Guided Forest Edit Distance: Better Structure Comparisons by Using Domain-knowledge

Z.S. Peng

H.F. Ting

The Forest Edit Distance

Edit distance of two ordered, labeled forests

Edit operations between E and F Relabling node i in E by the label of node j in F

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Edit distance of two ordered, labeled forests

Edit operations between E and F Relabling node i in E by the label of node j in F

Relabel (3,5)

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Edit distance of two ordered, labeled forests

Edit operations between E and F Relabling node i in E by the label of node j in F

Cost of the operation: (3,5)

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Edit distance of two ordered, labeled forests

Edit operations between E and F Delete node i from E

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Edit distance of two ordered, labeled forests

Edit operations between E and F Delete node i from E

Delete (2,-)

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Edit distance of two ordered, labeled forests

Edit operations between E and F Delete node i from E

Delete (2,-)

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Edit distance of two ordered, labeled forests

Edit operations between E and F Delete node i from E

Cost of the operation: (2,-)

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Edit distance of two ordered, labelled forests

Edit operations between E and F Delete node j from F

The cost of operation: (-,j)

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Edit distance of two ordered, labelled forests

The edit distance (E,F) between E and F is the minimum cost of edit operations that transform E to E' and F to F' such that E' = F'.

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Edit distance of two ordered, labelled forests

The edit distance (E,F) between E and F is the minimum cost of edit operations that transform E to E' and F to F' such that E' = F'.

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Edit distance of two ordered, labelled forests

The Guided edit distance (E,F,G) between E and F with respect to a third forest G is the minimum cost of edit operations that transform E to E' and F to F' such that E' = F' include G as a subforest.

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Application 1: RNA comparisons

Cherry small circular viroid-Like RNA GI:2347024 between base 287 and base 337. The Hammerhead motif of the RNA is printed in bold.

Application 2: Comparing XML documents

XML documents with same Document Type Descriptor should be aligned with this DTD to get more accurate results

The algorithms

(E,F)

Tai 1979:Zhang and Shasha 1989:

where Klein 1998:

(E,F,G):

This paper:

))()(|||(| 22 FdEdFEO

))()(|||(| FEFEO

|)|log|||(| 2 FFEO

))(|)()(|||||(| 2GLFEGFEO

)}(),(min{)( XdXLX

Special Cases

a

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Special Cases

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f

Longest Constraint Common Subsequence

Constrained Sequence Alignment

The algorithms

Constrained Longest Common Subsequent Tsai 2003:

Constrained Sequence Alignment Chin et al. :

This paper:

where

Since G has one leaf, the time becomes

|)||||(| gfeO

|)||||(| gfeO

))(|)()(|||||(| 2GLFEGFEO )}(),(min{)( XdXLX

|)||||(| GFEO

Our algorithm for computing (E,F,G)

Dynamic Programming

The sub-problems

Post-order numbering (naming) of the nodes

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The sub-problems

: A "consecutive" sub-forest

'..iiE

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The sub-problems

: A "consecutive" sub-forest

'..iiE

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21..4E

The sub-problems

),,( '..'..'.. kkjjii GFE

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E F G

The sub-problems

),,( 3..27..47..2 GFE

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7..2E 7..4F 3..2G

is equal to the minimum of the followings:

),()( ..,1)..(,)..( 00jGFE kjjsiis

1.

2.

3.

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5.

),()( ..,)..(,1)..( 00 iGFE kjjsiis

)],[],[()( ..,1)()..(,1)()..( 00 jEiEGFE kjsjsisis

),()()( )..(1)..(1)..(1)(..,1)()..(,1)()..( 00jiGFEGFE psjjsiispskjsjsisis

),()()( 1)..(1)..(1)..(1)(..,1)()..(,1)()..( 00jiGFEGFE sjjsiisskjsjsisis

)( ..,)..(,)..( 00 kjjsiis GFE

1. ),()( ..,)..(,1)..( 00 iGFE kjjsiis

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1. ),()( ..,)..(,1)..( 00 iGFE kjjsiis

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2. ),()( ..,1)..(,)..( 00jGFE kjjsiis

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)],[],[()( ..,1)()..(,1)()..( 00 jEiEGFE kjsjsisis

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)],[],[()( ..,1)()..(,1)()..( 00 jEiEGFE kjsjsisis

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),()()( )..(1)..(1)..(1)(..,1)()..(,1)()..( 00jiGFEGFE psjjsiispskjsjsisis

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),()()( )..(1)..(1)..(1)(..,1)()..(,1)()..( 00jiGFEGFE psjjsiispskjsjsisis

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),()()( 1)..(1)..(1)..(1)(..,1)()..(,1)()..( 00jiGFEGFE sjjsiisskjsjsisis

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),()()( 1)..(1)..(1)..(1)(..,1)()..(,1)()..( 00jiGFEGFE sjjsiisskjsjsisis

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),()()( 1)..(1)..(1)..(1)(..,1)()..(,1)()..( 00jiGFEGFE sjjsiisskjsjsisis

The order for solving the sub-problems

for i=1 to |E|

for j=1 to |F|

for h=1 to |G|

for k=1 to (|G|-h+1)

if k is a leaf then find ),,( )1..(..1..1 hkkji GFE

The time complexity

)|)(||||||(| 222 GLGFEO

Sparsify the dynamic program

using a clever trick of Zhang and Shasha

key-root: if it is the root, or has a left-slibling

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key-root: if it is the root, or has a left-slibling

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No. of key-roots ≤ no. of leaves

To compute (E,F,G)= (E||1..|E| ,F||1..|F| ,G||1..|G|)

for i=1 to |E|

for j=1 to |F|

for h=1 to |G|

for k=1 to (|G|-h+1)

if k is a leaf

find ),,( )1..(..1..1 hkkji GFE

To compute (E,F,G)= (E||1..|E| ,F||1..|F| ,G||1..|G|)

for i=1 to |E|

for j=1 to |F|

for h=1 to |G|

for k=1 to (|G|-h+1)

if k is a leaf and i and j are key-roots

find ),,( )1..(..1..1 hkkji GFE

The new running time

))(|)()(|||||(| 2GLFEGFEO

Thank you