Fast Exact String Matching Algorithms

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1 Fast Exact String Matching Algorithms Thierry Lecroq Laboratoire d Informatique, Traitement de l Information, Systèmes. Part of this work has been done with Maxime Crochemore LAW 2007 King s College London February 8th and 9th, 2007

2 Outline 1 Introduction 2 Best Matching Shift 3 Hashing q-grams 4 Experimental results Thierry Lecroq Fast String Matching 2/33

3 Outline 1 Introduction 2 Best Matching Shift 3 Hashing q-grams 4 Experimental results Thierry Lecroq Fast String Matching 3/33

4 Exact String Matching Problem Find one or more generally all the occurrences of a pattern x of length m in a text y of length n. Both x and y are build on an alphabet Σ of size σ. Thierry Lecroq Fast String Matching 4/33

5 Exact String Matching Solutions Many!! See Most famous : Knuth Morris Pratt and Boyer Moore, 1977 Thierry Lecroq Fast String Matching 5/33

6 Exact String Matching Sliding window mechanism KMP : from left to right ( ) BM : from right to left ( ) Thierry Lecroq Fast String Matching 6/33

7 Boyer-Moore Typical situation y x b = a A suffix u of the pattern is found and a mismatch occurs between a character a in the pattern x and a character b in the text y. u = u Thierry Lecroq Fast String Matching 7/33

8 Matching shift y x b = a The matching shift consists in aligning the substring u = x[i + 1..m 1] = y[i + j + 1..j + m 1] with one of its reoccurrences in x. u = u Thierry Lecroq Fast String Matching 8/33

9 Three Types of Matching Shift (I) Weak Matching Shift y x x b = a c No condition on the character c preceding u, it is then possible that c = a. u = u = u shift Thierry Lecroq Fast String Matching 9/33

10 Three Types of Matching Shift (II) Strong Matching Shift y b u = = x a u = = x c u c must be different from the character a. shift Thierry Lecroq Fast String Matching 10/33

11 Three Types of Matching Shift (III) Best Matching Shift y c must be equal to b. x x b = a c u = u = u shift Thierry Lecroq Fast String Matching 11/33

12 Three Types of Matching Shift weak and strong matching shift only depend on x best matching shift depends on x and the alphabet Thierry Lecroq Fast String Matching 12/33

13 Outline 1 Introduction 2 Best Matching Shift 3 Hashing q-grams 4 Experimental results Thierry Lecroq Fast String Matching 13/33

14 Computation of the best matching shift For 0 i m 1 : suff [i] = length of the longest suffix of x ending at position i in x. x a u = = x c u bmatch[m 1 suff[i],c] i suff[i] Thierry Lecroq Fast String Matching 14/33

15 Computation of the best matching shift Scan the position of the table suff from left to right Another proof of linearity Thierry Lecroq Fast String Matching 15/33

16 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

17 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

18 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

19 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

20 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

21 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

22 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

23 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

24 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

25 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

26 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

27 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 16/33

28 Best Matching Shift Degenerate case y x b = a x u = u = v u does not reoccur in x v is the longest prefix of x matching u which is a suffix of x v is a border that can be detected when suff [i] = i + 1 shift Thierry Lecroq Fast String Matching 17/33

29 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 18/33

30 Computation of the best matching shift Example a c g t i x[i] c a t a c a t a a a t a suff [i] Thierry Lecroq Fast String Matching 18/33

31 Outline 1 Introduction 2 Best Matching Shift 3 Hashing q-grams 4 Experimental results Thierry Lecroq Fast String Matching 19/33

32 Hashing q-grams Compute a hash value in [0; 255] for every q-grams of the pattern x. Compute a shift for every hash value. Unroll the loops as much as possible. Thierry Lecroq Fast String Matching 20/33

33 Hashing q-grams Example i x[i] c a t a c a t a a a t a shift[i] 10 i [0; 255] Thierry Lecroq Fast String Matching 21/33

34 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(cat) = ((rank(c) 2 + rank(a)) 2 + rank(t) = 194 shift[194] = 10 shift[194] 9 Thierry Lecroq Fast String Matching 21/33

35 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(ata) = ((rank(a) 2 + rank(a)) 2 + rank(a) = 205 shift[205] = 10 shift[205] 8 Thierry Lecroq Fast String Matching 21/33

36 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(tac) = ((rank(t) 2 + rank(a)) 2 + rank(c) = 245 shift[245] = 10 shift[245] 7 Thierry Lecroq Fast String Matching 21/33

37 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(aca) = ((rank(a) 2 + rank(c)) 2 + rank(a) = 171 shift[171] = 10 shift[171] 6 Thierry Lecroq Fast String Matching 21/33

38 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(cat) = ((rank(c) 2 + rank(a)) 2 + rank(t) = 194 shift[194] = 9 shift[194] 5 Thierry Lecroq Fast String Matching 21/33

39 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(ata) = ((rank(a) 2 + rank(t)) 2 + rank(a) = 205 shift[205] = 8 shift[205] 4 Thierry Lecroq Fast String Matching 21/33

40 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(taa) = ((rank(t) 2 + rank(a)) 2 + rank(a) = 243 shift[243] = 10 shift[243] 3 Thierry Lecroq Fast String Matching 21/33

41 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(aaa) = ((rank(a) 2 + rank(a)) 2 + rank(a) = 167 shift[167] = 10 shift[167] 2 Thierry Lecroq Fast String Matching 21/33

42 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(aat) = ((rank(a) 2 + rank(a)) 2 + rank(t) = 186 shift[186] = 10 shift[186] 1 Thierry Lecroq Fast String Matching 21/33

43 Hashing q-grams Example i x[i] c a t a c a t a a a t a h(ata) = ((rank(a) 2 + rank(t)) 2 + rank(a) = 205 shift[205] = 4 = sh1 4 shift[205] 0 Thierry Lecroq Fast String Matching 21/33

44 Hashing q-grams Algorithm Newq(x, m, y, n) for q = 3 Preprocessing for i 0 to 255 do shift[i] m 2 for i 2 to m 2 do h ((x[i 2] 2 + x[i 1]) 2) + x[i] shift[h mod 256] m 1 i h ((x[m 3] 2 + x[m 2]) 2) + x[m 1] sh1 shift[h mod 256] shift[h mod 256] 0 Thierry Lecroq Fast String Matching 22/33

45 Hashing q-grams Algorithm Newq(x, m, y, n) for q = 3 Searching y[n..n + m 1] x Sentinel j m 1 while True do sh 1 while sh 0 do h ((y[j 2] 2 + y[j 1]) 2) + y[j] sh shift[h mod 256] j j + sh if j < n then if x = y[j m + 1..j] then Report(j m + 1) j j + sh1 else Return Thierry Lecroq Fast String Matching 23/33

46 Outline 1 Introduction 2 Best Matching Shift 3 Hashing q-grams 4 Experimental results Thierry Lecroq Fast String Matching 24/33

47 Experimental results Conditions Intel Pentium processor at 1300MHz Texts Linux Red Hat version gcc with the full optimization option -O3 clock function 100 patterns randomly chosen in the texts Binary alphabet, random (uniform distribution), 4Mb E.coli from Large Canterbury Corpus, 4.6Mb Alphabet of size 8, random (uniform distribution), 4Mb world192.txt from Large Canterbury Corpus, 4.3Mb Thierry Lecroq Fast String Matching 25/33

48 Experimental results Algorithms BM2fast Boyer Moore with best matching shift and fast loop NEWq for q [3; 8] TBM Tuned Boyer Moore (Hume & Sunday, 1991) SSABS (Sheik, Aggarwal, Poddar, Balakrishnan & Sekar, 2004) SBNDM2 (Holub & Durian, 2005) Thierry Lecroq Fast String Matching 26/33

49 Experimental results Binary alphabet Thierry Lecroq Fast String Matching 27/33

50 Experimental results E.coli from the Large Canterbury corpus Thierry Lecroq Fast String Matching 28/33

51 Experimental results Alphabet of size 8 Thierry Lecroq Fast String Matching 29/33

52 Experimental results Natural language Thierry Lecroq Fast String Matching 30/33

53 Conclusions Binary alphabet : NEW5-8 for m [9; 256] Alphabet of size 4 : NEW3-5 for m [7; 128] Alphabet of size 8 : NEW3-5 for m [13; 64] Natural language : BM2fast for m [7; 15] Thierry Lecroq Fast String Matching 31/33

54 References Coming soon... Best matching shift and many other interesting things Thierry Lecroq Fast String Matching 32/33

55 References M. Crochemore and T. Lecroq A fast implementation of the Boyer Moore string matching algorithm Submitted T. Lecroq Fast string matching algorithms Information Processing Letters Accepted Thierry Lecroq Fast String Matching 33/33

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