1. MSc Bioinformatics for H15: Algorithms on strings and sequences
Master Course
14/04/2017
MSc Bioinformatics for
Health Sciences
H15: Algorithms on strings and sequences
Xavier Messeguer Peypoch (
Dep. de Llenguatges i Sistemes Informàtics
CEPBA-IBM Research Institute
Universitat Politècnica de Catalunya

2. Contents 1. (Exact) String matching of one pattern
14/04/2017
1. (Exact) String matching of one pattern
2. (Exact) String matching of many patterns
3. Approximate string matching (Dynamic programming)
4. Pairwise and multiple alignment
5. Suffix trees

3. Contents and bibliography
14/04/2017
1. (Exact) String matching of one pattern
2. (Exact) String matching of many patterns
3. Approximate string matching (Dynamic programming)
4. Pairwise and multiple alignment
5. Suffix trees
Flexible pattern matching in strings
G. Navarro and M. Raffinot, 2002, Cambridge Uni. Press
Algorithms on strings, trees and sequences
D. Gusfield, Cambridge University Press, 1997

4. String matching
14/04/2017
Definition: given a long text T and a set of k patterns
p1,p2,…,pk, the string matching problem is to find
all the ocurrences of all the patterns in the text T.
On-line algorithms: the patterns are known.
Off-line algorithms: the text is known.
Only one pattern (exact and approximated)
Five, ten, hundred, thusand,.. patterns (exact)
Extended patterns
Suffix trees

5. (Exact) string matching of one pattern
Master Course
14/04/2017
First lecture:
First part:
(Exact) string matching of one pattern

6. String matching: one pattern
14/04/2017
How does the string algorithms made the search?
For instance, given the sequence
CTACTACTACGTCTATACTGATCGTAGCTACTACATGC
search for the pattern ACTGA.
and for the pattern TACTACGGTATGACTAA
As you have seen this morning ....

7. String Matching: Brute force algorithm
14/04/2017
Example:
Given the pattern ATGTA, the search is
G T A C T A G A G G A C G T A T G T A C T G ...
A T G T A
A T G T A
A T G T A
A T G T A
A T G T A
A T G T A
As you have seen this morning ....

8. String Matching: Brute force algorithm
14/04/2017
Connect to
and open Brute Force algorithm
What is the meaning of the variables?
y: array with the text T n: length of the text
x: array with the pattern P m:length of the pattern
What is the meaning of the variables?
y: n:
x: m:
C code of the running file
Connect to

9. String Matching of one pattern
14/04/2017
The cost of Brute Force algorithm is O(nm).
Can the search be made with lower cost?
CTACTACTACGTCTATACTGATCGTAGCTACTACATGC
Prefix search
Suffix search
TACTACGGTATGACTAA
Factor search

10. String matching of one pattern
14/04/2017
How does the string algorithms made the search?
There is a sliding window along the text
against which the pattern is compared:
Pattern :
Text :
At each step the comparison is made and
the window is shifted to the right.
As you have seen this morning ....
Which are the facts that differentiate the algorithms?
How the comparison is made.
The length of the shift.

11. String Matching: Brute force algorithm
14/04/2017
Which is the next position of the window?
How the comparison is made?
Text :
Patern :
From left to right: prefix search
Patró :
Text :
The window is shifted only one cell
As you have seen this morning ....
The cost is O(mn).

12. String Matching: one pattern
14/04/2017
Most efficient algorithms (Navarro & Raffinot)
BNDM : Backward Nondeterministic Dawg Matching
| |
BOM : Backward Oracle Matching 64 32 16
Horspool 8
BOM
BNDM 4 2 w
Length of the pattern

13. String Matching: Horspool algorithm
14/04/2017
Which is the next position of the window?
How the comparison is made?
Text :
Pattern :
From right to left: suffix search
Pattern :
Text : a
It depends of where appears the last letter of the text, say it ‘a’, in the pattern:
As you have seen this morning .... a a a
Then it is necessary a preprocess that determines the length of the shift.

14. String Matching: Horspool algorithm
14/04/2017
Example:
A 4
C 5
G 2
T 1
Given the pattern ATGTA, the shift table is
And the search:
G T A C T A G A G G A C G T A T G T A C T G ...
A T G T A
A T G T A
A T G T A
A T G T A
A T G T A
A T G T A
As you have seen this morning ....

15. String Matching: Horspool algorithm
14/04/2017
Example:
A 4
C 5
G 2
T 1
Given the pattern ATGTA, the shift table is
And the search:
G T A C T A G A G G A C G T A T G T A C T G ...
A T G T A
A T G T A
A T G T A
A T G T A
A T G T A
A T G T A
As you have seen this morning ....
A T G T A …

16. String Matching: Horspool algorithm
14/04/2017
Connect to
and open the Horspool algorithm
C code
Connect to

17. String Matching: one pattern
14/04/2017
The most efficient algorithms (Navarro & Raffinot)
BNDM : Backward Nondeterministic Dawg Matching
| |
BOM : Backward Oracle Matching 64 32 16
Horspool 8
BOM
BNDM 4 2 w
Length of the pattern

18. BNDM algorithm ? How the comparison is made?
14/04/2017
How the shift is determined?
How the comparison is made?
Text :
Pattern :
Searches for suffixes of T that are factors of P x
This state is expressed with an array D of bits:
D2 =
How the next state can be obtained?
D = D<<1 & B(x)
Given the mask B(x) of x, the cells where character x appears into the pattern
D3 = ( ) & ( ) = ( )
If B(x) = ( ) then
As you have seen this morning .... ?

19. BNDM algorithm: example
14/04/2017
Given the pattern ATGTA,
the mask of characters is:
B(A) = ( )
B(C) =
B(G) =
B(T) =
As you have seen this morning ....

20. BNDM algorithm: example
14/04/2017
Given the pattern ATGTA,
the mask of characters is:
B(A) = ( )
B(C) = ( )
B(G) = ( )
B(T) = ( )
As you have seen this morning ....

21. BNDM algorithm: example
14/04/2017
Given the pattern ATGTA,
the mask of characters is:
B(A) = ( )
B(C) = ( )
B(G) = ( )
B(T) = ( )
Given the text :
G T A C T A G A G G A C G T A T G T A C T G ...
A T G T A
A T G T A
A T G T A
A T G T A
D1 = = ( )
D2 = ( ) & ( ) = ( )
D1 = = ( )
D2 = ( ) & ( ) = ( )
As you have seen this morning ....
D1 = = ( )
D2 = ( ) & ( ) = ( )
D3 = ( ) & ( ) = ( )
D4 = ( ) & ( ) = ( )

22. BNDM algorithm: example
14/04/2017
The pattern is ATGTA ,
the masks are:
and the text:
G T A C T A G A G G A C G T A T G T A C T G ...
A T G T A
B(A) = ( )
B(C) = ( )
B(G) = ( )
B(T) = ( )
D1 = = ( )
A T G T A
D2 = ( ) & ( ) = ( )
D3 = ( ) & ( ) = ( )
D4 = ( ) & ( ) = ( )
D5 = ( ) & ( ) = ( )
As you have seen this morning ....
D6 = ( ) & ( * * * * * ) = ( )
Pattern found! …

23. BNDM algorithm How the comparison is made?
14/04/2017
How the shift is determined?
How the comparison is made?
Text :
Pattern :
Searches for suffixes of T that are factors of P
This state is expressed with an array D of bits:
D =
As you have seen this morning .... ?

24. BNDM algorithm How the comparison is made?
14/04/2017
How the shift is determined?
How the comparison is made?
Text :
Pattern :
Searches for suffixes of T that are factors of P
This state is expressed with an array D of bits:
D =
As you have seen this morning ....
If the left bit is set to one in step i, it means that a prefix of P of length i is equal
to a suffix of T, then the window is shifted m-i cells; otherwise it is shifted m cells

25. String matching: one pattern
14/04/2017
The most efficient algorithms (Navarro & Raffinot)
BNDM : Backward Nondeterministic Dawg Matching
| |
BOM : Backward Oracle Matching 64 32 16
Horspool 8
BOM
BNDM 4 2
Long. patró w

26. BOM (Backward Oracle Matching)
14/04/2017
How the shifted is determined?
How the comparison is made?
Text :
Pattern :
Automaton: Factor Oracle(1999)
Checks if the suffix is a factor of the pattern ?
As you have seen this morning ....

27. Automaton Factor Oracle: properties
14/04/2017
Factor Oracle of the word G T A T G T A G A T
G T A T G
T A T G
A T G
T G G
As you have seen this morning ....
but the automaton also recognizes other strings as
G T G
then it is usefull only for discard words out as factors!

28. BOM: example How the comparison is made?
14/04/2017
How the comparison is made?
The Factor Oracle of the inverted pattern is built. Given the pattern ATGTATG G A T
Search:
G T A C T A G A A T G T G T A G A C A T G T A T G G T G A...
A T G T A T G
As you have seen this morning ....

29. BOM: example How the comparison is made?
14/04/2017
How the comparison is made?
The Factor Oracle of the inverted pattern is built. Given the pattern ATGTATG G A T
Search:
G T A C T A G A A T G T G T A G A C A T G T A T G G T G
A T G T A T G
A T G T A T G
As you have seen this morning ....

30. BOM: example How the comparison is made?
14/04/2017
How the comparison is made?
The Factor Oracle of the inverted pattern is built. Given the pattern ATGTATG G A T
Search
G T A C T A G A A T G T G T A G A C A T G T A T G G T G
A T G T A T G
A T G T A T G
A T G T A T G
As you have seen this morning ....

31. BOM: example How the comparison is made?
14/04/2017
How the comparison is made?
The Factor Oracle of the inverted pattern is built. Given the pattern ATGTATG G A T
Search :
G T A C T A G A A T G T G T A G A C A T G T A T G G T G
A T G T A T G
A T G T A T G
A T G T A T G
A T G T A T G
As you have seen this morning ....

32. BOM: example How the comparison is made?
14/04/2017
How the comparison is made?
The Factor Oracle of the inverted pattern is built. Given the pattern ATGTATG G A T
Search :
G T A C T A G A A T G T G T A G A C A T G T A T G G T G ...
A T G T A T G
A T G T A T G
A T G T A T G
A T G T A T G
A T G T A T G
As you have seen this morning ....

33. BOM: example How the comparison is made? …
14/04/2017
How the comparison is made?
Es construeix l’autòmata del patró invers: Suposem que el patró és ATGTATG G A T
Search :
G T A C T A G A A T G T G T A G A C A T G T A T G G T G ...
A T G T A T G
A T G T A T G
A T G T A T G
A T G T A T G
A T G T A T G
A T G T A T G
As you have seen this morning .... …

34. BOM (Backward Oracle Matching)
14/04/2017
How the shifted is determined?
How the comparison is made?
Text :
Pattern :
Automaton: Factor Oracle
Checks if the suffix is a factor of the pattern a
a is the first mismatch
As you have seen this morning ....

35. String Matching: BNDM and BOM
14/04/2017
Connect to
and open the BNDM and BOM algorithms
C code of BNDM
C code of BOM

36. (Exact) string matching of many patterns
Master Course
14/04/2017
First lecture:
Second part:
(Exact) string matching of many patterns

37. String matching: many patterns
14/04/2017
Given the sequence
CTACTACTACGTCTATACTGATCGTAGCTACTACATGC
Search for the patterns
ACTGACT
GTCT
AATT
ACTGATCTTT
GTAGC
AATACT
ACATGC
ACTGA.
As you have seen this morning ....

38. Trie of words GTATGTA,GTAT,TAATA,GTGTA
14/04/2017
Trie of words GTATGTA,GTAT,TAATA,GTGTA T G T A A T G G T A T A A T A A
As you have seen this morning ....
Which is the cost?

39. Horspool for many patterns
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
1. Build the trie of the inverted patterns
2. lmin=4
A 1
C 4 (lmin)
G 2
T 1
3. Table of shifts
As you have seen this morning ....
4. Start the search

40. Horspool for many patterns
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
A 1
C 4 (lmin)
G 2
T 1
The text ACATGCTATGTGACA…
As you have seen this morning ....

41. Horspool for many patterns
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
A 1
C 4 (lmin)
G 2
T 1
The text ACATGCTATGTGACA…
As you have seen this morning ....

42. Horspool for many patterns
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
A 1
C 4 (lmin)
G 2
T 1
The text ACATGCTATGTGACA…
As you have seen this morning ....

43. Horspool for many patterns
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
A 1
C 4 (lmin)
G 2
T 1
The text ACATGCTATGTGACA…
As you have seen this morning ....

44. Horspool for many patterns
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
A 1
C 4 (lmin)
G 2
T 1
The text ACATGCTATGTGACA…
As you have seen this morning ....

45. Horspool for many patterns
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
A 1
C 4 (lmin)
G 2
T 1
The text ACATGCTATGTGACA…
As you have seen this morning .... …

46. Horspool for many patterns
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
Short Shifts!
A 1
C 4 (lmin)
G 2
T 1
The text ACATGCTATGTGACA…
As you have seen this morning .... …

47. How do we can increase the length of the shifts?
Horspool to Wu-Manber
14/04/2017
How do we can increase the length of the shifts?
With a table shift of l-mers with the patterns ATGTATG,TATG,ATAAT,ATGTG
A 1
C 4 (lmin)
G 2
T 1
1 símbol
AA 1
AC (LMIN-L+1)
AG 3
AT 1
CA 3
CC 3
CG 3 …
2 símbols
AA 1
AT 1
GT 1
TA 2
TG 2
As you have seen this morning ....

48. … Wu-Manber algorithm Search for ATGTATG,TATG,ATAAT,ATGTG
14/04/2017
Search for ATGTATG,TATG,ATAAT,ATGTG T A G
AA 1
AT 1
GT 1
TA 2
TG 2
into the text: ACATGCTATGTGACATAATA
As you have seen this morning .... …
Experimental length: log|Σ| 2*lmin*r

49. String matching of many patterns
14/04/2017 8 4 2
| |
Wu-Manber
SBOM
Lmin
(5 patterns) 8 4 2
Wu-Manber
SBOM
(10 patterns) 8 4 2
Wu-Manber
SBOM
(100 patterns)

50. String matching of many patterns
14/04/2017
| |
(5 patterns) 8
Wu-Manber 4
SBOM
Lmin 2 8
Wu-Manber
(10 patterns) 4
SBOM 2
(1000 patterns)
SBOM 8
(100 patterns) 4 2

51. Horspool for a set of patterns
14/04/2017
How the comparison is made?
Comparison
Text :
Patrons:
Automaton with all the patterns
How the shift is determined? a
Segons l’aparició de l`últim carácter del text ‘a’into the s patrons,
concretament:
la primera aparició per la dreta no última i més curta que lmin, o lmin
As you have seen this morning ....

52. String matching of many patterns
14/04/2017 8 4 2
| |
Wu-Manber
SBOM
Long. mínima
(5 patterns) 8 4 2
Wu-Manber
SBOM
(10 patterns)
Ad AC 8 4 2
Wu-Manber
SBOM
(100 patterns)
Ad AC 8 4 2
Wu-Manber
SBOM
(1000 patterns)
Ad AC

53. SBOM ? How the comparison is made? How the shifted is determined?
14/04/2017
How the shifted is determined?
How the comparison is made?
Text :
Pattern :
Automaton: Factor Oracle
Checks if the suffix is a factor of any pattern ?
As you have seen this morning ....

54. Factor Oracle of many patterns
14/04/2017 G T A T G T A T G A
1,4 A A T A 3 2
As you have seen this morning ....
The AFO of GTATGTA, GTAA, TAATA i GTGTA

55. SBOM algorithm How the comparison is made?
14/04/2017
How the comparison is made?
Text :
Patrons:
Autòmaton………… of lenght lmin
How the shift is determined? a
If the a doesn’t appears in the AFO
As you have seen this morning ....
If lmin characters have been read

56. SBOM algorithm : example
14/04/2017
Search for the patterns ATGTATG, TAATG,TAATAAT i AATGTG G T A T G T A 1 4 G A T A A T A 2 3
ACATGCTAGCTATAATAATGTATG
As you have seen this morning ....

57. SBOM algorithm: example
14/04/2017
Search for the patterns ATGTATG, TAATG,TAATAAT i AATGTG G T A T G T A 1 4 G A T A A T A 2 3
ACATGCTAGCTATAATAATGTATG
As you have seen this morning ....

58. SBOM algorithm: example
14/04/2017
Search for the patterns ATGTATG, TAATG,TAATAAT i AATGTG G T A T G T A 1 4 G A T A A T A 2 3
ACATGCTAGCTATAATAATGTATG
As you have seen this morning ....

59. SBOM algorithm: example
14/04/2017
Search for the patterns ATGTATG, TAATG,TAATAAT i AATGTG G T A T G T A 1 4 G A T A A T A 2 3
ACATGCTAGCTATAATAATGTATG
As you have seen this morning ....

60. SBOM algorithm: example
14/04/2017
Search for the patterns ATGTATG, TAATG,TAATAAT i AATGTG G T A T G T A 1 4 G A T A A T A 2 3
ACATGCTAGCTATAATAATGTATG
As you have seen this morning ....

61. SBOM algorithm: example
14/04/2017
Search for the patterns ATGTATG, TAATG,TAATAAT i AATGTG G T A T G T A 1 4 G A T A A T A 2 3
ACATGCTAGCTATAATAATGTATG
As you have seen this morning ....

62. SBOM algorithm: example
14/04/2017
Search for the patterns ATGTATG, TAATG,TAATAAT i AATGTG G T A T G T A 1 4 G A T A A T A 2 3
ACATGCTAGCTATAATAATGTATG
As you have seen this morning ....

63. SBOM algorithm: example
14/04/2017
Search for the patterns ATGTATG, TAATG,TAATAAT i AATGTG G T A T G T A 1 4 G A T A A T A 2 3
ACATGCTAGCTATAATAATGT…
As you have seen this morning ....

64. Alg. Cerca exacta de molts patrons
14/04/2017 8 4 2
| |
Wu-Manber
SBOM
Long. mínima
(5 mots) 8 4 2
Wu-Manber
SBOM
(10 mots)
Ad AC 8 4 2
Wu-Manber
SBOM
(100 mots)
Ad AC 8 4 2
Wu-Manber
SBOM
(1000 mots)
Ad AC