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

String Matching with k Mismatches

Moshe Lewenstein Bar Ilan UniversityModified by Ariel Rosenfeld


Landau – Vishkin 1986

Galil – Giancarlo 1986

Abrahamson 1987

Amir - Lewenstein - Porat 2000

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…


P = p1 … pm


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



P = p1 … pm


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 7



P = p1 … pm


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 7 11



P = p1 … pm


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,..}



Problem: Matching not exact in applications of:

• Computational Biology

• Musicology

• Text Editing

• Meteorology

• etc.

Need other definitions of string matching!

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

String Matching with Mismatches


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…



P = p1 … pm


Example:

P = A B B A A C T = A B C A A B C A C… 2

Ham(P,T1) = 2



P = p1 … pm


Example:

P = A B B A A C T = A B C A A B C A C… 2, 4

Ham(P,T2) = 4



P = p1 … pm


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



P = p1 … pm


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



P = p1 … pm


Example:

P = A B B A A C T = A B C A A B C A C… 2, 4, 6, 2, …

Input: T = t1 . . . tn, P = p1 … pm


Output: Every i in T s.t. Ham(P, titi+1…ti+m-1) k

Example: k = 2


Input: T = t1 . . . tn, P = p1 … pm



Example: k = 2


Input: T = t1 . . . tn, P = p1 … pm



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,…

Naïve Algorithm(for counting mismatches or k-mismatches problem)

Running Time: O(nm) n = |T|, m = |P|

- Goto each location of text and compute hamming distance of P and Ti

The Kangaroo Method(for k-mismatches)

Landau – Vishkin 1986

Galil – Giancarlo 1986

Trie

• A tree representing a set of strings.

ab

c

e

e

f

d b

f

e g

{ aeef ad bbfe bbfg c }

Trie (Cont)

• Assume no string is a prefix of another

ab

c

e

e

f

d b

f

e g

Each string corresponds to a leaf.

Compressed Trie • Compress unary nodes, label edges by strings

ab

c

e

e

f

d b

f

e g

a

bbf

c

eefd

e g

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

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$ }

ab

ab$

ab$

b

$

$

$

Suffix Tree properties

- Succint in space - O(n).

- Can be built in O(n) time. McCreight, Weiner,

Ukkonen, Farach-Colton

b

12

ab

a

b$

a

b$

3

$ 4

$

5

$

Exact string matching

12

ab

ab$

ab$

b

3

$ 4

$

5

$

Given a pattern P = ab we traverse the tree according to the pattern.

s=abab$


12

ab

ab$

ab$

b

3

$ 4

$

5

$

Leaves correspond to locations of appearance!

s=abab$ 1 3


12

ab

ab$

ab$

b

3

$ 4

$

5

$

Prepare Tree: O(n) time

Find matches: O(m + occ) time occ = # of matches

s=abab$ 1 3

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

Why?The LCA of two leaves represents the longest common prefix (LCP) of these 2 suffixes

s = abbaab$

1

3

a

b

aab

ab$

b

5

$

2

b

4

b$a

6

$

7

$

b

$

aaab$


1

3

a

b

aab

ab$

b

5

$

2

b

4

b$a

6

$

7

$

b

$

aaab$

s = abbaab$ aab$


1

3

a

b

aab

ab$

b

5

$

2

b

4

b$a

6

$

7

$

b

$

aaab$

s = abbaab$ aab$ abbaab$


1

3

a

b

aab

ab$

b

5

$

2

b

4

b$a

6

$

7

$

b

$

aaab$

s = abbaab$

aab$ abbaab$

LCA/LCP propertiesa

1

3

b

aab

ab$

b

5

$

2

b

4

b$

a6

$

7

$

b

$

aaab$

Preprocesssing time : O(n)

Query Time: O(1)

Harel & Tarjan 1984, Schieber & Vishkin 1988, Berkman & Vishkin 1993


- 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 BT = A B B A C A B A B A B C A B B C A B C A … i




Example:





Example:





Example:





Example:





Example:





Example:





Example:





Example:




- Do up to k LCP queries for every text location

Example:



Preprocess:

Build suffix tree of both P and T - O(n+m) timeLCA preprocessing - O(n+m) time

Check P at given text location

Kangroo jump till next mismatch - O(k) time

Overall time: O(nk)

Documents

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