COT 5405: Fall 2006
Lecture 23
DFA for String matching
Finite Automaton
1. Set of states, Q.
2. Start state q ๎ Q.
3. Set of accepting states, A
๎
Q.
4. Alphabet,
๎
.
5. Transition function,
๎
: Q
๎
๎
๎
Q.
General Construction Scheme
Final state function:
๎
(w) is the state after scanning w.
โข
๎
(
๎
) = q0.
โข
๎
(wa) =
๎
(
๎
(w), a), w
๎
๎
*, a
๎
๎
.
Suffix function:
๎
(x) = max{k: P[1
โฆ
k] is a suffix of x}.
โข
๎
(x) is the length of the longest prefix of P that is also a suffix of x.
โข P0 =
๎
is a suffix of all strings.
Construction: Q = {0, 1,
โฆ
, m}, q0 = 0, A = {m},
๎
(q, a) =
๎
(Pqa).
โข Note:
๎
(x) = m iff P is a suffix of x, implying that a match has been found.
DFA-based Matching
FA-Matcher(T, ๎, m)
โข q ๎ 0
โข for i = 1 to n
o q ๎ ๎(q, T[i])
o if q == m
๎ Print i โ m
This takes
๎
(n) time and
๎
(m |
๎
|) space.
