Exact String Search Lecture 7: September 22, 2005 Algorithms in Biosequence Analysis Nathan Edwards - Fall, 2005

2 Boyer-Moore Method of choice for exact string search, for a single pattern Typically, examines fewer than m characters of the text (sublinear time) Linear worst case running time Conceptually very similar to K-M-P, but more complicated to running time proof Empirically, better for english text than DNA sequence

3 Boyer-Moore Three key ideas Right to left scan Bad character rule (Strong) good suffix rule The combination of these ideas can produce large pattern shifts. Provable O(n+m) running time when pattern is not in the text need extension for case when pattern is in the text to achieve linear running time.

4 Right to left scan / bad character rule T:xpbctbxabpqxctbpq P: tpabxab *^^^^

5 Right to left scan / bad character rule T:xpbctbxabpqxctbpq P: tpabxab *^^^^ P: tpabxab *

6 Right to left scan / bad character rule T:xpbctbxabpqxctbpqz P: tpabxab *^^^^ P: tpabxab * P: tpabxab

7 Bad character rule Comparing r-to-l, mismatch at i of P, k of T: If T(k) is absent from P shift left end of P to k+1 of T If right-most T(k) in P is to left of i shift pattern to align T(k) characters Otherwise shift pattern 1 position

8 Right to left scan / bad character rule T:xpbctbaabpqxctbpq P: tpabxab *^^

9 Right to left scan / bad character rule T:xpbctbaabpqxctbpq P: tpabxab *^^

10 Extended bad character rule Comparing r-to-l, mismatch at i of P, k of T: If T(k) is absent from P[1…i-1] shift left end of P to k+1 of T For right-most T(k) in P to left of i shift pattern to align T(k) characters Otherwise shift pattern 1 position

11 Right to left scan / extended bad character rule T:xpbctbaabpqxctbpq P: tpabxab *^^

12 Right to left scan / extended bad character rule T:xpbctbaabpqxctbpq P: tpabxab

13 (Extended) bad character rule For all x in Σ, R(x) is the position of the right-most occurrence of x in P. R(x) is zero if x is absent from P. Comp. r-to-l, mismatch i of P, k of T: shift P right max[1,i-R(T(k))] positions For extended bad character rule, need to lookup R(x,i)

14 (Strong) good suffix rule T:prstabstubabvqxrst P: qcabdabdab *

15 (Strong) good suffix rule T:prstabstubabvqxrst P: qcabdabdab *^^ P: qcabdabdab

16 (Strong) good suffix rule T:prstabstudabvqxrst P: abdubdab *^^^

17 (Strong) good suffix rule T:prstabstudabvqxrst P: abdubdab *^^^ P: abdabdab

18 (Strong) good suffix rule Substring t of T matches suffix of P: Find the right-most copy t’ in P s.t. t’ is not a suffix of P and char to left of t’ in P ≠ char to left of t in P shift P to align t’ in P with t in T If no such t’ shift P so that the longest proper prefix of P aligns with suffix of P

19 (Stong) good suffix rule Definitions: L(i) – max j < n such that P[i…n] matches suffix of P[1…j], 0 if no such j. L’(i) – max j < n such that P[i…n] matches suffix of P[1…j] and char. before suffix ≠ P(i-1), 0 if no such j. Weak and strong shifts for first part of good suffix rule.

20 Computing L’(i) Definition: N j (P) is the length of the longest suffix of P[1…j] that is also a suffix of P. compare with: Z i (S) is the length of the longest prefix of S[i…|S|] that is also a prefix of S.

21 Computing L’(i) Definition: N j (P) is the length of the longest suffix of P[1…j] that is also a suffix of P. (!) compare with: Z i (S) is the length of the longest prefix of S[i…|S|] that is also a prefix of S. Compute N j (P) as Z n-j+1 (reverse(P)).

22 Computing L’(i) L’(i) – max j < n s.t. N j (P) = |P[i…n]| = (n – i +1)

23 (Strong) good suffix rule Definition: l’(i) – length of the longest prefix of P that is also a suffix of P[i…n], 0 if no such prefix exists. l’(i) – max j < (n – i + 1) s.t. N j (P) = j

24 Boyer-Moore psuedo code Compute L’(i), l’(i), and R(x) for x in Σ. k = n while k ≤ n i = n, h = k while i > 0 and P(i) = T(h) i--; h-- if i = 0 occurrence of P in T k = k + n – l’(2) else If L’(i+1) > 0, λ = L’(i+1), λ = l’(i+1) k = k + max{ 1, i - R(T(h)), n – λ }

25 Running time analysis Notice that unlike K-M-P, we might re- compare text characters that matched in a previous iteration. Worst instance does Θ(nm) total comparisons, but only if P is in T If P is not in T, O(n+m) running time complicated proof! What goes wrong when P is in T?

26 Worst case instance, P in T T:aaaaaaaaaaaaaaaaa P: aaaaaaa ^^^^^^^ P: aaaaaaa ^^^^^^^

27 Galil’s Extention Comparing r-to-l, n of P aligned to k of T, matched at character s of T: If pos 1 of P shifts past s, then prefix of P matches in T up to pos k. skip these comparisons Sufficient for linear time bound, whether or not P is in T or not.

28 Worst case instance, P in T T:aaaaaaaaaaaaaaaaa P: aaaaaaa ^^^^^^^ P: aaaaaaa ^

29 Galil’s Extention T:prstabstudabvqxrst P: abdubdab *^^^ P: abdabdab

30 Lessons From B-M Sub-linear time is possible But we still need to read T from disk! Bad cases require periodicity in P or T matching random P with T is easy! Large alphabets mean large shifts Small alphabets make complicated shift data-structures possible B-M better for “english” and amino- acids than for DNA.