exact string search
DESCRIPTION
Exact String Search. Lecture 7: September 22, 2005 Algorithms in Biosequence Analysis Nathan Edwards - Fall, 2005. Boyer-Moore. Method of choice for exact string search, for a single pattern Typically, examines fewer than m characters of the text (sublinear time) - PowerPoint PPT PresentationTRANSCRIPT
Exact String SearchExact String Search
Lecture 7: September 22, 2005
Algorithms in Biosequence AnalysisNathan Edwards - Fall, 2005
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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
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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.
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Right to left scan / bad character rule
0 1 12345678901234567T:xpbctbxabpqxctbpqP: tpabxab *^^^^
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Right to left scan / bad character rule
0 1 12345678901234567T:xpbctbxabpqxctbpqP: tpabxab *^^^^P: tpabxab *
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Right to left scan / bad character rule
0 1 123456789012345678T:xpbctbxabpqxctbpqzP: tpabxab *^^^^P: tpabxab *P: tpabxab
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Bad character rule
Comparing r-to-l, mismatch at i of P, k of T:
If T(k) is absent from Pshift left end of P to k+1 of T
If right-most T(k) in P is to left of ishift pattern to align T(k) characters
Otherwiseshift pattern 1 position
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Right to left scan / bad character rule
0 1 12345678901234567T:xpbctbaabpqxctbpqP: tpabxab *^^
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Right to left scan / bad character rule
0 1 12345678901234567T:xpbctbaabpqxctbpqP: tpabxab *^^
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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 ishift pattern to align T(k) characters
Otherwiseshift pattern 1 position
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Right to left scan / extended bad character rule
0 1 12345678901234567T:xpbctbaabpqxctbpqP: tpabxab *^^
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Right to left scan / extended bad character rule
0 1 12345678901234567T:xpbctbaabpqxctbpqP: tpabxab
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(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)
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(Strong) good suffix rule
0 1 123456789012345678T:prstabstubabvqxrstP: qcabdabdab *
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(Strong) good suffix rule
0 1 123456789012345678T:prstabstubabvqxrstP: qcabdabdab *^^ P: qcabdabdab
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(Strong) good suffix rule
0 1 123456789012345678T:prstabstudabvqxrstP: abdubdab *^^^
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(Strong) good suffix rule
0 1 123456789012345678T:prstabstudabvqxrstP: abdubdab *^^^ P: abdabdab
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(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 Pshift 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
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(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.
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Computing L’(i)
Definition:
Nj(P) is the length of the longest suffix of P[1…j] that is also a suffix of P.
compare with:
Zi(S) is the length of the longest prefix of S[i…|S|] that is also a prefix of S.
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Computing L’(i)
Definition:
Nj(P) is the length of the longest suffix of P[1…j] that is also a suffix of P.
(!) compare with:
Zi(S) is the length of the longest prefix of S[i…|S|] that is also a prefix of S.
Compute Nj(P) as Zn-j+1(reverse(P)).
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Computing L’(i)
• L’(i) – max j < n s.t. Nj(P) = |P[i…n]| = (n – i +1)
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(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. Nj(P) = j
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Boyer-Moore psuedo code
Compute L’(i), l’(i), and R(x) for x in Σ.k = nwhile 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 }
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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?
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Worst case instance, P in T
0 1 12345678901234567T:aaaaaaaaaaaaaaaaaP: aaaaaaa ^^^^^^^P: aaaaaaa ^^^^^^^
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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, thenprefix 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.
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Worst case instance, P in T
0 1 12345678901234567T:aaaaaaaaaaaaaaaaaP: aaaaaaa ^^^^^^^P: aaaaaaa ^
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Galil’s Extention
0 1 123456789012345678T:prstabstudabvqxrstP: abdubdab *^^^ P: abdabdab
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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.