1. Recuperació de la informació
06/04/2019
Modern Information Retrieval (1999)
Ricardo-Baeza Yates and Berthier Ribeiro-Neto
Flexible Pattern Matching in Strings (2002)
Gonzalo Navarro and Mathieu Raffinot
Algorithms on strings (2001)
M. Crochemore, C. Hancart and T. Lecroq

2. String Matching
06/04/2019
String matching: definition of the problem (text,pattern)
Exact matching:
depends on what we have: text or patterns
The patterns ---> Data structures for the patterns
1 pattern ---> The algorithm depends on |p| and ||
k patterns ---> The algorithm depends on k, |p| and ||
Extensions
Regular Expressions
The text ----> Data structure for the text (suffix tree, ...)
Approximate matching:
Dynamic programming
Sequence alignment (pairwise and multiple)
Sequence assembly: hash algorithm
Probabilistic search:
Hidden Markov Models

3. Regular expression
06/04/2019
A regular expression ℛ is a string on the set of simbols
Σ U { ε, |, · , * , (, ) } which is recursively defined as:
ε (empty character) is a regular expression
A character of Σ is a regular expression
( ℛ ) is a regular expression
ℛ1 · ℛ2 is a regular expression
As you have seen this morning ....
ℛ1 | ℛ2 is a regular expression
ℛ * is a regular expression

4. Regular lenguage
06/04/2019
The lenguage defined by a regular expression ℛ is the set of strings generated by ℛ .
The problem of searching for a regular expression in the text T is to find all the factors in T that belong to the lenguage.
As you have seen this morning ....

5. Methods Regular expression Parse tree NFA DFA Strings found
06/04/2019
Regular expression
Parse tree
NFA
DFA
Search with deterministic finit automata
Search with bit-parallel Thompson automata
As you have seen this morning ....
Strings found

6. Methods Regular expression Parse tree NFA DFA Strings found
06/04/2019
Regular expression
Parse tree
NFA
DFA
Search with deterministic finit automata
Search with bit-parallel Thompson automata
As you have seen this morning ....
Strings found

7. Search with a deterministic finit automata
06/04/2019
Given the regular expression bb*(b|b*a) the NFA is b 1 a 3 2
As it’s not possible to spell the text out the NFA, the NFA is transformed into a DFA … b 1 a 3 12
As you have seen this morning ....
What is the cost?
And the search process…

8. Search example with DFA
06/04/2019 b 1 a 3 12
Given the regular expression bb*(b|b*a) and the NFA:
The search on the text:
b b b a a b a a b b …
As you have seen this morning .... …

9. Methods Regular expression Parse tree NFA DFA Strings found
06/04/2019
Regular expression
Parse tree
NFA
DFA
Search with deterministic finit automata
Search with bit-parallel Thompson automata
As you have seen this morning ....
Strings found

10. . Parse tree Is a tree such that:
06/04/2019
Is a tree such that:
- internal nodes are labeled by operators
- leaves are labeled by characters of Σ and ε .
ℛ ℛ2
( ℛ ) ℛ
ℛ1 · ℛ2
ℛ1 | ℛ2
As you have seen this morning .... |
ℛ ℛ2
ℛ * ℛ *

11. Parse tree: example
06/04/2019
Given the regular expression bb*(b|b*a) the parse tree is: . | b * b . b
As you have seen this morning .... a * b

12. NFA (Thompson automaton)
06/04/2019
From the regular expression or from the parse tree we define the automaton:
For a character a of Σ: a .
ℛ1 ℛ2 |
ℛ1 ℛ2 ε ε
As you have seen this morning .... ℛ * ε

13. Thompsom automaton construction
06/04/2019
bb*(b|b*a) . | b * b b . a * b b b a
As you have seen this morning .... b b

14. NFA: ε-closure (states ε-equivalents)
06/04/2019
bb*(b|b*a) b 6 7 b a 2 3 5 8 9 b 1 4 12 b 10 11 ε 1 3 4 5 7 9 11
1, 2, 4, 5, 6, 8, 10
2, 3, 4, 5, 6, 8,10
4, 5, 6, 8, 10
As you have seen this morning ....
5, 6, 8
6, 7, 8
9, 12
11, 12

15. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) ε
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 6 7 b 2 3 a 5 8 9 b D
-> 1 4 12 b 10 11
Text: ababbbaab
The bit-vector D mark the active states: at the begining
At every step we shift to the right followed by an “and” operator with the mask of the last read character… D
As you have seen this morning ....
The masks are B a b
(a)
…and the ε-closure extension of active states.

16. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 ε
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b
-> 1 4 12 b 10 11 B a b
Text: ababbbaab D D
->
As you have seen this morning ....
(a)

17. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 ε
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b
-> 1 4 12 b 10 11 B a b
Text: ababbbaab D D
->
As you have seen this morning ....
(a)
(b)

18. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D
->
As you have seen this morning ....

19. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D
->
As you have seen this morning ....
(a)

20. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D D
->
As you have seen this morning ....
(a)

21. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D D
->
->
As you have seen this morning ....
(a)

22. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D D
->
->
As you have seen this morning ....
(a)
(b)

23. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D D
->
->
As you have seen this morning ....
(a)
(b)

24. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D
->
As you have seen this morning ....

25. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D
->
As you have seen this morning ....
(a)

26. Bit-parallel Thompsom algorithm
06/04/2019
bb*(b|b*a) b 6 7 b 2 3 a 5 8 9 E
, 2, 4, 5, 6, 8, 10
, 3, 4, 5, 6, 8,10
, 5, 6, 8, 10
, 6, 8
, 7, 8
, 12
, 12 b 1 4 12 b 10 11 B a b
Text: ababbbaab D
->
As you have seen this morning ....
(a)