Tree Edit Distance

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Tree Edit Distance
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TED

• Minimum edits to transform one tree into another

3
The edit operations
Delete a node
Relabel a node
w
v
???
???
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The edit operations
Insert a node
v
???
???
???
5
Existing Algorithms
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Recursive Algorithm SZ89
Recurs on the rightmost root
Delete v d(F,G) min
Delete w Match v and
w
F
G
v
w
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Recursive Algorithm SZ89
Recurs on the rightmost root
Delete v d(F,G) min
Delete w Match v and
w
F
G
v
w
8
Recursive Algorithm SZ89
Recurs on the rightmost root
Delete v d(F,G) min
Delete w Match v and
w
F
G
v
w
9
Recursive Algorithm SZ89
Recurs on the rightmost root
Delete v d(F,G) min
Delete w Match v and
w
F
G
v
w
10
Recursive Algorithm SZ89
Recurs on the rightmost root
Delete v d(F,G) min
Delete w Match v and
w
F
G
v
w
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Recursive Algorithm SZ89
Recurs on the rightmost root
Delete v d(F,G) min
Delete w Match v and
w
F
G
v
w
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Time Complexity SZ89

• relevant subproblem if it shows up while
  computing d(F,G)
• relevant subproblems time complexity O(n2m2)
  O(n4)
• O(nm . minDepth(F),Leaves(F) .
  minDepth(G),Leaves(G))

F
G
v
w
Relevant subforests
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Klein98

• Same as previous algorithm, but recurs on a light
  child in F.
• relevant subproblems (relevant subforests of
  F) . m2
• O(nlogn
  . m2) O(n3logn)

F
G
By heavy path decomposition HT84
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Decomposition strategy DT03

• For every two subforests (F,G) a strategy says
  right or left.
• Zhang Shashas strategy right always.
• Kleins strategy right iff the rightmost tree
  in F is smaller than the leftmost tree in F.
• Lower bound of strategy algorithms ?(nm . logn
  . logm)
• Any strategy algorithm computes the edit distance
  between any two subtrees of F and G (without
  their roots).

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Our Results

• An O(m2n(log 1)) O(n3) time, O(nm) space
  algorithm. (Today O((nm)3/2
  )O(n3) time and space) DMRW ICALP07
• A strategy algorithm symmetrically dependant on
  the two input trees.
• A matching lower bound for all strategy
  algorithms. (Today A
  lower bound of ?(nm2))
• Local edit distance and affine gap penalties at
  the cost of one execution. (Today Local RNA edit
  distance) BHLW CPM06

n
m
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Our Algorithm
G
F

• Our algorithm to compute d(F,G)
• If FltG compute d(G,F).
• Recursively run d(Ki,G) for every Ki.
• Run Kleins strategy where master is F (no need
  to recurs).

K4
K1
K3
K2
K5
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Analysis
G
F

• Our algorithm to compute d(F,G)
• If FltG compute d(G,F).
• Recursively run d(Ki,G) for every Ki.
• Run Kleins strategy where master is F (no need
  to recurs).

K4
K1
K3
K2
K5
R(F, G) ?
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An O((nm)3/2) O(n3) Upper Bound

• We show that
  . Proof by induction
• R(F,G)

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An O((nm)3/2) O(n3) Upper Bound

• We show that
  . Proof by induction
• R(F,G)

By inductive assumption
By () and ()
We know GltF
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An O((nm)3/2) O(n3) Upper Bound

• We show that
  . Proof by induction
• R(F,G)

By inductive assumption
By () and ()
We know GltF
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An O((nm)3/2) O(n3) Upper Bound

• We show that
  . Proof by induction
• R(F,G)

By inductive assumption
By () and ()
We know GltF
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An O((nm)3/2) O(n3) Upper Bound

• We show that
  . Proof by induction
• R(F,G)

By inductive assumption
By () and ()
We know GltF
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An O((nm)3/2) O(n3) Upper Bound

• We show that
  . Proof by induction
• R(F,G)

By inductive assumption
By () and ()
We know GltF
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An O( ) Bound
n
m

• Proof idea
• At most log(n/m) nested recursive calls where F
  is master before all trees m.
• For all trees m use previous O(m3) bound . At
  most n/m such trees so total
• n/m.O(m3) O(nm2) .

F
G
K4
K1
K3
K2
K5
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A Matching Lower Bound for all decomposition
strategy algorithms
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A Matching Lower Bound for all decomposition
strategy algorithms

• An ?(nm2) lower bound

F
G
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A Matching Lower Bound for all decomposition
strategy algorithms

• An ?(nm2) lower bound
• Consider this computational path
• If the strategy says left delete from F,
  otherwise delete from G.
• For every two internal nodes v in F and w in G we
  get
• minFv,Gw new subproblems (Fv is the tree
  rooted at v).
• Summing over all such v,w

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A Matching Lower Bound for all decomposition
strategy algorithms

• An lower bound
• A careful counting argument on

G
F
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Thank you!