Strings matching in pattern recognition.ppt
- 2. Strings Let Σ be an alphabet, e.g. Σ = ( , a, b, c, …, z) A string is any member of Σ*, i.e. any sequence of 0 or more members of Σ ‘this is a string’ Σ* ‘this is also a string’ Σ* ‘1234’ Σ*
- 3. String operations Given strings s1 of length n and s2 of length m Equality: is s1 = s2? (case sensitive or insensitive) Running time O(n) where n is length of shortest string ‘this is a string’ = ‘this is a string’ ‘this is a string’ ≠ ‘this is another string’ ‘this is a string’ =? ‘THIS IS A STRING’
- 4. String operations Concatenate (append): create string s1s2 Running time Θ(n+m) ‘this is a’ . ‘ string’ → ‘this is a string’
- 5. String operations Substitute: Exchange all occurrences of a particular character with another character Running time Θ(n) Substitute(‘this is a string’, ‘i’, ‘x’) → ‘thxs xs a strxng’ Substitute(‘banana’, ‘a’, ‘o’) → ‘bonono’
- 6. String operations Length(‘this is a string’) → 16 Length(‘this is another string’) → 24
- 7. String operations Prefix(‘this is a string’, 4) → ‘this’ Suffix(‘this is a string’, 6) → ‘string’
- 8. String operations Substring(‘this is a string’, 4, 8) → ‘s is ’
- 9. Edit distance EXAM MUST (aka Levenshtein distance) Edit distance between two strings is the minimum number of insertions, deletions and substitutions required to transform string s1 into string s2 Insertion: ABACED ABACCED DABACCED Insert ‘C’ Insert ‘D’
- 10. Edit distance (aka Levenshtein distance) Edit distance between two strings is the minimum number of insertions, deletions and substitutions required to transform string s1 into string s2 Deletion: ABACED
- 11. Edit distance (aka Levenshtein distance) Edit distance between two strings is the minimum number of insertions, deletions and substitutions required to transform string s1 into string s2 Deletion: ABACED BACED Delete ‘A’
- 12. Edit distance (aka Levenshtein distance) Edit distance between two strings is the minimum number of insertions, deletions and substitutions required to transform string s1 into string s2 Deletion: ABACED BACED BACE Delete ‘A’ Delete ‘D’
- 13. Edit distance (aka Levenshtein distance) Edit distance between two strings is the minimum number of insertions, deletions and substitutions required to transform string s1 into string s2 Substitution: ABACED ABADED ABADES Sub ‘D’ for ‘C’ Sub ‘S’ for ‘D’
- 14. Edit distance examples Edit(Kitten, Mitten) = 1 Operations: Sub ‘M’ for ‘K’ Mitten
- 15. Edit distance examples Edit(Happy, Hilly) = 3 Operations: Sub ‘a’ for ‘i’ H I ppy Sub ‘l’ for ‘p’ Hi l py Sub ‘l’ for ‘p’ Hil l y
- 16. Edit distance examples Edit(Banana, Car) = 5 Operations: Delete ‘B’ anana Delete ‘a’ nana Delete ‘n’ naa Sub ‘C’ for ‘n’ Caa Sub ‘a’ for ‘r’ Car
- 17. Edit distance examples Edit(Simple, Apple) = 3 no of operation need Operations: Delete ‘S’ imple Sub ‘A’ for ‘i’ A mple Sub ‘m’ for ‘p’ A p ple
- 18. Is edit distance symmetric (reversibale)? that is, is Edit(s1, s2) = Edit(s2, s1)? Why? sub ‘i’ for ‘j’ sub ‘j’ for ‘i’ → delete ‘i’ insert ‘i’ → insert ‘i’ delete ‘i’ →
- 19. Calculating edit distance X = A B C B D A B Y = B D C A B A Ideas?
- 20. Calculating edit distance X = A B C B D A ? Y = B D C A B ? After all of the operations, X needs to equal Y
- 21. Calculating edit distance X = A B C B D A ? Y = B D C A B ? Operations: Insert Delete Substitute
- 22. Insert X = A B C B D A ? Y = B D C A B ?
- 23. Insert X = A B C B D A ? Y = B D C A B ? Edit ) , ( 1 ) , ( 1 ... 1 ... 1 m n Y X Edit Y X Edit
- 24. Delete X = A B C B D A ? Y = B D C A B ?
- 25. Delete X = A B C B D A ? Y = B D C A B ? ) , ( 1 ) , ( ... 1 1 ... 1 m n Y X Edit Y X Edit Edit
- 26. Substition X = A B C B D A ? Y = B D C A B ?
- 27. Substition X = A B C B D A ? Y = B D C A B ? Edit ) , ( 1 ) , ( 1 ... 1 1 ... 1 m n Y X Edit Y X Edit
- 28. Anything else? X = A B C B D A ? Y = B D C A B ?
- 29. Equal X = A B C B D A ? Y = B D C A B ?
- 30. Equal X = A B C B D A ? Y = B D C A B ? Edit ) , ( ) , ( 1 ... 1 1 ... 1 m n Y X Edit Y X Edit
- 31. Combining results ) , ( ) , ( 1 ... 1 1 ... 1 m n Y X Edit Y X Edit ) , ( 1 ) , ( 1 ... 1 1 ... 1 m n Y X Edit Y X Edit ) , ( 1 ) , ( ... 1 1 ... 1 m n Y X Edit Y X Edit ) , ( 1 ) , ( 1 ... 1 ... 1 m n Y X Edit Y X Edit Insert: Delete: Substitute: Equal:
- 32. Rabin-Karp algorithm P = ABA S = BABABBABABA - Use a function T to that computes a numerical representation of P , - Calculate T for all m symbol sequences of S and compare
- 33. P = ABA S = BABABBABABA Hash P T(P) Rabin-Karp algorithm - Use a function T to that computes a numerical representation of P - Calculate T for all m symbol sequences of S and compare
- 34. P = ABA S = BABABBABABA Hash m symbol sequences and compare T(P) Rabin-Karp algorithm - Use a function T to that computes a numerical representation of P - Calculate T for all m symbol sequences of S and compare T(BAB) =
- 35. P = ABA S = BABABBABABA Hash m symbol sequences and compare T(P) match Rabin-Karp algorithm - Use a function T to that computes a numerical representation of P - Calculate T for all m symbol sequences of S and compare T(ABA) =
- 36. P = ABA S = BABABBABABA Hash m symbol sequences and compare T(P) Rabin-Karp algorithm - Use a function T to that computes a numerical representation of P - Calculate T for all m symbol sequences of S and compare T(BAB) =
- 37. P = ABA S = BABABBABABA Hash m symbol sequences and compare T(P) … Rabin-Karp algorithm - Use a function T to that computes a numerical representation of P - Calculate T for all m symbol sequences of S and compare T(BAB) =
- 38. Rabin-Karp algorithm Given T(si…i+m-1) we must be able to efficiently calculate T(si+1…i+m) P = ABA S = BABABBABABA For this to be useful/efficient, what needs to be true about T? T(P) … T(BAB) =