Three new strategies for exact string matching
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1 Three new strategies for exact string matching Simone Faro 1 Thierry Lecroq 2 1 University of Catania, Italy 2 University of Rouen, LITIS EA 4108, France SeqBio 2012 November 26th-27th 2012 Marne-la-Vallée, France
2 Outline 1 Introduction to exact string matching 2 Multiple windows 3 Substring with no repetition 4 Multiple hashing SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
3 Outline 1 Introduction to exact string matching 2 Multiple windows 3 Substring with no repetition 4 Multiple hashing SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
4 Exact String Matching Definition Find all the occurrences of a pattern x of length m in a text y of length n. x, y Σ 2 instances x is given first y is given first SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
5 Exact String Matching Interests basic components of many softwares theoretical problems SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
6 Exact String Matching Theory linear time since Morris and Pratt 1970 linear time and constant space O((n log m)/m) in average [Yao 1979] SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
7 Exact String Matching Solutions Many!! see SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
8 Efficient solutions S. Faro and T. Lecroq The Exact Online String Matching Problem: a Review of the Most Recent Results ACM Computing Surveys 45(2) (2013) to appear. SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
9 Exact String Matching Classical solutions comparisons Knuth-Morris-Pratt (KMP) Boyer-Moore (BM) automata Backward DAWG Matching (with suffix automaton or oracle) (BDM) bit-parallelism Shift Or (SO) Backward Nondeterministic DAWG Matching (BNDM) filtering Karp-Rabin (KR) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
10 Sliding Window y window x beginning SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
11 Sliding Window y window x middle SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
12 Sliding Window y window x middle SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
13 Sliding Window y window x end SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
14 Boyer-Moore (1977) y x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
15 Boyer-Moore (1977) y x comparisons b v = a v SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
16 Boyer-Moore (1977) y x x comparisons b v = a v = c v good suffix shift (1) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
17 Boyer-Moore (1977) y x comparisons b v = a v = good suffix shift (2) v SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
18 Boyer-Moore (1977) y x comparisons b v = a v bad character shift b no b SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
19 Fast Search (Cantone-Faro, 2003) uses the bad character shift when a mismatch occurs with the pattern righmost character uses the good suffix shift otherwise SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
20 TVSBS (Thathoo-Virmani-Lakshmi-Balakrishnan-Sekar, 2006) compares first the rightmost character of the window then the leftmost, then all the others uses both the righmost character of the window and the following to perform the shift SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
21 TVSBS (Thathoo-Virmani-Lakshmi-Balakrishnan-Sekar, 2006) y x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
22 TVSBS (Thathoo-Virmani-Lakshmi-Balakrishnan-Sekar, 2006) y x a = a SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
23 TVSBS (Thathoo-Virmani-Lakshmi-Balakrishnan-Sekar, 2006) y x b = b a = a SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
24 TVSBS (Thathoo-Virmani-Lakshmi-Balakrishnan-Sekar, 2006) y x b = b comparisons u = u c d a = a SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
25 TVSBS (Thathoo-Virmani-Lakshmi-Balakrishnan-Sekar, 2006) forward character comparisons y x b = b u = u c d a = a e Berry-Ravindran shift a e no ae SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
26 BNDM (Navarro & Raffinot, 1998) x C A T A SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
27 BNDM (Navarro & Raffinot, 1998) x C A T A S A S C S G S T SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
28 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
29 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
30 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
31 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S T SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
32 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S T = SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
33 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S T = Shift SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
34 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
35 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S A SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
36 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S A = SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
37 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S A = Shift SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
38 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
39 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S C SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
40 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S C = SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
41 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S C = Shift SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
42 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
43 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S C SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
44 BNDM (Navarro & Raffinot, 1998) x C A T A y C C A T A C S A S C S G S T And with S C = SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
45 Outline 1 Introduction to exact string matching 2 Multiple windows 3 Substring with no repetition 4 Multiple hashing SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
46 Partition the text in k/2 substrings and use k windows m < n/k and k is even (A) s 0 }{{} w 0 (B) s 0 }{{} { w 1 }} { s 1 w 0 (C) s 0 }{{} w 0 s 1 w 2 {}}{ } {{ } w 1 s 2 { w 3 }} { s 3 A general scheme for the multiple sliding windows matcher with: (A) 1 window (B) 2 windows (C) 4 windows SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
47 A General Multiple Sliding Windows Approach Then process simultaneously the k different text windows where w 0, w 1,..., w k 1 w 2i = y[(2n/k)i.. (2n/k)i + m 1] and left windows, goes to right w 2i+1 = y[(2n/k)(i + 1) 1.. (2n/k)(i + 1) + m 2] right windows, goes to left for i = 0,..., (k 2)/2 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
48 Ending situation For each couple of windows (w 2i, w 2i+1 ) the sliding process ends when the window w 2i slides over the window w 2i+1 no occurrence can be missed due to the m 1 overlapping characters between adjacent substrings SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
49 SMART: String Matching Algorithm Research Tool SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
50 SMART: String Matching Algorithm Research Tool more than 80 string matching algorithms a corpus of 12 texts select/deselect string matching algorithms output experimental results in L A TEX, xml, html and txt formats easy to plug new algorithms SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
51 Experiments Random texts of length n = 4, 000, 000 on alphabets of size 16, 32 and 64 for m = 2, 4, 8, 16, 32, 64 with algorithms: Fast Search TVSBS SBNDM FSBNDM and k = 1, 2, 4, 6, 8 windows SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
52 Experimental results Fast Search TVSBS SBNDM FSBNDM SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
53 Experiments Random texts of length n = 4, 000, 000 on alphabets of size 16, 32 and 64 for m = 2, 4, 8, 16, 32, 64, 128, 256, 512 and with a protein and a natural language (English) files with algorithms: known EBOM HASH FSBNDM QF new FS-W FSBNDM-W SBNDM-W TVSBS-W SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
54 Experimental results Alphabet size 32, short patterns SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
55 Perspectives implement other known exact string matching algorithms in this multiple window framework truly parallelize apply it to other related problems SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
56 Reference S. Faro and T. Lecroq A Multiple Sliding Windows Approach to Speed Up String Matching Algorithms In: (R. Klasing editor, Proceedings of the 11th International Symposium on Experimental Algorithms (SEA 2012), Bordeaux, France, 2012), Lecture Notes in Computer Science 7276, Springer-Verlag, Berlin, SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
57 Outline 1 Introduction to exact string matching 2 Multiple windows 3 Substring with no repetition 4 Multiple hashing SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
58 Problem with bit-parallel algorithms Solutions LBNDM (Peltola & Tarhio, 2003) BXS (Durian, Peltola, Salmela & Tarhio, 2010) Factorized BNDM (Cantone, Faro & Giaquinta, 2010) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
59 LBNDM (Long patterns BNDM, Peltola & Tarhio, 2003) partition x = x 0 x 1 x m/k 1 where x j = k = (m 1)/ω + 1 construct a new pattern x where x [j] is the set of characters of x j search for x in y using the BNDM algorithm by considering every k-th character SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
60 LBNDM (Long patterns BNDM, Peltola & Tarhio, 2003) x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
61 LBNDM (Long patterns BNDM, Peltola & Tarhio, 2003) x ω ω ω SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
62 LBNDM (Long patterns BNDM, Peltola & Tarhio, 2003) x ω ω ω k = 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
63 LBNDM (Long patterns BNDM, Peltola & Tarhio, 2003) x k ω ω ω k = 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
64 LBNDM (Long patterns BNDM, Peltola & Tarhio, 2003) k x k = 4 x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
65 LBNDM (Long patterns BNDM, Peltola & Tarhio, 2003) k x k = 4 x y SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
66 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) partition x = x 0 x 1 x m/ω where x j = ω construct a new pattern x where x [j] is the set of characters of x[j + ω] search for x in y using the BNDM algorithm by using a circular automaton SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
67 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
68 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x ω ω ω SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
69 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
70 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
71 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
72 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
73 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
74 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
75 BXS (BNDM algorithm with Extended Shift, Durian, Peltola, Salmela & Tarhio, 2010) x x SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
76 Substring with no repetition Definition A substring s of x is a substring with no repetition (SNR) if any character c Σ appears at most once in s. SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
77 SNR SNR automaton Maximal SNR automaton SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
78 SNR SNR automaton Maximal SNR automaton SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
79 A New Fast Suffix Automaton Based Algorithm Properties s min{σ, m} an SNR admits a suffix automaton where there does not exist two states having incoming transitions labeled with the same symbol SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
80 SNR Suffix Automaton 5 u 4 t 3 o u 2 t m o 1 m a 0 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
81 The Preprocessing Phase Find a maximal SNR of the pattern, i.e. an SNR of maximal length. O(n) time Length of the maximal SNR In many practical cases the length of the maximal SNR is not large enough if compared with the pattern length. Average length of the maximal SNR m genome protein English SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
82 Longer SNRs In order to allow longer SNR it is convenient to use a condensed alphabet whose symbols are obtained by combining groups of q symbols. A hash function h : Σ q {0,..., MAX 1} can be used for combining the group of symbols, for a fixed constant value MAX. Let [u] q = h(u), for each u Σ q. Condensed alphabet x : agcctgatcg x 2 : [ag] 2 [gc] 2 [cc] 2 [ct] 2 [tg] 2 [ga] 2 [at] 2 [tc] 2 [cg] 2 x 3 : [agc] 3 [gcc] 3 [cct] 3 [ctg] 3 [tga] 3 [gat] 3 [atc] 3 [tcg] 3 x 4 : [agcc] 4 [gcct] 4 [cctg] 4 [ctga] 4 [tgat] 4 [gatc] 4 [atcg] 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
83 It turns out that the length of the maximal SNR, though quite less than m in most cases, is quite larger than the size of a computer word (which typically is 32 or 64). This leads to larger shift in a suffix automata based algorithm. Length of the maximal SNR using condensed characters The average length of the maximal SNR in patterns randomly extracted from a genome sequence. The SNR have been computed using condensed alphabets on q characters, where q ranges from 1 to 8. q/m SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
84 Experimental results on a genome sequence m BDM BNDM SBNDM BXS F-BNDM LBNDM BSDM BSDM BSDM BSDM BSDM BSDM BSDM BSDM SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
85 S. Faro and T. Lecroq A Fast Suffix Automata Based Algorithm for Exact Online String Matching In: (N. Moreira and R. Reis editors, Proceedings of the 17th International Conference on Implementation and Application of Automata (CIAA 2012), Porto, Portugal, 2012), Lecture Notes in Computer Science 7381, Springer-Verlag, Berlin, SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
86 Outline 1 Introduction to exact string matching 2 Multiple windows 3 Substring with no repetition 4 Multiple hashing SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
87 Multiple exact string matching Problem Find all the occurrences of the patterns in X = {x 0,..., x k 1 } in y. m = min{ x i } Classical solutions Aho & Corasick (1975) Commentz-Walter (1979) Wu & Manber (1994) Multi-Reverse-Factor (1999) MBNDM SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
88 Wu-Manber algorithm considers blocks of length l blocks are hashed using a function h into values less than maxvalue shift[h(x i [j l j])] = min{ x i, m l + 1} prefixes are also hashed implemented in agrep SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
89 Wu-Manber algorithm y x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
90 Wu-Manber algorithm y window x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
91 Wu-Manber algorithm window y B x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
92 Wu-Manber algorithm window y h(b) x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
93 Wu-Manber algorithm window y h(b) x 0 x 1 x 2 x 3 x 4 h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
94 Wu-Manber algorithm window y h(b) x 0 x 1 x 2 x 3 x 4 h(b) h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
95 Wu-Manber algorithm window y B x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
96 Wu-Manber algorithm window y h(b) x 0 x 1 x 2 x 3 x 4 h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
97 Wu-Manber algorithm window y h(b) x 0 x 1 x 2 x 3 x 4 h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
98 Wu-Manber algorithm window y B x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
99 Wu-Manber algorithm window y h(b) x 0 x 1 x 2 x 3 x 4 h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
100 Wu-Manber algorithm window y P h(b) x 0 x 1 x 2 x 3 x 4 h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
101 Wu-Manber algorithm window y h (P ) h(b) x 0 x 1 x 2 x 3 x 4 h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
102 Wu-Manber algorithm window y h (P ) h(b) x 0 x 1 x 2 x 3 x 4 h (P ) h (P ) h (P ) h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
103 Wu-Manber algorithm window y h (P ) h(b) x 0 x 1 x 2 x 3 x 4 h (P ) h (P ) h (P ) h(b) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
104 Our new method γ hashing functions SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
105 Our new method y x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
106 Our new method y window x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
107 Our new method window y B x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
108 Our new method window y h 1(B) x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
109 Our new method window y h 1(B) x 0 x 1 x 2 x 3 x 4 h 1(B) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
110 Our new method window y h 1(B) x 0 x 1 x 2 x 3 x 4 h 1(B) h 1(B) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
111 Our new method window y B x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
112 Our new method window y h 2(B) x 0 x 1 x 2 x 3 x 4 h 2(B) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
113 Our new method window y h 2(B) x 0 x 1 x 2 x 3 x 4 h 2(B) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
114 Our new method window y B x 0 x 1 x 2 x 3 x 4 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
115 Our new method window y h 1(B) x 0 x 1 x 2 x 3 x 4 h 1(B) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
116 Our new method window y h 2(B) x 0 x 1 x 2 x 3 x 4 h 1(B) h 2(B) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
117 Our new method window y h 2(B) x 0 x 1 x 2 x 3 x 4 F (B) F (B) h 1(B) h 2(B) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
118 Our new method window y h 2(B) x 0 x 1 x 2 x 3 x 4 F (B) F (B) h 1(B) h 2(B) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
119 Experimental results 10, 000 patterns on a genome sequence m MBNDM WM(2, 2) WM(2, 3) WM(3, 2) WM(3, 3) WM(4, 1) WM(4, 2) WM(4, 3) WM(6, 1) WM(6, 2) WM(6, 3) WM(8, 1) WM(8, 2) WM(8, 3) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
120 Experimental results Naive comparison per text symbol for 10, 000 patterns m MBNDM WM(2, 2) WM(2, 3) WM(3, 2) WM(3, 3) WM(4, 1) WM(4, 2) WM(4, 3) WM(6, 1) WM(6, 2) WM(6, 3) WM(8, 1) WM(8, 2) WM(8, 3) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
121 Experimental results Speed ups obtained via WM(q, γ) vs MBNDM k / m , SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
122 Reference S. Faro and T. Lecroq Fast Searching in Biological Sequences Using Multiple Hash Functions IEEE 12th International Conference on Bioinformatics and Bioengineering (BIBE 2012), Larnaca, Cyprus, 2012 SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
123 Thank you for your attention! SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
124 Multiple windows: experimental results Alphabet size 32, long patterns SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
125 Multiple windows: experimental results Protein alphabet (size 20), short patterns SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
126 Multiple windows: experimental results Protein alphabet (size 20), long patterns SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
127 Multiple windows: experimental results English text, short patterns SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
128 Multiple windows: experimental results English text, long patterns SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
129 BSDM Experimental results on a protein sequence m BDM BNDM SBNDM BXS F-BNDM LBNDM BSDM BSDM BSDM BSDM BSDM BSDM BSDM BSDM SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
130 BSDM Experimental results on a natural language text m BDM BNDM SBNDM BXS F-BNDM LBNDM BSDM BSDM BSDM BSDM BSDM BSDM BSDM BSDM SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
131 Multiple hashing: experimental results 100 patterns on a genome sequence m MBNDM WM(2, 2) WM(2, 3) WM(3, 2) WM(3, 3) WM(4, 1) WM(4, 2) WM(4, 3) WM(6, 1) WM(6, 2) WM(6, 3) WM(8, 1) WM(8, 2) WM(8, 3) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
132 Multiple hashing: experimental results 1000 patterns on a genome sequence m MBNDM WM(2, 2) WM(2, 3) WM(3, 2) WM(3, 3) WM(4, 1) WM(4, 2) WM(4, 3) WM(6, 1) WM(6, 2) WM(6, 3) WM(8, 1) WM(8, 2) WM(8, 3) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
133 Multiple hashing: experimental results Naive comparison per text symbol for 1000 patterns m MBNDM WM(2, 2) WM(2, 3) WM(3, 2) WM(3, 3) WM(4, 1) WM(4, 2) WM(4, 3) WM(6, 1) WM(6, 2) WM(6, 3) WM(8, 1) WM(8, 2) WM(8, 3) SF&TL (Catania & Rouen) Exact String Matching SeqBio / 63
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