23 Jan, 2008 SOFSEM 2008 1

A New Model to Solve the Swap Matching

Problem and Efficient Algorithms for Short

Patterns

Costas IliopoulosM. Sohel Rahman

23 Jan, 2008 SOFSEM 2008 2

Classic Pattern Matching

Input: A string T of length n (the text) A string P of length m (the

pattern).

Output Whether P occurs in T Occ = {i | P = T [i..i + m − 1]}

Existence Query

Computation of Occurrence

set

From Alphabet

23 Jan, 2008 SOFSEM 2008 3

Example

We have GAC at position 3 and 12 Occ = {3, 12}.

P = GAC

Occ = {5, 14}.

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Swap Matching

G CC TC T C A C G T TText

P = ACGCT1 109 112 3 4 5 6 7 8 12 13

A

C TC GA

1 2 3 4 5

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Swap Matching

G CC TC T C A C G T TText

P = ACGCT1 109 112 3 4 5 6 7 8 12 13

A

C TC GA

C TC GA

C TC GA

Occ = {1,5,6}

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Motivation Swap Error is a common error

during typing.

The phenomenon of swaps occurs in gene mutations and duplications.

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Existing results

O(nm1/3 log m log )

O(n log2 m)

O(n log m log )

= min(m,||)

(Some very special cases)

2000: Amir, Aumann, Landau,Lewenstein, Lewenstein.

1998: Amir, Landau,Lewenstein, Lewenstein.

2003: Amir, Cole, Hariharan,Lewenstein, Porat.

All results uses FFT

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Existing results Some related variants are also

investigated in the literature: Approximate version:

Amir, Lewenstein, Porat (2002) Weighted Version:

Zhang, Guo, Iliopoulos (2004)

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Our Contribution A new graph theoretic model O(m/w n logm) time.

For word-size patterns: O(n log m) The first non-FFT efficient algorithm

for swap matching

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The new Model

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T-Graph

c ca aa c b a c c b c

1 109 112 3 4 5 6 7 8 12 13

aT = b a

14 15

T-Graph

a c a abcacab a cc bc

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P-Graph

c b a b

1 2 3 4 5

aP = P-Graph

a c b

babca

b

b

a c

a

ab

1 2 3 4 5

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P-Graph

c c a b

1 2 3 4 5

aP = P-Graph

a c c

bacca

b

b

a c

a

ac

1 2 3 4 5

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So…

P swap matches T

P-Graph swap matches T-Graph

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An Efficient Algorithm

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Degenerate strings Let = {A, C, G, T} Then we can get 2^4 -1 = 15 non-

empty sets of letters. At each position of a degenerate

string we have one of those sets.

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Degenerate strings…

TGA C

GA C TA C TA G TC G

A C A G A T C G C TC G

A C G T

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Degenerate strings…

X=T

CCA

T

C

A

CA C

1 2 3 4 5 6 7

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Degenerate stringsEquality/Match

X=T

CCA

T

C

A

CA C

1 2 3 4 5 6 7

Y=T

CA

C

A

X[3] =d Y[1]. WHY?

Because, X[3] Y[1] = A

Y =d X[1..3]

Y =d X[3..5]

Y =d X[4..6]

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P-Graph => Degenerate String

a c b

babc

b

a c

a

ab

1 2 3 4 5

a

c

a

b

c

a

b

c

a

b

a

b

23 Jan, 2008 SOFSEM 2008 21

P =>

c ab a a a b c b

1 1092 3 4 5 6 7 8

bT =

According to Deg. Mat, OK!

According to Swap. Mat, NOT OK!

Swap Match vs Deg. Match

a

c

a

b

c

a

b

c

a

b

a

b

a

c

a

b

c

a

b

c

a

b

a

b

23 Jan, 2008 SOFSEM 2008 22

Why Doesn’t Work?

c ab a a a b c b

1 1092 3 4 5 6 7 8

bT =

a

c

a

b

c

a

b

c

a

b

a

b

a c b

babc

b

a c

a

ab

1 2 3 4 5

c c a b

1 2 3 4 5

a

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Forbidden Graph

a c c

bac

b

a

a

ac

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Our Algorithm

Shift-Or Algorithm

The concept of the Forbidden Graph

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D-Mask

c a baP = c => a

c

a

ba

b

a

c

a

cc

a c XD-> b

0 0 1ac 1

0 0 1ac 1

0 0 1ac 1

0 0 1abc 0

1

2

3

4

0 1 1ab 05

23 Jan, 2008 SOFSEM 2008 26

F-Mask

a c c

bac

b

a

a

ac

(a,a)

0

0

0

0

1

2

3

4

05

(a,b) (b,b) (c,c) (c,a) (X,X)

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

1 2 3 4 5

1 1

11

1

1

1 1

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Computing R matrix

c ca aa c b a c c b c

1 109 112 3 4 5 6 7 8 12 13

a b a

14 15

2

3

4

5

1

c

c

a

b

a

1

1

1

1

1

0

1

1

1

1

1

Shift

1

1

1

1

0

0

0

0

0

0

Da

0

0

0

0

0

F(X,a)

Or

1

1

1

1

01

1

1

1

0

X

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Computing R matrix

c ca aa c b a c c b c

1 109 112 3 4 5 6 7 8 12 13

a b a

14 15

2

3

4

5

1

c

c

a

b

a

1

1

1

1

1

0

1

1

1

1

0

Shift

0

1

1

1

0

0

0

0

1

0

Dc

0

0

0

0

0

F(a,c)

Or

0

1

1

1

01

1

1

1

0

X

0

1

1

1

0

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Computing R matrix

c ca aa c b a c c b c

1 109 112 3 4 5 6 7 8 12 13

a b a

14 15

2

3

4

5

1

c

c

a

b

a

1

1

1

1

1

0

0

1

1

1

0

Shift

0

0

1

1

0

0

0

0

0

0

Da

0

0

0

1

0

F(c,a)

Or

0

0

1

1

01

1

1

1

0

X

0

1

1

1

0

0

0

1

1

0

23 Jan, 2008 SOFSEM 2008 30

Computing R matrix

c ca aa c b a c c b c

1 109 112 3 4 5 6 7 8 12 13

a b a

14 15

2

3

4

5

1

c

c

a

b

a

1

1

1

1

1

0

0

0

0

1

0

Shift

0

0

0

0

0

1

1

0

0

1

Db

0

0

0

0

0

F(c,b)

Or

1

1

0

0

11

1

1

1

0

X

0

1

1

1

0

0

0

1

1

0

0

0

0

1

0

1

1

0

0

1

23 Jan, 2008 SOFSEM 2008 31

Computing R matrix

c ca aa c b a c c b c

1 109 112 3 4 5 6 7 8 12 13

a b a

14 15

2

3

4

5

1

c

c

a

b

a

1

1

1

1

1

0

1

1

1

1

0

X

0

1

1

1

0

0

0

1

1

0

0

0

0

1

0

1

1

0

0

1

1

1

1

0

0

0

1

1

1

0

1

0

1

1

0

1

1

0

1

1

1

1

1

0

0

0

1

1

1

0

0

0

1

1

0

0

0

0

1

0

1

1

0

0

1

1

1

1

0

0

23 Jan, 2008 SOFSEM 2008 32

Running Time

Computing D-Maks: O(m/w (m + ||))

Computing F-Maks: O(m/w m log m)

Computing R Values: O(m/w n log m)

O(m/w n log m)

O(n log m)short patterns (m~w)

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Future Works Explore the possibilities of using

Graph pattern matching Experimental works

Forthcoming paper contains experimental works using biological examples.

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The End

Thank you very much

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