1. Approximate k-edit-distance Approximative ( k -distance) matching, now for edit distance Given stringx = abbacbbbababacabbbbaandpatternp = bbbafind all almost-occurrences ofpindx x = a bba c bbba babacab b a bba 17 6 1

2. Edit distance Theedit-distancebetween stringsxandyisthe minimal number of - insertions - deletions - substitutions needed to translatexintoy 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 mind( 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 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j Main algorithm: fori=1..| x |: forj=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 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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+01+1 1+10 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 10 2 3 4 5 6 8. Example a b a b c a 1+12+1 0+11 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 9. Example a b a b c a 2+03+1 1+12 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 1012 2 3 4 5 6 10. Example a b a b c a 3+04+1 2+13 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 23 2 3 4 5 6 11. Example a b a b c a 4+15+1 3+14 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 34 2 3 4 5 6 12. Example a b a b c a 5+16+1 4+15 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 45 2 3 4 5 6 13. Example a b a b c a 6+17+1 5+16 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 56 2 3 4 5 6 14. Example a b a b c a 7+18+1 6+17 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 67 2 3 4 5 6 15. Example a b a b c a 1+10+1 2+11 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 7 21 3 4 5 6 16. Example a b a b c a 0+01+1 1+10 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 7 210 3 4 5 6 17. Example a b a b c a 1+12+1 0+11 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 7 21 01 3 4 5 6 18. Example a b a b c a 2+13+1 1+12 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 7 21 0 12 3 4 5 6 19. Example a b a b c a 3+14+1 2+13 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 7 21 0 1 23 3 4 5 6 20. Example a b a b c a 4+05+1 3+14 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 7 21 0 1 2 34 3 4 5 6 21. Example a b a b c a 5+16+1 4+15 a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 7 21 0 1 2 3 45 3 4 5 6 22. Example a b a b c a a b a a c b c c c[0,0] = 0 fori=1..| x |: c[i,0] = i forj=1..| p |: c[0,j] = j fori=1..| x |: forj=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 8 101 2 3 4 5 6 7 21 0 1 2 3 4 56 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 33 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' id( 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 fori=1..| x |: c[i,0] =0 forj=1..| p |: c[0,j] = j Main algorithm: fori=1..| x |: forj=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 indicesi,where c[i,| p |] k , correspond to one or more approximate matches. Some backtracking is needed to find the correspondingi'indices ... we can find onei'for eachiin time O(| p |) peri , for a total time of O(| x || p |). (Theorem 10.1.1) (More complicated to get all matches...) 27. Wu-Manber We define amatrix s the state of matching so far by: s [ q , j ] = 0iffd( x [ i-j+1 .. i ], p [1.. j ]) q forj =0..| p |, andq =0.. k i j 28. Wu-Manber As before, we use apre-calculated bit-matrix: t [h,j] = 0ifp [j] == h 1ifp [j] != h with rows indexed by the alphabet and columns indexed by indices inp 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 Theexpression: 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 forq=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 whens [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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 0123401234 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]] forq=1..k: s1 = old[q-1]s2 =s [q-1] >> 1s3 = s1 >> 1s4 = old[q] >> 1s [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...