The document describes the dynamic programming algorithm for calculating the edit distance between two strings. It defines edit distance as the minimum number of insertions, deletions or substitutions needed to change one string into the other. The algorithm uses a table c[i,j] to store the edit distance between the prefixes of the two strings x[1..i] and y[1..j], calculating values in the table using a recursive formula until it reaches the final edit distance value c[|x|,|y|].

1. Approximate k-edit-distance Approximative ( k -distance) matching, now for edit distance Given string x = abbacbbbababacabbbba and pattern p = bbba find all “almost”-occurrences of p ind x x = a bba c bbba babacab b a bba 17 6 1

2. Edit distance The edit-distance between strings x and y is the minimal number of - insertions - deletions - substitutions needed to translate x into y d( abab,acc ) = 3: abab -> aba -> aca -> acc d( abab,aac ) = 2: abab -> aab -> aac

3. Calculating the edit-distance Basis cases: - string vs empty string: d (x,””) = d( ””,x) = | x | - two single characters: d(a,b) = 1 if a!=b 0 if a==b

4. Calculating the edit-distance Recursion: - two non-empty strings: d( x [1..i], y [1..j]) = d( x [1..i-1], y [1..j])+1 min d( x [1..i], y [1..j-1])+1 d( x [1..i-1], y [1..j-1])+d( x [i], y [j]) { i j i j i j

5. Dynamic programming algorithm Use table c[i,j] = d( x [1..i], y [1..j]) Initialize: c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j Main algorithm: for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { c[i-1,j] c[i,j-1] c[i-1,j-1] c[i,j]

6. Example a b a b c a 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) {

7. Example a b a b c a 0+0 1+1 1+1 0 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 2 3 4 5 6

8. Example a b a b c a 1+1 2+1 0+1 1 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6

9. Example a b a b c a 2+0 3+1 1+1 2 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 2 3 4 5 6

10. Example a b a b c a 3+0 4+1 2+1 3 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 2 3 4 5 6

11. Example a b a b c a 4+1 5+1 3+1 4 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 2 3 4 5 6

12. Example a b a b c a 5+1 6+1 4+1 5 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 2 3 4 5 6

13. Example a b a b c a 6+1 7+1 5+1 6 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 2 3 4 5 6

14. Example a b a b c a 7+1 8+1 6+1 7 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 3 4 5 6

15. Example a b a b c a 1+1 0+1 2+1 1 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 3 4 5 6

16. Example a b a b c a 0+0 1+1 1+1 0 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 0 3 4 5 6

17. Example a b a b c a 1+1 2+1 0+1 1 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 0 1 3 4 5 6

18. Example a b a b c a 2+1 3+1 1+1 2 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 0 1 2 3 4 5 6

19. Example a b a b c a 3+1 4+1 2+1 3 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 0 1 2 3 3 4 5 6

20. Example a b a b c a 4+0 5+1 3+1 4 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 0 1 2 3 4 3 4 5 6

21. Example a b a b c a 5+1 6+1 4+1 5 a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 0 1 2 3 4 5 3 4 5 6

22. Example a b a b c a a b a a c b c c c[0,0] = 0 for i=1..| x |: c[i,0] = i for j=1..| p |: c[0,j] = j for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) { 0 1 2 3 4 5 6 7 8 1 0 1 2 3 4 5 6 7 2 1 0 1 2 3 4 5 6 3 2 1 0 1 2 3 4 5 4 3 2 1 1 2 2 3 4 5 4 3 2 2 1 2 2 3 6 5 4 3 2 2 2 3 3

23. Dynamic programming algorithm After filling out c, d( x , y )=c[| x |,| y |] Time and space complexity: O(| x || y |)

24. Approximate pattern matching j=0 1 2 3 4 5 i= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 c[i,j] = d( x [1..i], y [1..j]) j=0 1 2 3 4 5 i= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 c[i,j] = min i' ≤ i d( x [i'..i], y [1..j]) j j i i i' Edit distance Edit distance pattern matching

25. Approximate pattern matching Use table c[i,j] = min i' ≤ i d( x [1..i], y [1..j]) Initialize: c[0,0] = 0 for i=1..| x |: c[i,0] = 0 for j=1..| p |: c[0,j] = j Main algorithm: for i=1..| x |: for j=1..| p |: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) {

26. Approximate pattern matching After filling out, in time and space O(| x || p |) all indices i, where c[i,| p |] ≤ k , correspond to one or more approximate matches. Some backtracking is needed to find the corresponding i' indices ... we can find one i' for each i in time O(| p |) per i , for a total time of O(| x || p |). (Theorem 10.1.1) (More complicated to get all matches...)

27. Wu-Manber We define a matrix s – the state of matching so far – by: s [ q , j ] = 0 iff d( x [ i-j+1 .. i ], p [1.. j ]) ≤ q for j =0..| p |, and q =0.. k i j

28. Wu-Manber As before, we use a pre-calculated bit-matrix: t [h,j] = 0 if p [j] == h 1 if p [j] != h with rows indexed by the alphabet and columns indexed by indices in p

29. Wu-Manber The recursion: c[i-1,j] + 1 c[i,j] = min c[i,j-1] + 1 c[i-1,j-1] + d( x [i], y [j]) becomes: s i-1 [q-1, j] s i [q,j] = & s i [q-1,j-1] s i-1 [q-1,j-1] & ( s i-1 [q,j-1] | t [ x [i],j]) { {

30. Wu-Manber The expression: s i-1 [q-1, j] s i [q,j] = & s i [q-1,j-1] s i-1 [q-1,j-1] & ( s i-1 [q,j-1] | t [ x [i],j]) can be computed as: old = s s [0] = (old[0] >> 1) | t [ x [i]] // SHIFT-and-OR for q=1..k: s1 = old[q-1] // s1[j] = s i-1 [q-1, j] s2 = s [q-1] >> 1 // s2[j] = s i [q-1,j-1] s3 = s1 >> 1 // s3[j] = s i-1 [q-1,j-1] s4 = old[q] >> 1 // s4[j] = s i-1 [ q,j-1] s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) {

31. Wu-Manber Special case: -Initial matrix: s [q] = 01 | p | Match when s [k,| p |] == 0

32. Example x = bbacbbbababacabbbba i=0 p = bbba 01234 s 0 [0]: 01111 s 0 [1]: 01111 s 0 [2]: 01111

33. Example x = b bacbbbababacabbbba i=1 01234 01234 s 0 [0]: 01111 s 1 [0]: 00 111 = 00 111 | 0000 1 s 0 [1]: 01111 s 1 [1]: s 0 [2]: 01111 s 1 [2]: old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = bb b a

34. Example x = b bacbbbababacabbbba i=1 01234 01234 s 0 [0]: 01111 s 1 [0]: 0 0 111 = 00 111 | 0000 1 s 0 [1]: 01111 s 1 [1]: s 0 [2]: 01111 s 1 [2]: old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = bb b a 0 edit distance match

35. Example x = b bacbbbababacabbbba i=1 01234 01234 s 0 [0]: 01111 s 1 [0]: 00 111 = 00 111 | 0000 1 s 0 [1]: 01111 s 1 [1]: s 0 [2]: 01111 s 1 [2]: old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = bb b a Not 0 edit distance match

36. Example x = b bacbbbababacabbbba i=1 p = b bba 01234 01234 s 0 [0]: 01111 s 1 [0]: 00 111 s 0 [1]: 01111 s 1 [1]: 0 00 11 = 0 1111 & 000 11 s 0 [2]: 01111 s 1 [2]: & 00 111 & ( 00 111 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b b ba p = bb b a p = bb b a

37. Example x = b bacbbbababacabbbba i=1 p = b bba 01234 01234 s 0 [0]: 01111 s 1 [0]: 00 111 s 0 [1]: 01111 s 1 [1]: 0 00 11 = 0 1111 & 000 11 s 0 [2]: 01111 s 1 [2]: & 00 111 & ( 00 111 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b b ba p = bb b a p = bb b a 1 edit distance match

38. Example x = b bacbbbababacabbbba i=1 p = b bba 01234 01234 s 0 [0]: 01111 s 1 [0]: 00 111 s 0 [1]: 01111 s 1 [1]: 0 00 11 = 0 1111 & 000 11 s 0 [2]: 01111 s 1 [2]: & 00 111 & ( 00 111 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b b ba p = bb b a p = bb b a Not 1 edit distance match

39. Example x = b bacbbbababacabbbba i=1 p = b bba 01234 01234 s 0 [0]: 01111 s 1 [0]: 00 111 s 0 [1]: 01111 s 1 [1]: 000 11 s 0 [2]: 01111 s 1 [2]: 0000 1 = 0 1111 & 0000 1 & 00 111 & ( 00 111 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b b ba p = bb b a p = bb b a

40. Example x = b bacbbbababacabbbba i=1 p = b bba 01234 01234 s 0 [0]: 01111 s 1 [0]: 00 111 s 0 [1]: 01111 s 1 [1]: 000 11 s 0 [2]: 01111 s 1 [2]: 0000 1 = 0 1111 & 0000 1 & 00 111 & ( 00 111 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b b ba p = bb b a p = bb b a 2 edit distance match

41. Example x = b bacbbbababacabbbba i=1 p = b bba 01234 01234 s 0 [0]: 01111 s 1 [0]: 00 111 s 0 [1]: 01111 s 1 [1]: 000 11 s 0 [2]: 01111 s 1 [2]: 0000 1 = 0 1111 & 0000 1 & 00 111 & ( 00 111 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b b ba p = bb b a p = bb b a Not 2 edit distance match

42. Example x = bb acbbbababacabbbba i=2 p = b bba 01234 01234 s 1 [0]: 00111 s 2 [0]: 000 11 = 000 11 | 0000 1 s 1 [1]: 00011 s 2 [1]: s 1 [2]: 00001 s 2 [2]: old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = bb ba p = b bb a p = b bb a

43. Example x = bb acbbbababacabbbba i=2 p = b bba 01234 01234 s 1 [0]: 00111 s 2 [0]: 0 00 11 = 000 11 | 0000 1 s 1 [1]: 00011 s 2 [1]: s 1 [2]: 00001 s 2 [2]: old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = bb ba p = b bb a p = b bb a 0 edit distance match

44. Example x = bb acbbbababacabbbba i=2 p = b bba 01234 01234 s 1 [0]: 00111 s 2 [0]: 000 11 = 000 11 | 0000 1 s 1 [1]: 00011 s 2 [1]: s 1 [2]: 00001 s 2 [2]: old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = bb ba p = b bb a p = b bb a Not 0 edit distance match

45. Example x = bb acbbbababacabbbba i=2 01234 01234 s 1 [0]: 00111 s 2 [0]: 000 11 s 1 [1]: 00011 s 2 [1]: 0000 1 = 00 111 & 0000 1 s 1 [2]: 00001 s 2 [2]: & 000 11 & ( 0000 1 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = bb ba p = b bb a p = b bb a

46. Example x = bb acbbbababacabbbba i=2 01234 01234 s 1 [0]: 00111 s 2 [0]: 000 11 s 1 [1]: 00011 s 2 [1]: 0 000 1 = 00 111 & 0000 1 s 1 [2]: 00001 s 2 [2]: & 000 11 & ( 0000 1 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = bb ba p = b bb a p = b bb a 1 edit distance match

47. Example x = bb acbbbababacabbbba i=2 01234 01234 s 1 [0]: 00111 s 2 [0]: 000 11 s 1 [1]: 00011 s 2 [1]: 0000 1 = 00 111 & 0000 1 s 1 [2]: 00001 s 2 [2]: & 000 11 & ( 0000 1 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = bb ba p = b bb a p = b bb a Not 1 edit distance match

48. Example x = bb acbbbababacabbbba i=2 01234 01234 s 1 [0]: 00111 s 2 [0]: 000 11 s 1 [1]: 00011 s 2 [1]: 0000 1 s 1 [2]: 00001 s 2 [2]: 00000 = 000 11 & 00000 & 0000 1 & ( 00000 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = bb ba p = b bb a p = b bb a

49. Example x = bb acbbbababacabbbba i=2 01234 01234 s 1 [0]: 00111 s 2 [0]: 000 11 s 1 [1]: 00011 s 2 [1]: 0000 1 s 1 [2]: 00001 s 2 [2]: 0 0000 = 000 11 & 00000 & 0000 1 & ( 00000 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = bb ba p = b bb a p = b bb a 2 edit distance match

50. Example x = bb acbbbababacabbbba i=2 01234 01234 s 1 [0]: 00111 s 2 [0]: 000 11 s 1 [1]: 00011 s 2 [1]: 0000 1 s 1 [2]: 00001 s 2 [2]: 0 000 0 = 000 11 & 00000 & 0000 1 & ( 00000 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = bb ba p = b bb a p = b bb a 2 edit distance match

51. Example x = bba cbbbababacabbbba i=3 01234 01234 s 2 [0]: 00011 s 3 [0]: 0 1111 = 0000 1 | 0 111 0 s 2 [1]: 00001 s 3 [1]: s 2 [2]: 00000 s 3 [2]: old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = b bba

52. Example x = bba cbbbababacabbbba i=3 01234 01234 s 2 [0]: 00011 s 3 [0]: 0 1111 = 0000 1 | 0 111 0 s 2 [1]: 00001 s 3 [1]: s 2 [2]: 00000 s 3 [2]: old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = b bba Not 0 edit distance match

53. Example x = bba cbbbababacabbbba i=3 01234 01234 s 2 [0]: 00011 s 3 [0]: 0 1111 s 2 [1]: 00001 s 3 [1]: 00000 = 0000 1 & 00 111 s 2 [2]: 00000 s 3 [2]: & 00000 & ( 00000 | 0 111 0 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = b bba

54. Example x = bba cbbbababacabbbba i=3 01234 01234 s 2 [0]: 00011 s 3 [0]: 0 1111 s 2 [1]: 00001 s 3 [1]: 0 0000 = 0000 1 & 00 111 s 2 [2]: 00000 s 3 [2]: & 00000 & ( 00000 | 0 111 0 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = b bba 1 edit distance match

55. Example x = bba cbbbababacabbbba i=3 01234 01234 s 2 [0]: 00011 s 3 [0]: 0 1111 s 2 [1]: 00001 s 3 [1]: 00000 s 2 [2]: 00000 s 3 [2]: 00000 = 0000 1 & 00000 & 00000 & ( 00000 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = b bba

56. Example x = bba cbbbababacabbbba i=3 01234 01234 s 2 [0]: 00011 s 3 [0]: 0 1111 s 2 [1]: 00001 s 3 [1]: 00000 s 2 [2]: 00000 s 3 [2]: 0 0000 = 0000 1 & 00000 & 00000 & ( 00000 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = b bba 2 edit distance match

57. Example x = bba cbbbababacabbbba i=3 01234 01234 s 2 [0]: 00011 s 3 [0]: 0 1111 s 2 [1]: 00001 s 3 [1]: 00000 s 2 [2]: 00000 s 3 [2]: 0 000 0 = 0000 1 & 00000 & 00000 & ( 00000 | 0000 1 ) old = s s [0] = (old[0] >> 1) | t [ x [i]] for q=1..k: s1 = old[q-1] s2 = s [q-1] >> 1 s3 = s1 >> 1 s4 = old[q] >> 1 s [q] = s1 & s2 & s3 & (s4 | t [ x [i]]) p = b bba p = b b ba p = bb b a p = b bba 2 edit distance match

58. Exercise: Complete the example...