1. String Matching with k Mismatches
Moshe Lewenstein Bar Ilan University
Modified by Ariel Rosenfeld

2. String Matching with k Mismatches
Landau – Vishkin
Galil – Giancarlo
Abrahamson
Amir - Lewenstein - Porat
Clifford – Fontaine – Porat – Sach – Starikovskaya

3. Exact String Matching Input: T = t1 . . . tn P = p1 … pm
Output: All locations i of T where P appears
Example:
P = A B C A A B
T = A B A B C A A B C A A B C A A B A A…

4. Exact String Matching Input: T = t1 . . . tn P = p1 … pm
Output: All locations i of T where P appears
Example:
P = A B C A A B
T = A B A B C A A B C A A B C A A B A A… 3

5. Exact String Matching Input: T = t1 . . . tn P = p1 … pm
Output: All locations i of T where P appears
Example:
P = A B C A A B
T = A B A B C A A B C A A B C A A B A A…

6. Exact String Matching Input: T = t1 . . . tn P = p1 … pm
Output: All locations i of T where P appears
Example:
P = A B C A A B
T = A B A B C A A B C A A B C A A B A A…

7. Exact String Matching Input: T = t1 . . . tn P = p1 … pm
Output: All locations i of T where P appears
Example:
P = A B C A A B
T = A B A B C A A B C A A B C A A B A A…
Answer: {3,7,11,..}

8. Exact String Matching Problem: Matching not exact in applications of:
Computational Biology
Musicology
Text Editing
Meteorology
etc.
Need other definitions of string matching!

9. Approximate String Matching
Idea: 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 sj rj
Example: S = ABCABC
R = ABBAAC
Ham(S,R) = 2

10. String Matching with Mismatches
Input: T = t 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…

11. String Matching with Mismatches
Input: T = t 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

12. String Matching with Mismatches
Input: T = t 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

13. String Matching with Mismatches
Input: T = t 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

14. String Matching with Mismatches
Input: T = t 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

15. String Matching with Mismatches
Input: T = t 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, …

16. String Matching with k Mismatches
Input: T = t tn, P = p1 … pm
Output: Every i in T s.t. 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, …

17. String Matching with k Mismatches
Input: T = t tn, P = p1 … pm
Output: Every i in T s.t. 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, …

18. String Matching with k Mismatches
Input: T = t tn, P = p1 … pm
Output: Every i in T s.t. 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, …
Y,N,N,Y,…

19. 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|

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

21. Trie A tree representing a set of strings. c { a aeef b ad bbfe bbfg e

22. Trie (Cont) Assume no string is a prefix of another c a b e b d e f f
Each string corresponds to a leaf. a b e b d e f f e g

23. Compressed Trie Compress unary nodes, label edges by strings c c a a b c a a b e b d
bbf d
eef e f f e g e g

24. Suffix tree Suffix tree of string s:
a compressed trie of all suffixes of s
Prefix-free: add a special character, say $, at the end of s

25. Suffix tree (Example)
Let s = abab, a suffix tree of s is a compressed trie of all suffixes of s=abab$ $ { $ b$
ab$
bab$
abab$ } a b b $ a a $ b b $ $

26. Suffix Tree properties1 2 a b $ 3 4 5 b
Succint in space - O(n).
- Can be built in O(n) time. McCreight, Weiner,
Ukkonen, Farach-Colton

27. Exact string matching s=abab$ $ a b b $ a a $ b b $ $5 $ a a $ b 4 b $ $ 3 2 1
Given a pattern P = ab we traverse the tree according to the pattern.

28. Exact string matching s=abab$ 1 3 $ a b b $ a a $ b b $ $
1 3 a b b 5 $ a a $ b 4 b $ $ 3 2 1
Leaves correspond to locations of appearance!

29. Exact string matching s=abab$ 1 3 $ a b b $ a a $ b b $ $
1 3 a b b 5 $ a a $ b 4 b $ $ 3 2 1
Prepare Tree: O(n) time
Find matches: O(m + occ) time occ = # of matches

30. Lowest common ancestors
A lot more can be gained from the suffix tree if we preprocess it so that we can answer LCA queries on it

31. Why?
The LCA of two leaves represents the longest common prefix (LCP) of these 2 suffixes $
s = abbaab$ a b 7 $ a a b b b $ a a 6 b a b $ 4 b a $ $ a 3 b 5 $ 2 1

32. Why?
The LCA of two leaves represents the longest common prefix (LCP) of these 2 suffixes $
s = abbaab$
aab$ a b 7 $ a a b b b $ a a 6 b a b $ 4 b a $ $ a 3 b 5 $ 2 1

33. Why?
The LCA of two leaves represents the longest common prefix (LCP) of these 2 suffixes $
s = abbaab$
aab$
abbaab$ a b 7 $ a a b b b $ a a 6 b a b $ 4 b a $ $ a 3 b 5 $ 2 1

34. Why?
The LCA of two leaves represents the longest common prefix (LCP) of these 2 suffixes $
s = abbaab$
aab$
abbaab$ a b 7 $ a a b b b $ a a 6 b a b $ 4 b a $ $ a 3 b 5 $ 2 1

35. LCA/LCP properties Preprocesssing time : O(n) Query Time: O(1)3 b a $ 5 2 4 6 7
LCA/LCP properties a
Preprocesssing time : O(n)
Query Time: O(1)
Harel & Tarjan 1984, Schieber & Vishkin 1988,
Berkman & Vishkin 1993

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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)