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Title: CS 6293 Advanced Topics: Translational Bioinformatics


1
CS 6293 Advanced Topics Translational
Bioinformatics
  • Lecture 5
  • Exact String Matching Algorithms

2
Overview
  • Sequence alignment two sub-problems
  • How to score an alignment with errors
  • How to find an alignment with the best score
  • Today exact string matching
  • Does not allow any errors
  • Efficiency becomes the sole consideration
  • Time and space

3
Why exact string matching?
  • The most fundamental string comparison problem
  • Often the core of more complex string comparison
    algorithms
  • E.g., BLAST
  • Often repeatedly called by other methods
  • Usually the most time consuming part
  • Small improvement could improve overall
    efficiency considerably

4
Definitions
  • Text a longer string T (length m)
  • Pattern a shorter string P (length n)
  • Exact matching find all occurrences of P in T

length m
T
length n
P
5
The naïve algorithm
6
Time complexity
  • Worst case O(mn)
  • How to speedup?
  • Pre-processing T or P
  • Why pre-processing can save us time?
  • Uncovers the structure of T or P
  • Determines when we can skip ahead without missing
    anything
  • Determines when we can infer the result of
    character comparisons without doing them.

7
Cost for exact string matching
  • Total cost cost (preprocessing)
  • cost(comparison)
  • cost(output)

Overhead
Minimize
Constant
Hope gain gt overhead
8
String matching scenarios
  • One T and one P
  • Search a word in a document
  • One T and many P all at once
  • Search a set of words in a document
  • Spell checking (fixed P)
  • One fixed T, many P
  • Search a completed genome for short sequences
  • Two (or many) Ts for common patterns
  • Q Which one to pre-process?
  • A Always pre-process the shorter seq, or the one
    that is repeatedly used

9
Pre-processing algs
  • Pattern preprocessing
  • Knuth-Morris-Pratt algorithm (KMP)
  • Aho-Corasick algorithm
  • Multiple patterns
  • Boyer Moore algorithm (discuss if have time)
  • The choice of most cases
  • Typically sub-linear time
  • Text preprocessing
  • Suffix tree
  • Very useful for many purposes
  • Suffix array
  • Burrows-Wheeler Transformation

10
Algorithm KMP Intuitive example 1
abcxabc
T
mismatch
P
abcxabcde
Naïve approach
abcxabc
T
?
abcxabcde
  • Observation by reasoning on the pattern alone,
    we can determine that if a mismatch happened when
    comparing P8 with Ti, we can shift P by four
    chars, and compare P4 with Ti, without
    missing any possible matches.
  • Number of comparisons saved 6

11
Intuitive example 2
abcxabc
T
mismatch
P
abcxabcde
Naïve approach
abcxabc
T
?
abcxabcde
  • Observation by reasoning on the pattern alone,
    we can determine that if a mismatch happened
    between P7 and Tj, we can shift P by six
    chars and compare Tj with P1 without missing
    any possible matches
  • Number of comparisons saved 7

12
KMP algorithm pre-processing
  • Key the reasoning is done without even knowing
    what string T is.
  • Only the location of mismatch in P must be known.

x
t
T
y
z
t
t
P
i
j
y
z
t
t
P
i
j
Pre-processing for any position i in P, find
P1..is longest proper suffix, t Pj..i,
such that t matches to a prefix of P, t, and the
next char of t is different from the next char of
t (i.e., y ? z) For each i, let sp(i) length(t)
13
KMP algorithm shift rule
x
t
T
y
z
t
t
P
i
j
y
z
t
P
t
i
j
sp(i)
1
Shift rule when a mismatch occurred between
Pi1 and Tk, shift P to the right by i
sp(i) chars and compare x with z. This shift
rule can be implicitly represented by creating a
failure link between y and z. Meaning when a
mismatch occurred between x on T and Pi1,
resume comparison between x and Psp(i)1.
14
Failure Link Example
  • P aataac

If a char in T fails to match at pos 6,
re-compare it with the char at pos 3 ( 2 1)
a
a
t
a
a
c
sp(i) 0 1 0 0 2 0
aaat
aataac
15
Another example
  • P abababc

If a char in T fails to match at pos 7,
re-compare it with the char at pos 5 ( 4 1)
a
b
a
b
a
b
c
Sp(i) 0 0 0 0 0 4 0
ababaababc
abab
abababab
16
KMP Example using Failure Link
a
a
t
a
a
c
T aacaataaaaataaccttacta
aataac
  • Time complexity analysis
  • Each char in T may be compared up to n times. A
    lousy analysis gives O(mn) time.
  • More careful analysis number of comparisons can
    be broken to two phases
  • Comparison phase the first time a char in T is
    compared to P. Total is exactly m.
  • Shift phase. First comparisons made after a
    shift. Total is at most m.
  • Time complexity O(2m)

aataac .
aataac
aataac ..
aataac .
17
KMP algorithm using DFA (Deterministic Finite
Automata)
  • P aataac

If a char in T fails to match at pos 6,
re-compare it with the char at pos 3
Failure link
a
a
t
a
a
c
If the next char in T is t after matching 5
chars, go to state 3
a
t
t
a
a
c
a
a
1
2
3
4
5
0
6
DFA
a
a
All other inputs goes to state 0.
18
DFA Example
a
t
t
a
a
c
a
a
1
2
3
4
5
0
6
DFA
a
a
T aacaataataataaccttacta
1201234534534560001001
Each char in T will be examined exactly once.
Therefore, exactly m comparisons are made. But
it takes longer to do pre-processing, and needs
more space to store the FSA.
19
Difference between Failure Link and DFA
  • Failure link
  • Preprocessing time and space are O(n), regardless
    of alphabet size
  • Comparison time is at most 2m (at least m)
  • DFA
  • Preprocessing time and space are O(n ?)
  • May be a problem for very large alphabet size
  • For example, each char is a big integer
  • Chinese characters
  • Comparison time is always m.

20
The set matching problem
  • Find all occurrences of a set of patterns in T
  • First idea run KMP or BM for each P
  • O(km n)
  • k number of patterns
  • m length of text
  • n total length of patterns
  • Better idea combine all patterns together and
    search in one run

21
A simpler problem spell-checking
  • A dictionary contains five words
  • potato
  • poetry
  • pottery
  • science
  • school
  • Given a document, check if any word is (not) in
    the dictionary
  • Words in document are separated by special chars.
  • Relatively easy.

22
Keyword tree for spell checking
This version of the potato gun was inspired by
the Weird Science team out of Illinois
p
s
o
c
l
h
o
o
5
e
i
t
e
t
a
t
r
n
t
y
e
c
o
r
e
y
3
1
4
2
  • O(n) time to construct. n total length of
    patterns.
  • Search time O(m). m length of text
  • Common prefix only need to be compared once.
  • What if there is no space between words?

23
Aho-Corasick algorithm
  • Basis of the fgrep algorithm
  • Generalizing KMP
  • Using failure links
  • Example given the following 4 patterns
  • potato
  • tattoo
  • theater
  • other

24
Keyword tree
0
p
t
t
h
o
e
h
a
t
r
e
t
a
a
4
t
t
t
e
o
o
r
1
o
3
2
25
Keyword tree
0
p
t
t
h
o
e
h
a
t
r
e
t
a
a
4
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
26
Keyword tree
0
p
t
t
h
o
e
h
a
t
r
e
t
a
a
4
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
O(mn)
m length of text. n length of longest pattern
27
Keyword Tree with a failure link
0
p
t
t
h
o
e
h
a
t
r
e
t
a
a
4
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
28
Keyword Tree with a failure link
0
p
t
t
h
o
e
h
a
t
r
e
t
a
a
4
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
29
Keyword Tree with all failure links
0
p
t
t
h
o
e
h
a
t
r
e
t
4
a
a
t
t
t
e
o
o
r
1
o
3
2
30
Example
0
p
t
t
h
o
e
h
a
t
r
e
t
4
a
a
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
31
Example
0
p
t
t
h
o
e
h
a
t
r
e
t
4
a
a
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
32
Example
0
p
t
t
h
o
e
h
a
t
r
e
t
4
a
a
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
33
Example
0
p
t
t
h
o
e
h
a
t
r
e
t
4
a
a
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
34
Example
0
p
t
t
h
o
e
h
a
t
r
e
t
4
a
a
t
t
t
e
o
o
r
1
o
3
2
potherotathxythopotattooattoo
35
Aho-Corasick algorithm
  • O(n) preprocessing, and O(mk) searching.
  • n total length of patterns.
  • m length of text
  • k is of occurrence.

36
Suffix Tree
  • All algorithms we talked about so far preprocess
    pattern(s)
  • Boyer-Moore fastest in practice. O(m) worst
    case.
  • KMP O(m)
  • Aho-Corasick O(m)
  • In some cases we may prefer to pre-process T
  • Fixed T, varying P
  • Suffix tree basically a keyword tree of all
    suffixes

37
Suffix tree
  • T xabxac
  • Suffixes
  • xabxac
  • abxac
  • bxac
  • xac
  • ac
  • c

x
a
b
x
a
a
c
c
1
c
b
b
x
x
c
4
6
a
a
c
c
5
2
3
Naïve construction O(m2) using
Aho-Corasick. Smarter O(m). Very technical. big
constant factor Difference from a keyword tree
create an internal node only when there is a
branch
38
Suffix tree implementation
  • Explicitly labeling sequence end
  • T xabxa

x
a
x
a
b
x
b
a
a
x
a
a

1
1

b
b
b
b
x

x
x
x
4
a
a
a
a

5

2
2
3
3
  • One-to-one correspondence of leaves and suffixes
  • T leaves, hence lt T internal nodes

39
Suffix tree implementation
  • Implicitly labeling edges
  • T xabxa

12
x
a
3
b
x
22
a
a

1
1


b
b


x
x
3
3
4
4
a
a
5

5

2
2
3
3
  • Tree(T) O(T size(edge labels))

40
Suffix links
  • Similar to failure link in a keyword tree
  • Only link internal nodes having branches

x
a
b
P xabcf
a
b
c
f
c
d
d
e
e
f
f
g
g
h
h
i
i
j
j
41
ST Application 1 pattern matching
  • Find all occurrence of Pxa in T
  • Find node v in the ST that matches to P
  • Traverse the subtree rooted at v to get the
    locations

x
a
b
x
a
a
c
c
1
c
b
b
x
x
c
4
6
a
a
c
c
5
T xabxac
2
3
  • O(m) to construct ST (large constant factor)
  • O(n) to find v linear to length of P instead of
    T!
  • O(k) to get all leaves, k is the number of
    occurrence.
  • Asymptotic time is the same as KMP. ST wins if T
    is fixed. KMP wins otherwise.

42
ST application 2 repeats finding
  • Genome contains many repeated DNA sequences
  • Repeat sequence length Varies from 1 nucleotide
    to millions
  • Genes may have multiple copies (50 to 10,000)
  • Highly repetitive DNA in some non-coding regions
  • 6 to 10bp x 100,000 to 1,000,000 times
  • Problem find all repeats that are at least
    k-residues long and appear at least p times in
    the genome

43
Repeats finding
  • at least k-residues long and appear at least p
    times in the seq
  • Phase 1 top-down, count label lengths (L) from
    root to each node
  • Phase 2 bottom-up count of leaves descended
    from each internal node

For each node with L gt k, and N gt p, print all
leaves
O(m) to traverse tree
(L, N)
44
Maximal repeats finding
  • Right-maximal repeat
  • Si1..ik Sj1..jk,
  • but Sik1 ! Sjk1
  • Left-maximal repeat
  • Si1..ik Sj1..jk
  • But Si ! Sj
  • Maximal repeat
  • Si1..ik Sj1..jk
  • But Si ! Sj, and Sik1 ! Sjk1

acatgacatt
  • cat
  • aca
  • acat

45
Maximal repeats finding
5e
1234567890acatgacatt
5
t
a

c
10
a
5e
t
c
t
t
a
9
t
4
t
5e
5e
t
5e
t
7
3
6
8
1
2
  • Find repeats with at least 3 bases and 2
    occurrence
  • right-maximal cat
  • Maximal acat
  • left-maximal aca

46
Maximal repeats finding
5e
1234567890acatgacatt
5
t
a

c
10
a
5e
t
c
t
t
a
9
t
4
t
5e
5e
t
5e
t
7
3
6
8
1
2
Left char
g
c
c
a
a
  • How to find maximal repeat?
  • A right-maximal repeats with different left chars

47
Joint Suffix Tree (JST)
  • Build a ST for more than two strings
  • Two strings S1 and S2
  • S S1 S2
  • Build a suffix tree for S in time O(S1 S2)
  • The separator will only appear in the edge ending
    in a leaf (why?)

48
Joint suffix tree example
  • S1 abcd
  • S2 abca
  • S abcdabca

a b c d
(2, 0) useless
a
d

c
b c d a b c a
a
b
c
b
c
d

d
d

a

a
a
a
2,4
b
1,4
a
c
2,3
a
b
2,1
c
2,2
d
1,1
Seq ID
1,3
Suffix ID
1,2
49
To Simplify
a b c d
useless
a
d

c
b c d a b c a
a
a
b
d
c
b
c
c
b c d

b
d
d
d
c


a

d
a
d
a
1,4
a
2,4
b
a
1,4
a
a
c
a
2,4
2,3
a
b
1,1
2,3
2,1
c
2,1
2,2
1,3
d
1,1
2,2
1,2
1,3
1,2
  • We dont really need to do anything, since all
    edge labels were implicit.
  • The right hand side is more convenient to look at

50
Application 1 of JST
  • Longest common substring between two sequences
  • Using smith-waterman
  • Gap mismatch -infinity.
  • Quadratic time
  • Using JST
  • Linear time
  • For each internal node v, keep a bit vector B
  • B1 1 if a child of v is a suffix of S1
  • Bottom-up find all internal nodes with B1
    B2 1 (green nodes)
  • Report a green node with the longest label
  • Can be extended to k sequences. Just use a bit
    vector of size k.

a
d
c
b c d
b
c

d
d
1,4
a
a
a
2,4
1,1
2,3
2,1
1,3
2,2
1,2
51
Application 2 of JST
  • Given K strings, find all k-mers that appear in
    at least (or at most) d strings
  • Exact motif finding problem

Llt k
cardinal(B) gt 3
B BitOR(1010, 0011) 1011
L gt k
cardinal(B) 3
B 0011
B 1010
4,x
3,x
1,x
3,x
52
Application 3 of JST
  • Substring problem for sequence databases
  • Given A fixed database of sequences (e.g.,
    individual genomes)
  • Given A short pattern (e.g., DNA signature)
  • Q Does this DNA signature belong to any
    individual in the database?
  • i.e. the pattern is a substring of some sequences
    in the database
  • Aho-Corasick doesnt work
  • This can also be used to design signatures for
    individuals
  • Build a JST for the database seqs
  • Match P to the JST
  • Find seq IDs from descendents

a
d
c
b c d
b
c

d
1,4
a
d
a
a
2,4
1,1
2,3
2,1
Seqs abcd, abca P1 cd P2 bc
1,3
2,2
1,2
53
Suffix Tree Memory Footprint
  • The space requirements of suffix trees can become
    prohibitive
  • Tree(T) is about 20T in practice
  • Suffix arrays provide one solution.

54
Suffix Arrays
  • Very space efficient (m integers)
  • Pattern lookup is nearly O(n) in practice
  • O(n log2 m) worst case with 2m additional
    integers
  • Independent of alphabet size!
  • Easiest to describe (and construct) using suffix
    trees
  • Other (slower) methods exist

x
a
b
x
a
a

1

b

5 2 3 4 1
1. xabxa 2. abxa 3. bxa 4. xa 5. a
b
x
x
4
a
a
5
a
abxa
bxa
xa
xabxa


2
3
55
Suffix array construction
  • Build suffix tree for T
  • Perform lexical depth-first search of suffix
    tree
  • output the suffix label of each leaf encountered
  • Therefore suffix array can be constructed in O(m)
    time.

56
Suffix array pattern search
  • If P is in T, then all the locations of P are
    consecutive suffixes in Pos.
  • Do binary search in Pos to find P!
  • Compare P with suffix Pos(m/2)
  • If lexicographically less, P is in first half of
    T
  • If lexicographically more, P is in second half of
    T
  • Iterate!

57
Suffix array pattern search
  • T xabxa
  • P abx

R
M
L
x
a
b
x
a
a

1

5 2 3 4 1
b

b
x
x
4
a
abxa
bxa
xa
xabxa
a
a
5


2
3
58
Suffix array binary search
  • How long to compare P with suffix of T?
  • O(n) worst case!
  • Binary search on Pos takes O(n log m) time
  • Worst case will be rare
  • occur if many long prefixes of P appear in T
  • In random or large alphabet strings
  • expect to do less than log m comparisons
  • O(n log m) running time when combined with LCP
    table
  • suffix tree suffix array LCP table

59
Summary
  • One T, one P
  • Boyer-Moore is the choice
  • KMP works but not the best
  • One T, many P
  • Aho-Corasick
  • Suffix Tree (array)
  • One fixed T, many varying P
  • Suffix tree (array)
  • Two or more Ts
  • Suffix tree, joint suffix tree

Alphabet independent
Alphabet dependent
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