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String Matching with Mismatches

String Matching with Mismatches. Some slides are stolen from Moshe Lewenstein (Bar Ilan University). String Matching with Mismatches. Landau – Vishkin 1986 Galil – Giancarlo 1986 Abrahamson 1987

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String Matching with Mismatches

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  1. String Matching with Mismatches Some slides are stolen from Moshe Lewenstein (Bar Ilan University)

  2. String Matching with Mismatches Landau – Vishkin 1986 Galil – Giancarlo 1986 Abrahamson 1987 Amir - Lewenstein - Porat 2000

  3. Approximate String Matching problem: Find all text locations where distance from pattern is sufficiently small. distance metric: HAMMING DISTANCE Let S = s1s2…sm R = r1r2…rm Ham(S,R) = The number of locations j where sjrj Example: S = ABCABC R = ABBAAC Ham(S,R)= 2

  4. Problem 1: Counting mismatches Input: T = t1 . . . tn P = p1 … pm Output: For each i in T Ham(P,titi+1…ti+m-1) Example: P = A B B A A C T = A B C A A B C A C…

  5. Counting mismatches Input: T = t1 . . . tn P = p1 … pm Output: For each i in T Ham(P,titi+1…ti+m-1) Example: P = A B B A A C T = A B C A A B C A C… 2 Ham(P,T1)= 2

  6. Counting mismatches Input: T = t1 . . . tn P = p1 … pm Output: For each i in T Ham(P,titi+1…ti+m-1) Example: P = A B B A A C T = A B C A A B C A C… 2, 4 Ham(P,T2)= 4

  7. Counting mismatches Input: T = t1 . . . tn P = p1 … pm Output: For each i in T Ham(P,titi+1…ti+m-1) Example: P = A B B A A C T = A B C A A B C A C… 2, 4, 6 Ham(P,T3)= 6

  8. Counting mismatches Input: T = t1 . . . tn P = p1 … pm Output: For each i in T Ham(P,titi+1…ti+m-1) Example: P = A B B A A C T = A B C A A B C A C… 2, 4, 6, 2 Ham(P,T4)= 2

  9. Counting mismatches Input: T = t1 . . . tn P = p1 … pm Output: For each i in T Ham(P,titi+1…ti+m-1) Example: P = A B B A A C T = A B C A A B C A C… 2, 4, 6, 2, …

  10. Problem 2: k-mismatches Input: T = t1 . . . tn P = p1 … pm Output: Every i where Ham(P,titi+1…ti+m-1) ≤ k Example: k = 2 P = A B B A A C T = A B C A A B C A C… 2, 4, 6, 2, …

  11. Problem 2: k-mismatches Input: T = t1 . . . tn P = p1 … pm Output: Every i where Ham(P,titi+1…ti+m-1) ≤ kh Example: k = 2 P = A B B A A C T = A B C A A B C A C… 2, 4, 6, 2, … 1, 0, 0, 1,

  12. Naïve Algorithm (for counting mismatches or k-mismatches problem) - Goto each location of text and compute hamming distance of P and Ti Running Time: O(nm) n = |T|, m = |P|

  13. The Kangaroo Method (for k-mismatches) Landau – Vishkin 1986 Galil – Giancarlo 1986

  14. The Kangaroo Method (for k-mismatches) • Create suffix tree (+ lca) for: s = P#T • Check P at each location i of T by kangrooing • Example: • P = A B A B A A B A C A B • T = A B B A C A B A B A B C A B B C A B C A … • i

  15. The Kangaroo Method (for k-mismatches) • Create suffix tree for: s = P#T • Check P at each location i of T by kangrooing • Example: • P = A B A B A A B A C A B • T = A B B A C A B A B A B C A B B C A B C A … • i

  16. The Kangaroo Method (for k-mismatches) • Create suffix tree for: s = P#T • Check P at each location i of T by kangrooing • Example: • P = A B A B A A B A C A B • T = A B B A C A B A B A B C A B B C A B C A … • i

  17. The Kangaroo Method (for k-mismatches) • Create suffix tree for: s = P#T • Check P at each location i of T by kangrooing • Example: • P = A B A B A A B A C A B • T = A B B A C A B A B A B C A B B C A B C A … • i

  18. The Kangaroo Method (for k-mismatches) • Create suffix tree for: s = P#T • Check P at each location i of T by kangrooing • Example: • P = A B A B A A B A C A B • T = A B B A C A B A B A B C A B B C A B C A … • i

  19. The Kangaroo Method (for k-mismatches) • Create suffix tree for: s = P#T • Check P at each location i of T by kangrooing • Example: • P = A B A B A A B A C A B • T = A B B A C A B A B A B C A B B C A B C A … • i

  20. The Kangaroo Method (for k-mismatches) • Create suffix tree for: s = P#T • Check P at each location i of T by kangrooing • Example: • P = A B A B A A B A C A B • T = A B B A C A B A B A B C A B B C A B C A … • i

  21. The Kangaroo Method (for k-mismatches) • Create suffix tree for: s = P#T • Do up to k LCP queries for every text location • Example: • P = A B A B A A B A C A B • T = A B B A C A B A B A B C A B B C A B C A … • i

  22. The Kangaroo Method (for k-mismatches) Preprocess: Build suffix tree of both P and T - O(n+m) time LCA preprocessing - O(n+m) time Check P at given text location Kangroo jump till next mismatch - O(k) time Overall time: O(nk)

  23. How do we do counting in less than O(nm) ?

  24. If we pad the pattern and the text with zeros this is like a convolution of two vectors of length m+n We can do this using FFT in O(nlog(n)) time Lets start with binary strings P = 0 1 0 1 1 0 1 1 T = 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 ... ...

  25. a b a c c a c b a c a b a c c And if the strings are not binary ? P = T = a b b b c c c a a a a b a c b ... ...

  26. a b a c c a c b a c a b a c c P = a-mask a b a c c a c b a c a b a c c T = a b b b c c c a a a a b a c b ... ...

  27. a b a c c a c b a c a b a c c 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 P = a-mask a b a c c a c b a c a b a c c Pa T = a b b b c c c a a a a b a c b ... ...

  28. a b a c c a c b a c a b a c c 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 P = Pa T = a b b b c c c a a a a b a c b ... ...

  29. a b a c c a c b a c a b a c c 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 P = Pa T = a b b b c c c a a a a b a c b ... ... not-a mask a b b b c c c a a a a b a c b ... ...

  30. a b a c c a c b a c a b a c c 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 P = Pa T = a b b b c c c a a a a b a c b ... ... not-a mask a b b b c c c a a a a b a c b ... ... Tnot a 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1

  31. a b a c c a c b a c a b a c c 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 P = Pa T = a b b b c c c a a a a b a c b ... ... Tnot a 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1

  32. 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 P = a b a c c a c b a c a b a c c Pa T = a b b b c c c a a a a b a c b ... ... Tnot a 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 Multiply Pa and T(not a) to count mismatches using “a” Pa 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 Tnot a ... 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 ...

  33. 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 P = a b a c c a c b a c a b a c c Pa T = a b b b c c c a a a a b a c b ... ... Tnot a 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 Multiply Pa and Tnot a to count mismatches using a Pa 1 0 1 0 0 1 0 0 1 0 1 0 1 0 0 Tnot a ... 0 1 1 1 1 1 1 0 0 0 0 1 0 1 1 ...

  34. Boolean Convolutions (FFT) Method

  35. Boolean Convolutions (FFT) Method Running Time: One boolean convolution - O(n log m) time # of matches of all symbols - O(n| | log m) time

  36. How do we do counting in less than O(nm) ? Lets count matches rather than mismatches

  37. ... ... For each character you have a list of offsets where it occurs in the pattern, When you see the char in the text, you increment the appropriate counters. P = a b c d e b b d b g d e f h d c c a b g h h ... T = ... counter increment

  38. ... ... For each character you have a list of offsets where it occurs in the pattern, When you see the char in the text, you increment the appropriate counters. P = a b c d e b b d b g d e f h d c c a b g h h ... T = ... counter increment

  39. ... ... This is fast if all characters are “rare” P = a b c d e b b d b g d e f h d c c a b g h h ... T = ... counter increment

  40. Partition the characters into rare and frequent Rare: occurs ≤ c times in the pattern For rare characters run this scheme with the counters Takes O(nc) time

  41. Frequest chars You have at most m/c of them Do a convolution for each Total cost O(m/c n log(n)).

  42. Fix c Complexity:

  43. Back to the k-mismatch problem Want to beat the O(nk) kangaroo bound Frequent Symbol: a symbol that appears at least times in P.

  44. Few (≤√k) frequent symbols Do the counters scheme for non-frequent Convolve for each frequent O(n ) O(n log n)

  45. (≥√k) frequent symbols Intuition: There cannot be too many places where we match

  46. (≥√k) frequent symbols - Consider frequent symbols. - For each of them consider the first appearances. Do the counters scheme just for these symbols and occurrences

  47. a b a c c a c b a c a b a c c a b a c c a c b a c a b a c c a b b b c c c a a a a b a c b ... ... k = 4, = 4 P = a-mask a b a c c a c b a c a b a c c c-mask T = use a-mask

  48. a b a c c a c b a c a b a c c a b a c c a c b a c a b a c c Example of Masked Counting k = 4, = 4 P = a-mask a b a c c a c b a c a b a c c c-mask a b a c c a c b a c a b a c c T = d a b b b c c c a a a a b a c b ... ... 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 counter ... ...

  49. a b a c c a c b a c a b a c c a b a c c a c b a c a b a c c Example of Masked Counting k = 4, = 4 P = a-mask a b a c c a c b a c a b a c c c-mask a b a c c a c b a c a b a c c T = d a b b b c c c a a a a b a c b ... ... 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 counter ... ...

  50. Counting Stage: Run through text and count occurrences of all marks. Time: O(n ). Important Observations: 1) Sum of all counters 2 n 2) Every counter whose value is less than k already has more than k errors. Why? The total # of elements in all masks is 2 = 2k. For location i of T, if counteri < k then no match at location i.

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