etri linear-time search in suffix arrays july 14, 2003 jeong seop sim, dong kyue kim heejin park,...

ETRIETRI

Linear-Time Search in Suffix Arrays

July 14, 2003

Jeong Seop Sim, Dong Kyue Kim

Heejin Park, Kunsoo Park

ETRIETRI

Suffix arrays

Suffix array of text TThe lexicographically sorted list of all suffixes of text T

ETRIETRI

Suffix arraysExample for T = abbabaababbb#

The suffixes of T abbabaababbb# (1)

bbabaababbb# (2) abaababbb# (3)

… b# (12) # (13)

are stored in lexicographical order.

1 #

2 a a b a b b b #

3 a b a a b a b b b #

4 a b a b b b #

5 a b b a b a a b a b b b #

6 a b b b #

7 b #

8 b a a b a b b b #

9 b a b a a b a b b b #

10 b a b b b #

11 b b #

12 b b a b a a b a b b b #

13 b b b ## is the lexicographically smallest special character.

ETRIETRI

Suffix arraysExample for T = abbabaababbb#

The suffixes of T are abbabaababbb# (1) bbabaababbb# (2)

abaababbb# (3) … b# (12) # (13)

In actual suffix arrays, we store only the starting positions of suffixes in T but for convenience, we assume that suffixes themselves are stored.

1 13 #

2 6 a a b a b b b #

3 4 a b a a b a b b b #

4 7 a b a b b b #

5 1 a b b a b a a b a b b b #

6 9 a b b b #

7 12 b #

8 5 b a a b a b b b #

9 3 b a b a a b a b b b #

10 8 b a b b b #

11 11 b b #

12 2 b b a b a a b a b b b #

13 10 b b b #

ETRIETRI

Suffix arrays

Definition: s-suffixesSuffixes starting with string sa-suffixes, ba-suffixes, …

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

ETRIETRI

Suffix arrays vs. Suffix treesConstruction time

Suffix Array = Suffix Tree

Space Suffix Array = Suffix Tree

In practice , suffix arrays are more space efficient than suffix trees.

Search timeSuffix Array: , (p=|P|, n=|T|)Suffix Tree:

|)|log( p)log( np |)|( p

ETRIETRI

ContributionConstruction time




Search timeSuffix Array: , , Suffix Tree: |)|log( p

)log( np |)|( p |)|log( p

ETRIETRI

The meaning of our contributionConstruction time






Search time: SA ST

ETRIETRI

The meaning of our contributionConstruction time






Search time: SA ST

Suffix arrays are more powerful than suffix trees.

ETRIETRI

Our search algorithm

Our search algorithm

ETRIETRI

Search in a suffix array

Definition: Search in a suffix arrayInput

A pattern P A suffix array of T

Output

All P-suffixes of T

ETRIETRI


All ab-suffixes are neighbors.

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

P = ab

T = abbabaababbb#

Find all ab-suffixes.

A search example

ETRIETRI


We have only to find

the first and the last ab-suffixes.

Because the other ab-suffixes are

stored between them.

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

P = ab

T = abbabaababbb#

A search example

ETRIETRI

Related workIn developing our search algorithm, we adopt the idea suggested by Ferragina and Manzini (FOCS 2001).

Search a pattern in a file compressed by the Burrows-Wheeler compression algorithm

Search P from the last character to the first character of PP = ababaaabbabaaabb

We adopt this backward pattern searching idea.

ETRIETRI

Algorithm outline

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

P = aba

T = abbabaababbb#

Outline of our search algorithm

We find all aba-suffixes

by searching P backward.

Our algorithm has p stages

(In this case, there are 3 stages.)

ETRIETRI

Algorithm outline

P = aba

T = abbabaababbb#




Stage 1: find all a-suffixes.

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

ETRIETRI

Algorithm outline

P = aba

T = abbabaababbb#




stage 1: find all a-suffixes.

stage 2: find all ba-suffixes.

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

ETRIETRI

Algorithm outline

P = aba

T = abbabaababbb#




stage 1: find all a-suffixes.

stage 2: find all ba-suffixes.

stage 3: find all aba-suffixes.

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

ETRIETRI

Elaborate stage 2

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

P = aba

A stage by elaborating stage 2

We find the first ba-suffix from the

first a-suffix and the last ba-suffix

from the last a-suffix.

We find all ba-suffixes

using a-suffixes found in stage 1.

ETRIETRI

Elaborate stage 2

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

P = aba

Only explain how to find the first

ba-suffix from the first a-suffix.

Finding the last ba-suffix is similar.

A stage by elaborating stage 2

ETRIETRI

Elaborate stage 2

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

To find the first ba-suffix, we count the number of suffixes that precede ba-suffixes in this suffix array.

P = aba

ETRIETRI

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

Suffixes preceding ba-suffixes are

divided into two categories.

- A-type: Suffixes starting with

characters lexicographically smaller than b. (#-suffixes, a-suffixes)

- B-type: Suffixes starting with the same

character b and preceding ba-suffixes.

We count A-type and B-type suffixes in different ways.

Elaborate stage 2

A-type

B-type

ETRIETRI

Count the number of A-type suffixes

Count the number of A-type suffixes 1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

The number of A-type suffixes = The number of #-suffixes and a-suffixes = The position of the last a-suffix.

A-type

ETRIETRI


We generate an array that stores the positions of the last #-suffix, the last a-suffix, and the last b-suffix.

With this array, we can count A-type suffixes in O(1) time.

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

# 1

a 6

b 13

ETRIETRI


1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

Array S pace:Time: O(n) (one scan)

|)(|

# 1

a 6

b 13

ETRIETRI

Count the number of B-type suffixes

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

Count B-type suffixesb-suffixes preceding ba-suffixes.

B-type

ETRIETRI


1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

B-type suffixesb-suffixes preceding ba-suffixes.

A suffix generated by removing the leftmost b from a B-type suffix appears in a suffix subarray preceding a-suffixes found in stage 1.

B-type

ETRIETRI

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #


The number of B-type suffixes are the number of suffixes

being in a suffix subarray that precedes a-suffixes

whose previous characters are bs B-type

We count this with array N.

b

b

b

a

#

b

b

a

b

a

b

a

a

Let U be the conceptual array of

previous characters of suffixes.

U

ETRIETRI

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

b

b

b

a

#

b

b

a

b

a

b

a

a

Count the number of B-type suffixes # a b

0 0 1

0 0 2

0 0 3

0 1 3

1 1 3

1 1 4

1 1 5

1 2 5

1 2 6

1 3 6

1 3 7

1 4 7

1 5 7

5],7[ bN

Array N

entries|| n

N[i,b] stores the number of suffixes whose previous characters are bs in a suffix subarray SA[1,i].

U

ETRIETRI

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

b

b

b

a

#

b

b

a

b

a

b

a

a

Count the number of B-type suffixes # a b

0 0 1

0 0 2

0 0 3

0 1 3

1 1 3

1 1 4

1 1 5

1 2 5

1 2 6

1 3 6

1 3 7

1 4 7

1 5 7

U

We can count B-type

suffixes in O(1) time

by accessing an entry of N.

ETRIETRI

Array NSpace:

An alternative way Space: O(n) time for counting B-type suffixes.

Array N

|)| (O n

|)|(logO

# a b

0 0 1

0 0 2

0 0 3

0 1 3

1 1 3

1 1 4

1 1 5

1 2 5

1 2 6

1 3 6

1 3 7

1 4 7

1 5 7

ETRIETRI

Query for N[i,b]Counting B-type suffixes

O(log n) time

O(log ) time||

ETRIETRI

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

b

b

b

a

#

b

b

a

b

a

b

a

a

UQuery for N[i,b] O(log n) time

In an O(log n) time algorithm,

we generate an array

whose ith entry stores

the location of the ith b in U.

1 1

2 2

3 3

4 6

5 7

6 9

7 11

ETRIETRI

b

b

b

a

#

b

b

a

b

a

b

a

a

UQuery for N[i,b]: O(log n) time

1 1

2 2

3 3

4 6

5 7

6 9

7 11

To count suffixes whose previous

characters are bs in SA[1,8].

= To count bs in U[1,8]

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

ETRIETRI

b

b

b

a

#

b

b

a

b

a

b

a

a


1 1

2 2

3 3

4 6

5 7

6 9

7 11

Find the largest value not

exceeding 8 in this array.

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

ETRIETRI

b

b

b

a

#

b

b

a

b

a

b

a

a


1 1

2 2

3 3

4 6

5 7

6 9

7 11

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

To find 7 in this array,

we perform binary search.

O(log n) time.

ETRIETRI

b

b

b

a

#

b

b

a

b

a

b

a

a


1 1

2 2

3 3

4 6

5 7

6 9

7 11

The index of 7 (5) is

the number of b’s in U[1,8].1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

ETRIETRI

b

b

b

a

#

b

b

a

b

a

b

a

a


1 1

2 2

3 3

4 6

5 7

6 9

7 11

1 4

2 8

3 10

4 12

5 13

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

1 5

Generally, we require arrays for

all characters. #

a

b

O(n) space

ETRIETRI

Query for N[i,b]

O(log n) time

O(log ) time||

ETRIETRI

For the last characters

of each block,

we compute the entries

of N.

b

b

b

a

#

b

b

a

b

a

b

a

a

UQuery for N[i,b]: time

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

Divide U into

-sized blocks.

|)|(log O# a b

0 0 3

1 1 4

1 2 6

1 4 7

||

ETRIETRI

For the other entries

in each block,

we generate a similar

data structure used

in O(log n) time alg.

O(log ) time

for binary search.

Still O(n) space in total.

b

b

b

a

#

b

b

a

b

a

b

a

a

UQuery for N[i,b]: time

1 #

2 a a b a b b b #


4 a b a b b b #


6 a b b b #

7 b #

8 b a a b a b b b #


10 b a b b b #

11 b b #


13 b b b #

|)|(log O# a b

0 0 3

1 1 4

1 2 6

1 4 7

||

ETRIETRI

Summaryp stages

Each stageCount A-type suffixes

Time: O(1) Space: O(n) for M array

Count B-type suffixes Time: Space: O(n) for computing the value of an entry N

In total, time with O(n) space.|)|log( p

|)|(log

ETRIETRI

Conclusion

In a suffix array, one can choose or search time algorithm depending on the alphabet

size.

Suffix arrays are more powerful than suffix trees.

|)|log( p )log( np

etri linear-time search in suffix arrays july 14, 2003 jeong seop sim, dong kyue kim heejin park,...

Documents