1. String Matching of Regular Expression

2. Introduction Regular Expression (RE)
A generalized string description with
Basic string
Kleene star (*)
Concatenation
Union (|)
Nondeterministic Finite Automata (NFA)
More then one next transition
RE to NFA require m state
Deterministic Finite Automata (DFA)
Only one next transition
RE to DFA may 2m state
Using (m+1)(2m+1|Σ|) bits

3. RE to NFA Construction Thompson’s construction Glushkov’s construction
Produce up to 2m states
Not null-free NFA
Using (m)(2m+1+|Σ|) bits
Glushkov’s construction
Produce exactly m+1 states
null-free NFA
Using (m+1)(2m+1+|Σ|) bits

4. Thompson’s Construction

5. Thompson’s Construction
Example

6. Glushkov Construction
RE = ((AT|GA((AG|AAA)∗))
Marked RE = (A1T2|G3A4((A5G6|A7A8A9)∗))
Used in Glushkov construction
First(RE)
The set of positions at which the reading can start.
Ex: First (A1T2|G3A4((A5G6|A7A8A9)∗))= {1 ,3 }.
Last(RE)
The set of positions at which a string read can be recognized.
Ex: Last (A1T2|G3A4((A5G6|A7A8A9)∗))={2 ,4 ,6 ,9 }.
Follow(RE,x)
All the positions in RE accessible from x
Ex: Follow ((A1T2|G3A4((A5G6|A7A8A9)∗)),6)= {7,5}.
EmptyRE is {ε} if ε belongs to L(RE) and ∅ otherwise.

7. Glushkov Construction
Initial set of m+1 states
Marked final states, use Last (RE)
Create transition link by Follow (RE,x)
RE = (A1T2|G3A4((A5G6|A7A8A9)∗))

8. Bit Parallel Automata Ex: Shift-And Automata Update Function
State Mask
Occurrence Table

9. Thompson BPA |Σ| Notation D : State mask E: null-closure of D
B: Precomute Table
S: string length
Tj: current char
null-closure,
reachable state from D with null input
B Table:
bit mask of the state reachable by each letter
|Σ|
Alphabet
m+1
Pattern

10. Glushkov BPA |Σ| & D Notation D : State mask T[D}: Follow of D
B: Build by Glushkov
Tj: current char
T Table:
Which states can be reached from an active state
B Table:
bit mask of the state reachable by each letter
Active states
D=2m+1
|Σ|
Alphabet
m+1
m+1
Pattern
States
& D

11. Glushkov Search Algorithm
Build B Table

12. Glushkov Search Algorithm
Build T Table
Initial to zero
Active states
D=2m+1
m+1
States

13. Glushkov Search Algorithm
Compute First, Last,
Follow and Empty

14. Performance Comparison
Forward Algorithm
DFA
Glushkov
with BuildT
Thompson ’s
Construction
Glushkov
with BuildTree
Test Pattern
Preprocessing time
Searching time

15. Reference
G. Navarro and M. Raffinot. Compact DFA representation for fast regular expression search . In Proceedings of the 5th Workshop on Algorithm Engineering , number 2141 in Lecture Notes in Computer Science, pages 1-12, 2001.