Fast String Correction With l Even Shte in Automata

8/8/2019 Fast String Correction With l Even Shte in Automata

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W

V

0 1 i i+2

xi+1 xi+2

xi+1 xi+2 (i+1)# (i+2)#

i+1

i#i0 1

W

V

0 1 i i+2

xi+1 xi+2

xi+2 (i+1)# (i+2)#

i+1

i0 1

W

V

0 1 i i+2

xi+1 xi+2

xi+2 (i+2)#

i+1

0 1 i

Insertion

Substitution

Deletion

1

2

3


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i

{i+1}

{i#,(i+1)#,(i+2)#}

{i#,(i+1)#}

0 i n-2

n-1

{n}

{(n-1)#,n#}

xi+1

xi+2

xn

n {n#}i

{i+1}

{i#,(i+1)#}

0 i n-2

xi+1

i = n-1 i = n

i#

{(i+1)#}xi+1

0 i n-1

pr(xi+1,xi+2) = (12)

pr(xi+1,xi+2) = (11)

n#


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x1

x2

x2

x3

x3

x4

x4

x5

x50 1 2 3 4 5

1# 2# 3# 4#

5#

(012)#

(01)# (12)# (23)#

(34)#(45)#(123)# (234)# (345)#

x2 x3 x4 x5

x2 x3 x4 x5x1

x1

x2

x2 x2

x3

x3

x4

x40 1 2 3 4 5

1# 2# 3# 4# 5#

(012)#

(01)#

(12)#

(23)# (34)#(45)#(234)# (345)#

x2 x2 x3 x4

x2

x2

x3x4

x1

x2

x2


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i

{i+1}

{#: i,i+1,i+2}

{#: i,i+1}

0 i n-3

pr(xi+1,xi+2,xi+3) = (1,2,3)

xi+1xi+2

{#: i,i+1

##: i+3}

xi+3i

{i+1}

{#: i,i+1}

pr(xi+1,xi+2,xi+3) = (1,1,2)

xi+1

{#: i,i+1

##: i+3}

xi+3 i

{i+1}

{#: i,i+1}

pr(xi+1,xi+2,xi+3) = (1,2,2)

xi+1

{#: i,i+1,i+2}xi+2 i

{i+1}

{#: i,i+1}

pr(xi+1,xi+2,xi+3) = (1,2,1)

xi+1

{#: i,i+1,i+2}xi+2

i

{i+1}

{#: i,i+1}

pr(xi+1,xi+2,xi+3) = (1,1,1)

xi+1

0 i n-3 0 i n-3 0 i n-3

0 i n-3

n-2

{n-1}

{#: n-2,n-1}

xn-1

xn{#: n-2,n-1,n} n-2

{n-1}

{#: n-2,n-1}

xn-1

i = n-2 i = n-2

pr(xn-1,xn) = (1,2) pr(xn-1,xn) = (1,1)

n-1

{n}

{#: n-1,n}

i = n-1

xn

n {#: n}

i = n


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Ins(?) after x5

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

Del(x5)

Del(x3)

Subst(x3,x3)

Ins(x3) before x3

Subst(x5,x5)

Ins(x5) before x5

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

x2

x2

x3

x3

x4

x4

x5

x5

x3

x3

x3

x4

x4

x4

x5

x5

x5

?


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x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

x2

x2

x4 x5

x5

x3

x3

x3

x4

x4

x4

x5

x5

x5

?

x3 x4 x5

x3 deletion-deletion-transitions

deletion of x3 followed by

deletion of x4

deletion of x4 followed by deletion of x5

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

x2

x2

x3

x3

x4

x4

x5

x5

x3

x3

x3

x4x4

x4

x5x5

x5

?

x2 x3 x4 x5

deletion-substitution-transitions

deletion of x4 followed by

substitution of x5

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4x4

x5

x5

x5

x2

x2

x3

x3

x4

x4

x5

x5

x3

x3

x3

x4x4

x4

x5x5

x5

?

x3 x4 x5

substitution-deletion-transitions

substitution of x3 followed by

deletion of x4


deletion of x5


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x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

x2

x2

x3

x3

x4

x4

x5

x5

x3

x3

x3

x4x4

x4

x5x5

x5

?

x2 x3 x4x5

substitution-substitution-transitions


substitution of x5

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

?

x2

x2

x3

x3

x4

x4

x5

x5

x3

x3

x3

x4

x4

x4

x5

x5

x5

?

x2x1

x1 x2 x3 x4 x5

insertion-insertion-transitions

two insertions before x5

two insertions after x5

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

x2

x2

x3

x3

x4

x4

x5

x5

x3

x3

x3

x4x4

x4

x5x5

x5

?

x2

x1 x2 x3 x4 x5

insertion-substitution-transitions

insertion before x5 followed by

substitution of x5

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

x2

x2

x3

x3

x4

x4

x5

x5

x3

x3

x3

x4x4

x4

x5x5

x5

?

x2

x3x2 x4 x5

substitution-insertion-transitions


insertion before x5?


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x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5

x2

x2

x3

x3

x4

x4

x5

x5

x3

x3

x3

x4

x4

x4

x5

x5

x5

x3

x3

x4

x4

x5

x5

x4

x4

x4

x5

x5

x5

?

?


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x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 Transpose(x4x5)

x2x1 x3x2 x4x3 x5x4

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2x2

x2

x3

x3

x3

x4

x4

x4

x5x5

x5

Split(x5,x5x5)

x1 x2 x3 x4 x5

x1

x1

x1

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 ?

x2

x2

x2

x3

x3

x3

x4

x4

x4

x5

x5

x5 Merge(x4x5,x4&x5)

x1&x2 x2&x3 x3&x4 x4&x5


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Documents

Fast String Correction With l Even Shte in Automata