String Matching of Regular Expression

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String Matching of Regular Expression
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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 (m1)(2m1S) bits

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RE to NFA Construction

• Thompsons construction
• Produce up to 2m states
• Not null-free NFA
• Using (m)(2m1S) bits
• Glushkovs construction
• Produce exactly m1 states
• null-free NFA
• Using (m1)(2m1S) bits

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Thompsons Construction
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Thompsons Construction
Example
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Glushkov Construction

• RE ((ATGA((AGAAA)))
• Marked RE (A1T2G3A4((A5G6A7A8A9)))
• Used in Glushkov construction
• First(RE)
• The set of positions at which the reading can
  start.
• Ex First (A1T2G3A4((A5G6A7A8A9))) 1 ,3 .
• Last(RE)
• The set of positions at which a string read can
  be recognized.
• Ex Last (A1T2G3A4((A5G6A7A8A9)))2 ,4 ,6 ,9
  .
• Follow(RE,x)
• All the positions in RE accessible from x
• Ex Follow ((A1T2G3A4((A5G6A7A8A9))),6)
  7,5.
• EmptyRE is e if e belongs to L(RE) and Ø
  otherwise.

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Glushkov Construction

• Initial set of m1 states
• Marked final states, use Last (RE)
• Create transition link by Follow (RE,x)

RE (A1T2G3A4((A5G6A7A8A9)))
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Bit Parallel Automata

• Ex Shift-And
• Automata
• Update Function

State Mask
Occurrence Table
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Thompson BPA
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
Alphabet
S
m1
Pattern
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Glushkov BPA
Notation D State mask TD Follow of D B
Build by Glushkov Tj current char
B Table bit mask of the state reachable by each
letter
T Table Which states can be reached from an
active state
Alphabet
Active states
S
D2m1
m1
m1
Pattern
States

D
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Glushkov Search Algorithm
Build B Table
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Glushkov Search Algorithm
Build T Table
Initial to zero
Active states
D2m1
m1
States
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Glushkov Search Algorithm
Compute First, Last, Follow and Empty
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Performance Comparison
Forward Algorithm DFA
Glushkov with BuildT
Thompson s Construction
Glushkov with BuildTree
Test Pattern
Preprocessing time
Searching time
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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.