Approximate String Matching Using Compressed Suffix Arrays Trinh N. D. Huynh, W. K. Hon, T. W. Lam and W. K. Sung, Theoretical Computer Science, Vol. 352, 2006, pp. 240-249

1
Approximate String Matching Using Compressed
Suffix ArraysTrinh N. D. Huynh, W. K. Hon, T.
W. Lam and W. K. Sung, Theoretical Computer
Science, Vol. 352, 2006, pp. 240-249

• Advisor Prof. R. C. T. Lee
• Speaker C. W. Lu

2

• Let x and y be two strings. Edit distance d(x,
  y) is the minimum number of character insertions,
  deletions, and replacements to covert string x to
  y.
• k-difference string matching problem
• Given a text T with length n, a pattern P with
  length m, and an error bound k.
• Find all position i of T such that there exists
  an suffix S of T(1, i), d(S, P) ? k.

3

• The approach of this paper is as the follows
• Given a pattern P and an error bound k, we
  generate all possible Ps which contain (?k)
  errors deduced from P.
• Then we conduct an exact match of all such Ps
  against T.

4

• Example
• Tabbaaa,
• Paba and k1.
• From P and k, we generate the following Ps
• ba, aaba, baba, bba, aa, abba, aaa, ab, abaa,
  abb, aba.

5

• Then we conduct an exact matching of all Ps
  against T. Any success indicates that there is a
  substring S in T such that d(S,T)?k.
• How can we generate all Ps which we want?
• We use the following observation.

6
S
S2
S1
T
P
P1
P2
Let S be a substring of T, and S S1S2. P
P1P2. If d(S1, P1) ?k, and Dist(S2, P2) 0, d(S,
P) ? k.
7

• Example

k 2
1
2
3
4
5
6
7
8
9
10
11
12
13
T
A
C
A
C
A
A
A
A
A
C
A
C
C
S1
S2
1
2
3
4
5
6
P
A
G
A
B
C
A
P1
P2
Consider the substring S T(6, 11) AAAACA, Let
S1 T(6, 9) AAAA, and S2 T(10, 11)
CA. Dist(S1, P1) 2 ?k, and Dist(S2, P2) 0. We
have Dist(S, P) 2 ?k.
8

• Example

k 2
1
2
3
4
5
6
7
8
9
10
11
12
13
T
A
C
A
C
A
A
A
A
A
C
A
C
C
S1
S2
1
2
3
4
5
6
P
A
G
A
B
C
A
P1
P2
Consider the substring S T(8, 11) AACA, Let
S1 T(8, 9) AA, and S2 T(10, 11)
CA. Dist(S1, P1) 2 ?k, and Dist(S2, P2) 0. We
have Dist(S, P) 2 ?k.
9

• Based upon the above observation, we can generate
  all edited pattern Ps by editing the prefix and
  keeping the suffix untouched, in some manner.
• Consider Paba, k1.

10

• Paba, k1.

ba (Deletion) k 1
aaba (Insertion) k 1
i 1
baba (Insertion) k 1
P aba
bba (Substution) k 1
aa (Deletion) k 1
aba k 0
aaba (Insertion) k 1
abba (Insertion) k 1
i 2
aaa (Substution) k 1
ab (Deletion) k 1
aba k 0
abaa (Insertion) k 1
abba (Insertion) k 1
i 3
abb (Substution) k 1
aba k 0
abaa (Insertion) k 1
abab (Insertion) k 1
i 4
11

• Paba, k2.

ba (Deletion) k 1
aaba (Insertion) k 1
i 1
baba (Insertion) k 1
P aba
bba (Substution) k 1
aa (Deletion) k 1
aba k 0
aaba (Insertion) k 1
abba (Insertion) k 1
i 2
aaa (Substution) k 1
ab (Deletion) k 1
aba k 0
abaa (Insertion) k 1
abba (Insertion) k 1
i 3
abb (Substution) k 1
aba k 0
abaa (Insertion) k 1
abab (Insertion) k 1
i 4
12

• Paba, k2.

a (Deletion) k 2
i 2
aba (Insertion) k 2
bba (Insertion) k 2
ba (k 1)
aa (Substution) k 2
b (Deletion) k 2
ba k 1
baa (Insertion) k 2
bba (Insertion) k 2
i 3
bb (Substution) k 2
ba k 1
baa (Insertion) k 2
bab (Insertion) k 2
i 4
13
PR
PL
i
For i1 to m1
Deletion, k
P
PR
PL
i
P
PL
PR
P
PL
PR
A
Replacement , k
P
P
C

PL
PR
kDist(PL, PL)?k. Dist(PR, PR) 0
P
Insertion, k
A
P
C

PL
PR
No operation.
P
i
Terminate if k gt k.
14

• Our problem now becomes the following Given a
  pattern P, we produce a modified pattern P. Our
  job is to determine whether P exactly matches
  some substring of T or not.
• For example, Suppose Paba. We have ba as one of
  the modified patterns. So, we like to find out
  whether ba matches exactly with a substring in T.

15

• This exact matching can be found by using the
  suffix array and the inverse suffix array.

16
Suffix Array

• Let , where t0, t1, tn-1 an alphabet
  A and tn is a special symbol that is not in A
  and smaller than any symbol in A.
• The jth suffix of T is defined as T(j, n) tjtn
  and is denoted by Tj.
• The suffix array SA0..n of T is an array of
  integers j that represent suffix Tj and the
  integers are sorted in lexicographic order of
  corresponding suffixes.

17
Example
0
1
2
3
4
5
6
7
8
9
T
G
A
C
A
G
T
T
C
G

Suffixes of T GACAGTTCG, ACAGTTCG, CAGTTCG,
AGTTCG, GTTCG, TTCG, TCG, CG, G,
Lexicographic order , ACAGTTCG, AGTTCG,
CAGTTCG, CG, G, GACAGTTCG, GTTCG, TCG,
TTCG. T9, T1, T3, T2, T7, T8, T0, T4, T6, T5
0
1
2
3
4
5
6
7
8
9
i
?
SAi
9
1
3
2
7
8
0
4
6
5
18
Inverse Suffix Array

• The inverse suffix array of T is denoted as
  SA-1i.
• SA-1i equals the number of suffix which are
  lexicographically smaller then Ti.

19
Example
0
1
2
3
4
5
6
7
8
9
T
G
A
C
A
G
T
T
C
G

Lexicographic order (T9) ACAGTTCG (T1) AGT
TCG (T3) CAGTTCG (T2) CG (T7) G (T8) GAC
AGTTCG (T0) GTTCG (T4) TCG (T6) TTCG. (T5)
SAi
i
SA-1i
SA-106 because there are 6 suffixes smaller
than T0 GACAGTTCG.
9
0
6
1
1
1
3
2
3
2
3
2
7
4
7
8
5
9
0
6
8
4
7
4
SA-1SAx x.
6
8
5
5
9
0
20

• The size of SA and SA-1 are O(nlogn) bits. Both
  data structures can be constructed in linear
  time13, 15, 17.

21

• In this paper, an interval st..ed is called the
  range of the suffix array of T corresponding to a
  string P if st..ed is the largest interval such
  that P is a prefix of every suffix Tj for j
  SAst, SAst1, , SAed.
• We write st..ed range(T, P).

22
Example
0
1
2
3
4
5
6
7
8
9
T
G
A
C
A
G
T
T
C
G

Lexicographic order (T9) ACAGTTCG (T1) AGT
TCG (T3) CAGTTCG (T2) CG (T7) G (T8) GAC
AGTTCG (T0) GTTCG (T4) TCG (T6) TTCG. (T5)
SAi
i
P G.
9
0
G is a prefix of T8, T0 and T4.
1
1
3
2
T8 TSA5 T0 TSA6 T4 TSA7 ? st5,
ed7, range(T, P) 5..7.
2
3
7
4
8
5
0
6
4
7
6
8
5
9
23

• Lemma 1 (Gusfild 12)
• Given a text T together with its suffix array,
  assume st..ed range(T, P). Then, for any
  character c, the intervalst..ed range(T,
  Pc) can be computed in O(logn) time.

24

• Lemma 2
• Given the interval st1..ed1 range(T , P1) and
  the interval st2..ed2 range(T , P2), we can
  find the interval st..ed range(T , P1P2) in
  O(logn) time using the suffix array and the
  inverse suffix array of T.

25

• Let st1..ed1 range(T , P1),
• st2..ed2 range(T , P2),
• st..ed range(T , P1P2).
• st..ed is a subinterval of st1..ed1.

26
Example
0
1
2
3
4
5
6
7
8
9
T
G
A
C
A
G
T
T
C
G

Lexicographic order (T9) ACAGTTCG (T1) AGT
TCG (T3) CAGTTCG (T2) CG (T7) G (T8) GAC
AGTTCG (T0) GTTCG (T4) TCG (T6) TTCG. (T5)
SAi
i
P1 G. P2 A.
9
0
1
1
range(T, P1) 5..7.
3
2
2
3
range(T, P1P2) must be within 5..7. How can we
find the exact interval with 5..7?
7
4
8
5
0
6
4
7
6
8
5
9
27

• By the definition of suffix array, the
  lexicographic order of are increasing.
• The lexicographic order of
• are also increasing.

28
T2 CAGTTCG T21 T3 AGTTCG T21 is
obtained by deleting the prefix with length 1
from T2. In general, Ti1 can be obtained by
deleting the prefix with length 1 from Ti.
Lexicographic order (T9) ACAGTTCG (T1) AGT
TCG (T3) CAGTTCG (T2) CG (T7) G (T8) GAC
AGTTCG (T0) GTTCG (T4) TCG (T6) TTCG. (T5)
29
Example
0
1
2
3
4
5
6
7
8
9
T
G
A
C
A
G
T
T
C
G

P1 G. P2 A.
SAi
i
Lexicographic order (T9) ACAGTTCG (T1) AG
TTCG (T3) CAGTTCG (T2) CG (T7) G (T8) GA
CAGTTCG (T0) GTTCG (T4) TCG (T6) TTCG. (T5
)
9
0
range(T, P1) 5..7.
1
1
3
2
2
3
? T8 lt T0 lt T4
7
4
8
5
0
6

• T81, T01, T41
• T9 lt T1 lt T5

4
7
6
8
5
9
30

• The lexicographic order of
• are also increasing.
• Thus
• To find st and ed, we find the smallest st such
  that
  and the largest ed such that

31
Example
0
1
2
3
4
5
6
7
8
9
T
G
A
C
A
G
A
T
C
G

P1 G. P2 A.
SAi
i
SA-1i
Lexicographic order (T9) ACAGTTCG (T1) AG
TTCG (T3) ATCG. (T5) CAGTTCG (T2) CG (T7)
G (T8) GACAGTTCG (T0) GATCG (T4) TCG (T6
)
9
0
range(T, P1) 6..8.
7
1
1
1
range(T, P2) 1..3.
3
2
4
range(T, P1P2) st..ed.
5
3
2
2
4
8
6 ? st, ed ? 8
7
5
3
8
6
9
0
7
5
4
8
6
6
9
0
? st 7 and ed 8.
32

• To find the interval of the first character of P
• We construct an array C such that for any c in
  A, Cc stores the total number of occurrences of
  all c in T, where c ? c.
• range(T, p1) Cc21 Cc where c2 is a
  character immediately before c in A.

33
Example
0
1
2
3
4
5
6
7
8
9
T
G
A
C
A
G
T
T
C
G

SAi
i
CA 2 CC 4 CG 7 CT 9
Lexicographic order (T9) ACAGTTCG (T1) AG
TTCG (T3) CAGTTCG (T2) CG (T7) G (T8) GA
CAGTTCG (T0) GTTCG (T4) TCG (T6) TTCG. (T5
)
9
0
1
1
3
2
2
3
7
4
8
5
P GACAGCA
0
6
4
7
range(T, p1) CC1CG 57.
6
8
5
9
34

• Lemma 3
• Given the suffix array and the inverse suffix
  array of T, assume st..ed range(T, P). For
  any character c, assume we have in advance the
  array C, we can find the interval st..ed
  range(T, cP) in O(logn) time.

35

• I Construct Fst 1..m1 and Fed 1..m1 such
  that Fst i..Fed i range(T ,Pi..m).
• II Call kapproximate(0..n, 1, 0, e, e).
• kapproximate(s..e, i, k, PL, ? )
• begin
• 1. Given Fst i..Fed i range(T ,
  Pi..m) and s..e range(T , PL), by
• Lemma 2 find st..ed range(T ,
  PLPi..m).
• 2. Report occurrences of P PLPi..m in
  st..ed if the interval exists.
• 3. If (k k) return.
• 4. For j i to m1
• (a) (when j ?m, deletion at j)
• Call kapproximate(s..e, j1,
  k1, PL, d?).
• (b) (when j? m, replacement at j ) for
  each c in A
• i. Given s..e range(T ,
  PL), by Lemma 1 find s..e range(T ,
  PLc).
• ii. Call kapproximate(s..e,
  j1, k1, PLc, r?).
• (c) (insertion at j) for each c in A
• i. Given s..e range(T ,
  PL), by Lemma 1 find s..e range(T ,
  PLc).
• ii. Call kapproximate(s..e,
  j, k1, PLc, i?).
• (d) (when j?m) Given s..e range(T
  , PL), by Lemma 1 find s..e

36

• After an O(n) time preprocessing the text T into
  an O(nlogn)-bit data structure, the algorithm
  solves the k-difference problem in O(Akmklogn
  outputtime) time.

37

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40

• Thank you!