1. 1 Approximate Boyer-Moore String Matching Source : SIAM Journal on Computing, Vol. 22, No. 2, 1993, pp.243-260 J. Tarhio and E. Ukkonen Advisor: Prof. R. C. T. Lee Speaker: Kuei-hao Chen

2. 2 The k mismatches problem The k differences problem

3. 3 Definition of the k mismatches problem Given a pattern string P of length m and a text string T of length n, we would like to find all approximate occurrences P in T with at most k mismatches. If k=1, then

4. 4 Consider the following situation where a pattern P is matching with a windows W of T and there are already (k+1) mismatches:

5. 5 Since there are already (k+1) mismatches, we must move the pattern. The following is obvious: P must be moved to such an extent that there are at most k mismatches between a suffix S of W and a substring S of P.

6. 6 Our trick is as follows: Consider the (k+1)- suffix of W. There are two cases:

7. 7 Case 1: There is one character in this (k+1)- suffix which exists in P in such a way as shown below. Move the pattern to match these characters. Note that in such a situation, there are at most k mismatches between the (k+1)-suffix and its corresponding substring in P.

8. 8 Case 2: No such a character exists. Move the pattern in such a way that the k-prefix of P aligns with the k-suffix of W as shown below. Under such a situation, again, there are at most k-mismatches between the k-suffix of W and k- prefix of P.

9. 9 The generalization of the BM algorithm for the k mismatches problem will be very natural: for k=0 the generalized algorithm is exact string matching. Recall that the k mismatches problem asks for finding all occurrences of P in T such that in at most k positions of P, T and P have different characters.

10. 10 We just scan the pattern from right to the left until we have found k+1 mismatches (unsuccessful search) or the pattern ends (successful search).

11. 11 Preprocessing phase for approximate matching D k table The value D k for a particular alphabet is defined as the rightmost position of that character in the pattern – 1 and the end position i where i=[m..m-k]. ΣA C G * D 1 [i=8, a]1 6 2 8 Example : Let k=1, m=8, a j 12345678 P:GCAGAGAG i 12345678 GCAGAGAG ΣA C G * D 1 [i=7, a]2 5 1 8 j 1234567 P:GCAGAGA ΣA C G * D 1 [i=8, a]1 6 2 8 D 1 [i=7, a]2 5 1 8

12. 12 P = p 1 p 2 …p m,T = t 1 t 2 …t n Preprocessing For a Do For j=m downto m-k Do Begin d k [j,a] m Find a character a that it is close to p j. If it is found, we calculate the distance between the position of the character a and j and insert it into d k [j,a]. Algorithm for preprocessing phase

13. 13 P = p 1 p 2 …p m,T = t 1 t 2 …t n Searching j=m; While j n+ k Do Begin h=j; i=m; mismatch=0; While i>0 and mismatch k Do Begin d=min(d k [i, t h ], d k [i-1, t h-1 ]); If t hp i Then mismatch=mismatch+1; i= i- 1; h= h-1 End of while; If mismatch k Then report match at position j; j= j+ d Endof while Algorithm for searching phase

14. 14 Complete example for approximate string matching Example 1: Let k=1, m=4, n=17 T:TTAACGTAATGCAGCTA P:AGCT ΣA C G T D 1 [i=4, a]3 1 2 4 D 1 [i=3, a]2 3 1 4

15. 15 Example 1 (1/6) T:TTAACGTAATGCAGCTA P:AGCT ΣA C G T D 1 [i=4, a]3 1 2 4 D 1 [i=3, a]2 3 1 4

16. 16 Example 1 (2/6) T:TTAACGTAATGCAGCTA P:AGCT ΣA C G T D 1 [i=4, a]3 1 2 4 D 1 [i=3, a]2 3 1 4

17. 17 Example 1 (3/6) T:TTAACGTAATGCAGCTA P:AGCT ΣA C G T D 1 [i=4, a]3 1 2 4 D 1 [i=3, a]2 3 1 4

18. 18 Example 1 (4/6) T:TTAACGTAATGCAGCTA P:AGCT ΣA C G T D 1 [i=4, a]3 1 2 4 D 1 [i=3, a]2 3 1 4

19. 19 Example 1 (5/6) T:TTAACGTAATGCAGCTA P:AGCT ΣA C G T D 1 [i=4, a]3 1 2 4 D 1 [i=3, a]2 3 1 4

20. 20 Example 1 (6/6) T:TTAACGTAATGCAGCTA P:AGCT ΣA C G T D 1 [i=4, a]3 1 2 4 D 1 [i=3, a]2 3 1 4 j 16 + p, j 16+ 3, j 19 jump out of while loop

21. 21 Example 2: Let k=1, m=8, n=24 T:GCATCGCAGAGAGTATACAGTACG P:GCAGAGAG ΣA C G * D 1 [i=8, a]1 6 2 8 D 1 [i=7, a]2 5 1 8

22. 22 Example 2 (1/14) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D 1 [i=8, a]1 6 2 8 D 1 [i=7, a]2 5 1 8

23. 23 Example 2 (3/14) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8

24. 24 Example 2 (4/14) ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG

25. 25 Example 2 (5/14) ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG Then report match at position j; j 13 + p, j 13+ 2, j 15

26. 26 Example 2 (6/14) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8

27. 27 Example 2 (7/14) ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG

28. 28 Example 2 (8/14) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8

29. 29 Example 2 (9/14) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8

30. 30 Example 2 (11/14) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8

31. 31 Example 2 (13/14) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8

32. 32 Example 2 (14/14) ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG If h = 0 Then report match at position j; j 24 + p, j 24+ 2, j 26 jump out of while loop

33. 33 Time complexity preprocessing phase in O(m+ kc) time and O(kc) space complexity. searching phase in O(mn) time complexity.

34. 34 Definition of the k differences problem Given a pattern string P of length m and a text string T of length n, we would like to find all approximate occurrences P in T with edit distance not larger than k.

35. 35 The basic approach to solve the problem is to find the edit distance for T(1, i) and P for every i [Ukk85b] : Let Edit be an m+1 by n+1 table such that Edit(i, j) is the minimum edit distance between p 1 p 2 …p j and any substring of T ending at t i.

36. 36 Table Edit must be completely evaluated column-by-column in time O(mn).

37. 37 If we can find out all occurrences of i where Edit(T(1, i), P) cannot be smaller than k. We may skip this i. This paper is based upon Rule 7 proposed by Professor Lee.

38. 38 Rule 7 If k characters in String A do not appear in String B, Distance(A,B) is not smaller than k.

39. 39 In the scanning phase, we define some terms first. A diagonal h of Edit for h=-m,…, n, consists of all Edit(i, j) such that i- j=h. For every Edit(i, j), there is a minimizing arc from Edit(i-1, j) to Edit(i, j) if Edit(i, j)=Edit(i-1, j)+1, from Edit(i, j-1) to Edit(i, j) if Edit(i, j-1)+1, and from Edit(i-1, j-1) to Edit(i, j) if Edit(i, j)=Edit(i-1, j-1) where p j =t i or if Edit(i, j)=Edit(i-1, j-1)+1 where p jt i. The costs of the arcs are 1, 1, 0 and 1, respectively.

40. 40 A minimizing path is any path that consists of minimizing arcs and leads from an entry Edit(i, 0) on the first row of Edit to an entry Edit(h, m) on the last row of Edit. A minimizing path is successful if it leads to an entry Edit(h, m)k.

41. 41 Lemma 1: The entries on a successful minimizing path M are contained in k+1 successive diagonals of Edit. Proof : Each addition of a diagonal comes from either an insertion or deletion. If there are more than (k+1) diagonals, there must be more than (k+1) operations, either deletions or insertions. Thus there cannot be more than (k+1) diagonals.

42. 42 ABCABBA C B A B A C 00000000 11101111 22111112 32221221 43232122 54333222 65434333 T:ABCABBA P:CBABAC S:C-AB-- P:CBAB AC EDIT(P, S)=3 There are (k+1)=3+1=4 successive diagonals because there are three deletions. Successive diagonals

43. 43 BCABDAB C B A D B 00000000 11011111 21111221 32212222 43322233 54433333 T:BCABDAB P:CBADB k =3 S:C-ABDABP:CBA-D-BS:C-ABDABP:CBA-D-B EDIT(P, S)=3 There are 1+2=3 <(k+1) =3+1=4 successive diagonals because there are one deletion and two insertions. Successive diagonals

44. 44 By Lemma 1, for each diagonal d, any successful minimizing path starting at the top of this diagonal will have a bandwidth of 1+k+k=2k+1

45. 45 ABCABBA C B A B A C 00000000 11101111 22111112 32221221 43232122 54333222 65434333 T:ABCABBA P:CBABAC k=3 S:C-AB-- P:CBAB AC EDIT(P, S)=3 Result Successive diagonals The successful minimizing path is only in the bandwidth 7 of Edit. k=3

46. 46 For the width of bandwidth k of Edit, we give it a name, call k-environment. For each j=1, …, m, let the k-environment of the pattern symbol p j be the string C j =p j-k …p j+k, where p a =ε for a m.

47. 47 The longest vertical path in any minimizing path has length not greater than 2k+1. We only have to determine whether t i appears in the k environment of p j.

48. 48 Given T=ATGCGAGAGAT, P=GCAGAGAGATG, and k=2. We select t 5, t 8 and t 11 three characters. The 2-environment of t 5 is C 5 =p 3 p 4 p 5 p 6 p 7 =AGAGA. The 2-environment of t 8 is C 8 =p 6 p 7 p 8 p 9 p 10 =GAGAT. The 2-environment of t 11 is C 11 =p 9 p 10 p 11 =ATG.

49. 49 We now obtain a stronger version of Rule 7. Lemma 2: Let a successful minimizing path M go through some entry on a diagonal h of Edit. Then for at most k indexes j, 1j m, character t h+j does not occur in the k environment of C j. A formal proof can be found in the paper. In the following, we give some physical feeling of it.

50. 50 In this case, although there are two mismatches, by deleting a which mismatches x, we may achieve a perfect match. Thus the edit distance between T and P may still be 1. k=1

51. 51 In this case, it can be seen that deleting one character in P will not result in a perfect match. Thus, the edit distance between T and P must be larger than 1. k=1

52. 52 The shift table is based on table D k. We determine the first diagonal after h, say h+d, where at least one of the characters t h+m, t h+m-1, …, t h+m-k matches with corresponding character of P. Finally, the maximum of k+1 and d is the length of the shift.

53. 53 The algorithm explains when a possible occurrence of P in T was found, DP approach is immediately used to find alignment result.

54. 54 Input: P = p 1 p 2 …p m,T = t 1 t 2 …t n and k Output: All occurrence P in T Initially, the start position h of T =0, i=h+m; While i n+ k do begin j=m; bad=0; While i>k and bad k do begin If t i does not occur in C j then bad=bad+1 j=j-1;i=i-1 end; If bad k then W is a sequence from t h-k to t h+m. Using dynamic programming to align W with P Output alignment result. We calculate shift steps d=min(D k [i, t r ], D k [i-1, t r-1 ],); h=h+max(k+1,d) end; Algorithm

55. 55 Complete example for approximate string matching For example : Let k=1, m=8, n=24 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D 1 [i=8, a]1 6 2 8 D 1 [i=7, a]2 5 1 8

56. 56 Example(1/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 k=1 >k>k t 8 =A appears in P(7,8) t 7 =C does not appear in P(6,8) t 6 =G appears in P(5,7) t 5 =C does not appear in P(4,6) Shifting is needed now.

57. 57 Example(2/15) ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG >k>k k=1 t 9 =G appears in P(7,8) t 8 =A appears in P(6,8) t 7 =C does not appear in P(5,7) t 6 =G appears in P(4,6) t 5 =C does not appear in P(3,5) Shifting is needed now.

58. 58 Example(3/15) ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG >k>k k=1 t 11 =G appears in P(7,8) t 10 =A appears in P(6,8) t 9 =G appears in P (5,7) t 8 =A appears in P(4,6) t 7 =C does not appear in P(3,5) t 6 =G appears in P(2,4) t 5 =C appears in P(1,3) t 4 =T does not appear in P(1,2) Shifting is needed now.

59. 59 Example(4/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG ΣA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 GCAGAGAG Output : CGCAGAGAGT G C A G A G A G 00000000000 11011010101 21101111111 32210112122 421011212 52101122 6210112 721012 82101 W= CGCAGAGAGT P= GCAGAGAG k=1

60. 60 Example(5/15) GCAGAGA- GCAGAGAG Output : CGCAGAGAGT G C A G A G A G 00000000000 11011010101 21101111111 32210112122 421011212 52101122 6210112 721012 82101 W= CGCAGAGAGT P= GCAGAGAG

61. 61 Example(6/15) - CAGAGAG GCAGAGA G Output : CGCAGAGAGT G C A G A G A G 00000000000 11011010101 21101111111 32210112122 421011212 52101122 6210112 721012 82101 W= CGCAGAGAGT P= GCAGAGAG

62. 62 Example(7/15) CGCAGAG AG - GCAGAGA G Output : CGCAGAGAGT G C A G A G A G 00000000000 11011010101 21101111111 32210112122 421011212 52101122 6210112 721012 82101 W= CGCAGAGAGT P= GCAGAGAG

63. 63 Example(8/15) GCAGAGAGT GCAGAGAG- Output : CGCAGAGAGT G C A G A G A G 00000000000 11011010101 21101111111 32210112122 421011212 52101122 6210112 721012 82101 W= CGCAGAGAGT P= GCAGAGAG

64. 64 Example(9/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG aA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 >k>k k=1 t 15 =A appears in P(7,8) t 14 =T does not appear in P(6,8) t 13 =G appears in P(5,7) t 12 =A appears in P(4,6) t 11 =G appears in P(3,5) t 10 =A appears in P(2,4) t 9 =G appears in P(1,3) t 8 =A does not appear in P(1,2) Shifting is needed now.

65. 65 Example(10/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG aA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 >k>k k=1 t 16 =T does not appear in P(7,8) t 15 =A appears in P(6,8) t 14 =T does not appear in P(5,7) Shifting is needed now.

66. 66 Example(11/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG aA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 >k>k k=1 t 18 =C does not appear in P(7,8) t 17 =G appears in P(6,8) t 16 =T does not appear in P(5,7) Shifting is needed now.

67. 67 Example(12/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG aA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 >k>k k=1 t 19 =A appears in P(7,8) t 18 =C does not appear in P(6,8) t 17 =G appears in P(5,7) t 16 =T does not appear in P(4,6) Shifting is needed now.

68. 68 Example(13/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG aA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 >k>k k=1 t 20 =G appears in P(7,8) t 19 =A appears in P(6,8) t 18 =C does not appear in P(5,7) t 17 =G appears in P(4,6) t 16 =T does not appear in P(3,5) Shifting is needed now.

69. 69 Example(14/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG aA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 >k>k k=1 t 22 =G appears in P(7,8) t 21 =A appears in P(6,8) t 20 =G appears in P(5,7) t 19 =A appears in P(4,6) t 18 =G does not appear in P(3,5) t 17 =G appears in P(2,4) t 16 =T does not appear in P(1,3) Shifting is needed now.

70. 70 Example(15/15) 123456789101112131415161718192021222324 T:GCATCGCAGAGAGTATGCAGAGCG P:GCAGAGAG aA C G * D[i=8, a]1 6 2 8 D[i=7, a]2 5 1 8 GCAGAGCG GCAGAGAG jump out of while loop TGCAGAGCG G C A G A G A G 0000000000 1101101010 2210111101 321011211 42101121 5210122 621012 72112 8221 W= TGCAGAGCG P= GCAGAGAG Result : k=1

71. 71 Time complexity preprocessing phase and searching phase in O(mn/k) time and O(| Σ |n) space complexity.

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75. 75 THANK YOU