Suffix Trees and Suffix Arrays: Efficient String Searching Algorithms, Lecture notes of Bioinformatics

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Suffix Trees andSuffix

Trees and

Suffix Arrays

y

Some problems

p

• Given a pattern P = P[1..m], find all

p

[^

]

occurrences of P in a text S = S[1..n]

• Another problem:• Another problem:
  • Given two strings S

[1..n 1

] and S 1

[1..n 2

] find their 2

l^

t^

b t i

longest common substring.• find i, j, k such that S

[i .. i+k-1] = S 1

[j .. j+k-1] and k is^2

l^

ibl

as large as possible.

• Any solutions? How do you solve these

problems (efficiently)?

App

lications in Bioinformatics

pp

• Multiple genome alignment

p

g

g

• Michael Hohl et al. 2002– Longest common substring problem– Longest common substring problem– Common substrings of more than two strings
  • Selection of signature oligonucleotides for

microarrays

y

• Kaderali and Schliep, 2002Identification of sequence repeats
  • Identification of sequence repeats
    • Kurtz and Schleiermacher, 1999

Suffix trees• Any string of length m can be degenerated

i t

ffi

into m suffixes.– abcdefgh (length: 8)

g

(^

g

)

• 8 suffixes:
  • h gh fgh efgh defgh cdefgh bcefgh abcdefgh

h, gh, fgh, efgh, defgh, cdefgh, bcefgh, abcdefgh

• The suffixes can be stored in a suffix-tree and

thi

t^

b

t d i

O( ) ti

this tree can be generated in O(

n

)^

time

• A string pattern of length

m

can be searched

g p

g

in this suffix tree in O(

m

) time.

Whereas a regular sequential search would take

• Whereas, a regular sequential search would take

O(

n

) time.

Definition of a suffix tree• Let

S

S

[1..

n

]^

be a string of length

n

over a

[^

]^

g

g

fixed alphabet

. A suffix tree for

S

is a tree

with

n

leaves (representing

n

suffixes) and

with

n

leaves (representing

n

suffixes) and

the following properties:

Every internal node other than the root has at least 2

• Every internal node other than the root has at least 2

children

• Every edge is labeled with a nonempty substring of

S

• Every edge is labeled with a nonempty substring of

S

.

• The edges leaving a given node have labels starting with

different letters.different letters.

• The concatenation of the labels of the path from the root

to leaf

i^

spells out the

i-

th suffix

S

[ i ..

n ]

of

S

. We denote

p^

[^

]

S

[i..n] by

S

. i

A

n example suffix tree

p

• The suffix tree for string: 1 2 3 4 5 6

g

x a b x a c

Does a suffix treealways exist?always exist?

Problem• Note that if a suffix is a prefix of another suffix

p

we cannot have a tree with the propertiesdefined in the previous slidesdefined in the previous slides.– e.g.

xabxa

The fourth suffix

xa

or the fifth suffix

a

won’t be

represented by a leaf node.

Solution: the terminal character $• Note that if a suffix is a prefix of another suffix

p

we cannot have a tree with the propertiesdefined in the previous slidesdefined in the previous slides.– e.g.

xabxa

The fourth suffix

xa

or the fifth suffix

a

won’t be

represented by a leaf node.

• Solution: insert a special terminal character at

the end such as $ Therefore xa$ will not be athe end such as $. Therefore xa$ will not be aprefix of the suffix xabxa.

Suffix tree construction•^

Start with a root and a leaf numbered 1, connectedby an edge labeled

S

$

.

-^

Enter suffixes

S

[2..

n

]$;

S

[3...

n

]$; ... ;

S

[ n

]$ into the

[^

] ;

[^

] ;

;^

[ ]

tree as follows:

-^

To insert

K

=i

S

[ i

n

]$ follow the path from the root

To insert

K

i^

S

[ i

..

n

]$, follow the path from the root

matching characters of

K

until the first mismatch ati

character

K

[^ i

j^

] (which is bound to happen)

character

K

[^ i

j^

]^

(which is bound to happen)

(a) If the matching cannot continue from a node, denotethat node by

w

that node by

w

(b) Otherwise the mismatch occurs at the middle of anedge, which has to be splitedge, which has to be split

Suffix tree construction - 2• If the mismatch occurs at the middle of an

edge

e

S

[ u

v

]

• let the label of that edge be

a

1

a

l

let the label of that edge be

a

... 1

a

l

• If the mismatch occurred at character

a

, then k

create a new node

w

and replace

e

by two edges

create a new node

w

, and replace

e

by two edges

S[u ... u+k-1] and S[u+k ... v] labeled by

a

... 1

a

k and

a

a

a

k+

...

a

l

• Finally, in both cases (a) and (b), create a

new leaf numbered

i

, and connect

w

to it by

an edge labeled with

K

[ i

j^

K

|] i

g

[ i j^

|^

|] i

Example contd...

p

• Inserting the fourth suffix xac$ will cause the

g

first edge to be split:

$

$

$

$

S

thi

h

f^

th

d

d

• Same thing happens for the second edge

when ac$ is inserted.

Example contd...

p

• After inserting the remaining suffixes the tree

g

g

will be completed:

Storing the edge labels efficiently

g

g

y

• Note that, we do not store the actual

substrings

S

[ i

j

] of

S

in the edges, but only

their start and end indices (

i^

j )

their start and end indices (

i ,

j

• Nevertheless we keep thinking of the edge

l b l

b t i

f^

S

labels as substrings of

S

• This will reduce the space complexity to O(

n

p

p

y

(^

Suffix tree applet

pp

• http://pauillac.inria.fr/~quercia/documents-

p

p

q

info/Luminy-98/albert/JAVA+html/SuffixTreeGrow html98/albert/JAVA html/SuffixTreeGrow.html