 3: Exact paaern matching II

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Text Algorithms (6EA) Lecture 3: Exact paaern matching II Jaak Vilo 2012 fall Jaak Vilo MTAT.03.

1 Text Algorithms (6EA) Lecture 3: Exact paaern matching II Jaak Vilo 2012 fall Jaak Vilo MTAT Text Algorithms 1 2 Algorithms Brute force O(nm) Knuth- Morris- raa O(n) Karp- Rabin hir- OR, hir- AND R. Boyer,.Moore: A fast string searching algorithm. CACM 20 (1977), [DF] Boyer- Moore Factor searches 3 4 ABCDE BBCDE Have we missed anything? What have we learned if we test for a potenaal match from the end? 5 6 1

2 A B 7 8 Bad character heuris2cs maximal shir on [i] A B X X X First x in pattern (from end) delta 1 ( [i] ) m if pattern does not contain [i] patlen-j max j so that [j] == [i] [i] void bminitocc() { char a; int j; for(a=0; a<alphabetsize; a++) occ[a]=-1; for (j=0; j<m; j++) { a=p[j]; occ[a]=j; } } 9 10 Good suffix heuris2cs Boyer- Moore algorithm delta 2 ( [i] ) A B minimal shift so that matched region is fully covered or that the sufix of match is also a prefix of Input: Text [1..n] and pattern [1..m] Output: Occurrences of in preprocess_bm() // delta1 and delta2 i=m while i <= n for( j=m; j>0 and [j]==[i-m+j]; j-- ) ; if j==0 report match at position i-m+1 i = i+ max( delta1[ [i] ], delta2[ j ] )

3 implificaaons of BM hap:// flensburg.de/lang/algorithmen/paaern/bmen.htm hap://biit.cs.ut.ee/~vilo/edu/ /text_algorithms/aracles/exact/boyer- Moore- original- p762- boyer.pdf Animaaon: hap://www- igm.univ- mlv.fr/~lecroq/string/ There are many variants of Boyer- Moore, and many scienafic papers. On average the ame complexity is sublinear Algorithm speed can be improved and yet simplify the code. It is useful to use the last character heurisacs (Horspool (1980), Baeza- Yates(1989), Hume and unday(1991)) Algorithm BMH (Boyer- Moore- Horspool) RN Horspool - racacal Fast earching in trings o&ware - rac/ce and Experience, 10(6): Input: Text [1..n] and pattern [1..m] Output: occurrences of in 1. for a in Σ do delta[a] = m 2. for j=1..m-1 do delta[[j]] = m-j 3. i=m 4. while i <= n 5. if [i] == [m] 6. j = m-1 7. while ( j>0 and [j]==[i-m+j] ) j = j-1 ; 8. if j==0 report match at i-m+1 9. i = i + delta[ [i] ] 15 tring Matching: Horspool algorithm How the comparison is made? Text : attern : From right to left: suffix search Which is the next position of the window? Text : a attern : It depends of where appears the last letter of the text, say it a, in the pattern: a a a a a a a a a Then it is necessary a preprocess that determines the length of the shift. Algorithm Boyer- Moore- Horspool- Hume- unday (BMHH) Use delta in a aght loop If match (delta==0) then check and apply original delta d Input: Text [1..n] and pattern [1..m] Output: occurrences of in 1. for a in Σ do delta[a] = m 2. for j=1..m-1 do delta[[j]] = m-j 3. d = delta[ [ m ] ]; // memorize d on [m] 4. delta[ [ m ] ] = 0; // ensure delta on match of last char is 0 5. for ( i=m ; i<= n ; i = i+d ) 6. repeat // skip loop 7. t=delta[ [i] ] ; i = i + t 8. until t==0 9. for( j=m-1 ; j> 0 and [j]==[i-m+j] ; j = j-1 ) ; 10. if j==0 report match at i-m+1 Daniel M. unday: A very fast substring search algorithm [DF] Communica/ons of the ACM August 1990, Volume 33 Issue 8 Loop unrolling: Avoid too many loops (each loop requires tests) by just repeaang code within the loop. Line 7 in previous algorithm can be replaced by: 7. i += delta[ [i] ]; i += delta[ [i] ]; i += (t = delta[ [i] ]) ; BMHH requires that the text is padded by : [n+1]..[n+m] = (in order for the algorithm to finish correctly at least one occurrence!)

Forward- Fast- earch: Another Fast Variant of the Boyer- Moore tring Matching Algorithm The rague tringology Conference '03 Domenico Cantone and imone Faro Abstract: We present a variaaon of the

Horspool, Quick earch, Tuned Boyer- Moore, Reverse Factor, Berry- Ravindran, and Fast- earch itself.

4 Forward- Fast- earch: Another Fast Variant of the Boyer- Moore tring Matching Algorithm The rague tringology Conference '03 Domenico Cantone and imone Faro Abstract: We present a variaaon of the Fast- earch string matching algorithm, a recent member of the large family of Boyer- Moore- like algorithms, and we compare it with some of the most effecave string matching algorithms, such as Horspool, Quick earch, Tuned Boyer- Moore, Reverse Factor, Berry- Ravindran, and Fast- earch itself. All algorithms are compared in terms of run- ame efficiency, number of text character inspecaons, and number of character comparisons. It turns out that our new proposed variant, though not linear, achieves very good results especially in the case of very short paaerns or small alphabets. hap://cs.felk.cvut.cz/psc/event/2003/p2.html.gz (local copy) Factor based approach Opamal average- case algorithms Assuming independent characters, same probability Factor based approach Opamal average- case algorithms Assuming independent characters, same probability Factor a substring of a paaern Any substring (how many?) Factor searches Examples X u Backward DAWG Matching (BDM) Crochemore et al 1994 Backward Nondeterminisac DAWG Matching (BNDM) Navarro, Raffinot 2000 Backward Oracle Matching (BOM) Allauzen, Crochermore, Raffinot 2001 Do not compare characters, but find the longest match to any subregion of the pattern

Backward DAWG Matching BDM uffix automaton recognises all

bitparallelism Bits show where the factors have occurred

Determinisac version of the same Backward Factor Oracle 27

Backward Oracle matching tring Matching of one paaern

5 Backward DAWG Matching BDM uffix automaton recognises all factors (and suffixes) in O(n) BNDM simulate using bitparallelism Bits show where the factors have occurred so far Do not compare characters, but find the longest match to any subregion of the pattern BNDM matches an NDA NDA on the suffixes of announce Determinisac version of the same Backward Factor Oracle BNDM Backward Non-Deterministic DAWG Matching BOM - Backward Oracle matching tring Matching of one paaern CTACTACTACGTCTATACTGATCGTAGC TACTACGGTATGACTAA uffix search refix search Factor search 29 5

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Text Algorithms (6EAP) Lecture 3: Exact pa;ern matching II

Text Algorithms (6EAP) Lecture 3: Exact pa;ern matching II Jaak Vilo 2010 fall Jaak Vilo MTAT.03.190 Text Algorithms 1 Find occurrences in text P S 2 Algorithms Brute force O(nm) Knuth- Morris- Pra; O(n)