1. Tree edit distance1 Tree Edit Distance

2.  Minimum edits to transform one tree into another Tree edit distance2 TED

3. Tree edit distance3 Delete a node: The edit operations w ˙˙˙ v Relabel a node:

4. Tree edit distance4 The edit operations ˙˙˙ Insert a node: ˙˙˙ v

5. Tree edit distance5 Existing Algorithms

6. Tree edit distance6 Recursive Algorithm [SZ89] v w FG Recurs on the rightmost root: Delete v d(F,G) = min Delete w Match v and w

7. Tree edit distance7 Recursive Algorithm [SZ89] v w FG Recurs on the rightmost root: Delete v d(F,G) = min Delete w Match v and w

8. Tree edit distance8 Recursive Algorithm [SZ89] v w FG Recurs on the rightmost root: Delete v d(F,G) = min Delete w Match v and w

9. Tree edit distance9 Recursive Algorithm [SZ89] v w FG Recurs on the rightmost root: Delete v d(F,G) = min Delete w Match v and w

10. Tree edit distance10 Recursive Algorithm [SZ89] v w FG Recurs on the rightmost root: Delete v d(F,G) = min Delete w Match v and w

11. Tree edit distance11 Recursive Algorithm [SZ89] v w FG Recurs on the rightmost root: Delete v d(F,G) = min Delete w Match v and w

12. Tree edit distance 12 Time Complexity [SZ89]  relevant subproblem: if it shows up while computing d(F,G)  #relevant subproblems = time complexity = O(n 2 m 2 ) = O(n 4 )  O(nm. min{Depth(F),Leaves(F)}. min{Depth(G),Leaves(G)}) v w F G Relevant subforests

13. Tree edit distance13 Klein98  Same as previous algorithm, but recurs on a light child in F.  #relevant subproblems = (#relevant subforests of F). m 2 = = O(nlogn. m 2 ) = O(n 3 logn) FG By heavy path decomposition [HT84]

14. Tree edit distance14 Decomposition strategy [DT03]  For every two subforests (F,G) a strategy says right or left.  Zhang & Shasha’s strategy = right always.  Klein’s 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).

15. Tree edit distance15 Our Results  An O ( m 2 n(log + 1) ) = O(n 3 ) time, O(nm) space algorithm. (Today: O((nm) 3/2 )=O(n 3 ) 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  (nm 2 ))  Local edit distance and affine gap penalties at the cost of one execution. (Today: Local RNA edit distance) [BHLW CPM06] n m

16. Tree edit distance16 Our Algorithm  Our algorithm to compute d(F,G): 1.If F<G compute d(G,F). 2.Recursively run d(K i,G) for every K i. 3.Run Klein’s strategy where “master” is F (no need to recurs). K5K5 K3K3 K4K4 F K2K2 K1K1 G

17. Tree edit distance17 Analysis  Our algorithm to compute d(F,G): 1.If F<G compute d(G,F). 2.Recursively run d(K i,G) for every K i. 3.Run Klein’s strategy where “master” is F (no need to recurs). K5K5 K3K3 K4K4 F K2K2 K1K1 G R(F, G) = ?

18. Tree edit distance18 An O((nm) 3/2 ) = O(n 3 ) Upper Bound  We show that. Proof by induction: R(F,G)

19. Tree edit distance19  We show that. Proof by induction: R(F,G) By inductive assumption By (*) and (**) We know G<F An O((nm) 3/2 ) = O(n 3 ) Upper Bound

20. Tree edit distance20 An O((nm) 3/2 ) = O(n 3 ) Upper Bound  We show that. Proof by induction: R(F,G) By inductive assumption By (*) and (**) We know G<F

21. An O((nm) 3/2 ) = O(n 3 ) Upper Bound Tree edit distance21  We show that. Proof by induction: R(F,G) By inductive assumption By (*) and (**) We know G<F

22. Tree edit distance22  We show that. Proof by induction: R(F,G) By inductive assumption By (*) and (**) We know G<F An O((nm) 3/2 ) = O(n 3 ) Upper Bound

23. Tree edit distance23 An O((nm) 3/2 ) = O(n 3 ) Upper Bound  We show that. Proof by induction: R(F,G) By inductive assumption By (*) and (**) We know G<F

24. Tree edit distance 24 An O( ) Bound  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(m 3 ) bound. At most n/m such trees so total = n/m. O(m 3 ) = O(nm 2 ). n m K5K5 K3K3 K4K4 F K2K2 K1K1 G

25. Tree edit distance25 A Matching Lower Bound for all decomposition strategy algorithms

26. Tree edit distance26 A Matching Lower Bound for all decomposition strategy algorithms  An  (nm 2 ) lower bound: F G

27. Tree edit distance27 A Matching Lower Bound for all decomposition strategy algorithms  An  (nm 2 ) 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:  min{|F v |,|G w |} new subproblems (F v is the tree rooted at v).  Summing over all such v,w:

28. Tree edit distance28 A Matching Lower Bound for all decomposition strategy algorithms  An lower bound  A careful counting argument on: F G

29. Tree edit distance29 Thank you!