String Matching

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Finite Automaton
Definition 3 A finite automaton is a 5-tuple, M = (Q, q0 , A, Σ, δ ), where • Q is a finite set of states. • q0 ∈ Q is the start state. • A ⊆ Q is the distinguished set of accepting states. • Σ is a finite input alphabet. • δ : Q × Σ → Q is the transition function of M . A finite automaton M induces a function φ : Σ∗ → Q (called final-state function) such that φ( ) = q0
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Homework
Homework 4 How would you extend the RABINKARP-MATCHER method to the problem of searching a text string for an occurrence of any one of a given set of k patterns? Start by assuming that all k patterns have the same length. Then generalize your solution to allow the pattern to have different lengths. Homework 5 Show hoe to extend the RABINKARP-MATCHER method to handle the problem of looking for a given m × m pattern in an n × n array of characters. (The pattern may be shifted vertically and horizontally, but it may not be rotated.)
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Expected Cost of RK Algorithm
Property 2 The expected matching time of RABIN-KARP-MATCHER is O(n). Proof The probability of ts ≡ p (mod q ) is 1/q , so the expected number of matchings is O(m(n − m + 1)/q ). If we choose q ≥ m, then the expected time of matchings is O(n − m + 1) = O(n) because m ≤ n.
a b c a b a a b c a b a c s=3
a b a a
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Algorithms of String Matching
Algorithm Naive Rabin-Karp Finite Automaton Knuth-Morris-Pratt Boyer-Moore Preprocessing Time 0 Θ(m) O ( m|Σ |) Θ(m) Θ(m) Matching Time O((n − m + 1)m) O((n − m + 1)m) Θ(n) Θ(n) Θ(n)
x z y z y xching
NAIVE-STRING-MATCHER(T, P ) 1 n ← length[A] 2 m ← length[P ] 3 for s ← 0 to n − m 4 do if P [1 . . . m] = T [s + 1 . . . s + m] 5 then print “Occur with shift” s NAIVE-STRING-MATCHER takes time O((n−m+ 1)m), and the bound is tight in the worst case. For example, let T = an and P = am. Example 3 If P = aaab and we find s = 0 is valid, then none of s = 1, 2, 3 is valid.
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Idea of Rabin-Karp Algorithm
Assume Σ = {0, 1, 2, · · · , 9} and T, P ∈ Σ∗.
1. Turn T [1 . . . n] and P [1 . . . m] to decimal numbers.
2. Do we have ts ≡ p (mod q )? where s = 0, 1, · · · , n − m and q is a prime. (a) If yes, T [s + 1 . . . s + m] = P [1 . . . m]. (b) If no, maybe T [s+1 . . . s+m] = P [1 . . . m].
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Example of Rabin-Karp Algorithm
2359023141526739921
...
... ...mod 13 Valid Spurious match hit
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8 9 3 11 0 1 7 8 4 5 10 11 7 9 11
How to Get Decimal Integers?
3. String matching with finite automaton
4. Knuth-Morris-Pratt algorithm
5. Conclusion
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Problem of String Matching
Definition 1 Given a text string T [1 . . . n] of length n and a pattern string P [1 . . . m] of length m. We say P occurs with shift s in T if T [s + 1 . . . s + m] = P [1 . . . m]. Example 1 Given T = abcabaabcabac and P = abaa, the string matching is
• We can compute the remaining values t1, t2, · · · , tn−m in time Θ(n − m): ts+1 = 10(ts −10m−1T [s+1])+T [s+m+1] So, the preprocessing time is Θ(m) for p and the matching time is Θ(n − m + 1).
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Example
14152 ≡ ≡ ≡ (31415 − 3 · 10000) · 10 + 2 (mod 13) (7 − 3 · 3) · 10 + 2 (mod 13) 8 (mod 13)
314152
78
Usually, 10q just fits within one computer word, which allows all the necessary computations to be performed with single-precision arithmetic. If q is large enough, then we hope that spurious hits occur infrequently enough that the cost of the extra checking is low.
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Example
a c a a b c a a b
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Example
a c a a b c
s=1
a a b
6-a
Example
a c a a b c s=2 a a b
6-b
Example
a c a a b c s=3 a a b
6-c
Homework
Homework 1 Show the comparisons NAIVESTRING-MATCHER makes for the pattern P = 0001 in the text T = 000010001010001. Homework 2 Suppose that all characters in the pattern P are different. Show how to accelerate NAIVE-STRING-MATCHER to run in time O(n) on an n-character text T .
Definition 2 A string w is a prefix of a string x, denoted by w x if x = wy for some string y ∈ Σ∗. Similarly, a string w is a suffix of a string x, denoted by w x if x = yw for some string y ∈ Σ∗. Example 2 ab abcca and cca abcca.
• We can compute p and t0 in Θ(m) time using Horner’s Rule:
p = P [m]+10(P [m−1]+· · ·+10(P [2]+10P [1]) · · · ))
Similarly, t0 is computed from T [1 . . . m].