String Matching. String Matching. Naive String Matching Rabin-Karp Matcher String Matching with Finite Automata Knuth-Morris-Pratt Algorithm

Transcription

1 String Mtching String Mtching Nive String Mtching Rin-Krp Mtcher String Mtching with Finite Automt Knuth-Morris-Prtt Algorithm Input is pttern P[1..m] nd text T[1..n]. Find shift s s.t. P[i] = T[s+i] for 1 i m. Applictions: document serch, grep, DNA Nive-String-Mtcher(T, P) for s 0 to n m if P[1..m] = T[s+1..s+m] pttern occurs with shift s Nive-String-Mtcher is O(nm). Consider P = nd T =... CS 5633 Anlysis of Algorithms Chpter 32: Slide 2 String Mtching Rin-Krp String Mtching Rin-Krp Exmple Rin-Krp Mtcher Averge Cse Anlysis of Rin-Krp Finite Stte Automt FSA Ide for String Mtching FSA Mtcher (Incomplete) Delt Trnsition for String Mtching FSA for Mtching FSA for Mtching FSA for Mtching Knuth-Morris-Prtt Algorithm Computing the Prefix Function Prefix Function Anlysis KMP Mtcher KMP Mtcher Anlysis Rin-Krp String Mtching P[1..m] cn e converted to numer: m 1 p = P[m i] d i i=0 where ech chrcter is in rdix-d nottion. Similrly, T[s+1...s+m] cn e converted: m 1 t = T[s+m i] d i i=0 Ide: test p mod q = t mod q efore the more expensive P[1...m] = T[s+1...s+m]. Use prime numer for q. CS 5633 Anlysis of Algorithms Chpter 32: Slide 3 1 2

2 Rin-Krp Exmple Averge Cse Anlysis of Rin-Krp Proility Model: Suppose there re v vlid shifts. If s is not vlid shift, then p mod q = t mod q with proility 1/q. O(m) per vlid shift. O(m) per spurious hit. Expected numer of spurious hits is O(n/q). O(1) per itertion otherwise. Expected running time is O(m(v +n/q)+n), which is O(mv +n) if q m. CS 5633 Anlysis of Algorithms Chpter 32: Slide 6 CS 5633 Anlysis of Algorithms Chpter 32: Slide 4 Rin-Krp Mtcher Finite Stte Automt A finite stte utomton is defined y: Q, set of sttes q 0 Q, the strt stte A Q, the ccepting sttes Σ, the input lphet δ, the trnsition function, from Q Σ to Q Rin-Krp-Mtcher(T, P, d, q) p 0, t 0, h d m mod q for i 1 to m p (d p+p[i]) mod q t (d t+t[i]) mod q for s 0 to n m if s > 0 t (d t+t[s+m] T[s] h) mod q if p = t nd P[1..m] = T[s+1..s+m] pttern occurs with shift s CS 5633 Anlysis of Algorithms Chpter 32: Slide 7 CS 5633 Anlysis of Algorithms Chpter 32: Slide 5 3 4

3 FSA Ide for String Mtching Strt in stte q o. Perform trnsition from q 0 to q 1 if next chrcter of T = P[1]. Stte q i mens first i chrcters of P mtch. Trnsition from q i to q i+1 if the next chrcter of T = P[i+1]. Trnsition Function for P = Stte Inputs 1? 3?? 2?? CS 5633 Anlysis of Algorithms Chpter 32: Slide 8 Delt Trnsition for String Mtching q i+1 if i < m nd T[s] = P[i+1] if j is the mximum vlue δ(q i,t[s]) = q j+1 such tht T[s] = P[j +1] nd T[s j..s 1] = P[1..j] q 0 otherwise q i+1 if i < m nd x = P[i+1] if j is the mximum vlue δ(q i,x) = q j+1 such tht x = P[j +1] nd P[i j +1..i] = P[1..j] otherwise q 0 CS 5633 Anlysis of Algorithms Chpter 32: Slide 10 FSA Mtcher (Incomplete) FSA-Mtcher(T, P) q 0 // q is the stte of the FSA. for s 1 to n if q < m nd T[s] = P[q +1] q q +1 else q??? if q = m pttern occurs with shift s m Cnnot simply reset stte to 0. Consider P = nd T = CS 5633 Anlysis of Algorithms Chpter 32: Slide 9 FSA for Mtching Trnsition Function Sttes Inputs CS 5633 Anlysis of Algorithms Chpter 32: Slide

4 FSA for Mtching Trnsition Function Stte Inputs CS 5633 Anlysis of Algorithms Chpter 32: Slide 12 Knuth-Morris-Prtt Algorithm The Knuth-Morris-Prtt lgorithm efficiently implements finite stte utomtons. It is sed on computing prefix function: π[q] = mx{k : k < q nd P k is suffix of P q } where 0 k < q m nd P k = P[1...k] nd P q = P[1...q] P k is suffix of P q if P k = P[q k +1..q] CS 5633 Anlysis of Algorithms Chpter 32: Slide 14 FSA for Mtching Trnsition Function Stte Inputs CS 5633 Anlysis of Algorithms Chpter 32: Slide 13 Computing the Prefix Function Compute-Prefix-Function(P) m P.length π[1] 0 k 0 for q 2 to m while k > 0 nd P[k +1] P[q] k π[k] if P[k +1] = P[q] k k +1 π[q] k return π CS 5633 Anlysis of Algorithms Chpter 32: Slide

5 Prefix Function Anlysis Running time is Θ(m). Count chnges to k. π[k] < k, so k π[k] decreses k. k is incremented m 1 times nd k 0, so k cn e decresed t most m 1 times. If P[q] = P[k +1], then π[q] = k +1. If P[q] P[k +1], then check π[k] next ecuse P π[k] is suffix of oth P k nd P q 1. CS 5633 Anlysis of Algorithms Chpter 32: Slide 16 KMP Mtcher Anlysis Running time is O(n+m). Count chnges to q. π[q] < q, so q π[q] decreses q. q is incremented O(n) times nd q 0, so q cn e decresed t most O(n) times. Show correctness of computtion. Loop invrint is P q = T[i q...i 1]. This is true efore the first itertion. In while loop, If P[q +1] = T[i], then q is incremented. If P[q] T[i], then check π[q] next ecuse P π[q] is lso suffix of T i 1. CS 5633 Anlysis of Algorithms Chpter 32: Slide 18 KMP Mtcher KMP-Mtcher(T, P) π Compute-Prefix-Function(P) q 0 // q is the stte of the FSA. for i 1 to n while q > 0 nd P[q +1] T[i] q π[q] if P[q +1] = T[i] then q q +1 if q = m pttern occurs with shift i m q π[q] CS 5633 Anlysis of Algorithms Chpter 32: Slide