1. A New Top-down Algorithm for Tree Inclusion Dr. Yangjun Chen Dept. Applied Computer Science, University of Winnipeg 515 Portage Ave. Winnipeg, Manitoba, Canada R3B 2E9

2. Outline Motivation Basic algorithm for tree inclusion problem -Definition -Algorithm description Improvements Summary

3. Given two ordered labeled trees P and T, called the pattern and the target, respectively. An interesting problem is: Can we obtain pattern P by deleting some nodes from target T? That is, is there a sequence v 1,..., v k of nodes such that for T 0 = T and T i+1 = delete(T i, v i +1 ) for i = 0,..., k - 1, we have T k = P. If this is the case, we say, P is included in T, T contains P, or say, T covers P. Motivation a b d ef T:T: cb d ef T:T: a delete(T, c)

4. Motivation s vp vnadv “reads”“book” s np vp detnvnpadv “The”“student”“reads”detadjn “the”“interesting”“book” “again and again” Linguistic analysis

5. Definition 1 Let F and G be labeled ordered forests. We define an ordered embedding (, G, F) as an injective function  : V(G)  V(F) such that for all nodes v, u  V(G), i)label(v) = label((v)); (label preservation condition) ii)v is an ancestor of u iff (v) is an ancestor of (u);(ancestor condition) iii)v is to the left of u iff (v) is to the left of (u); (Sibling condition) Tree inclusion algorithm Definition a b b G:G: a d b ebb F:F:

6. Algorithm Tree inclusion algorithm 1.Let T = (k  1) be a tree and G = (l  1) be a forest. We handle G as a tree P =, where p v represents a virtual node, matching any node in T. 2.Consider a node in P with children v 1,..., v j. We use a pair (i  j) to represent an ordered forest containing the first i subtrees of v:. Then, represents the first j trees in G. P:P: v1v1 vivi vkvk … … v

7. Algorithm Tree inclusion algorithm 3.In addition, h(v) represents the height of v in a tree; and (v) represents a link from v in P to the leaf node on the left-most path in P[v]. Let v’ be a leaf node in P. We denote by  -1 (v’) a set of nodes x such that for each v  x (v) = v’. -1(v 3 ) = {v 1, v 2, v 3 } v1v1 v5v5 v4v4 v2v2 v3v3 (v1)(v1) (v2)(v2) P:P:

8. The tree inclusion checking is done by calling two functions recursively: top-down(T, G), bottom-up(T’, G), where T is a tree, and T’ and G are two forests. Algorithm Tree inclusion algorithm Each of the two functions returns a pair with v being p v or a node on the left-most path in P 1. T = T’ = G =

9. Function: top-down(T, G) Tree inclusion algorithm Case 1: G = ; or G = (l > 1), but |T |  |P 1 | + |P 2 |. In this case, we try to find a pair such that T contains the first i subtrees of v, where v = p v, or v   -1 (v’) and v’ is the leaf node on the left-most path in P 1. T:T: G:G: P1P1 pvpv G:G: …… P1P1 P2P2 pvpv |T|  |P 1 | + |P 2 |. T:T: t t PlPl p1p1 In top-down(T, G), two cases will be handled. p1p1

10. Function: top-down(T, G) Tree inclusion algorithm i)If t is a leaf node, we will check whether label(t) = label((p 1 )), where p 1 is the root of P 1. If it is the case, return. Otherwise, return. T = : G:G: P1P1 pvpv G:G: …… P1P1 P2P2 pvpv |T |  |P 1 | + |P 2 |. t t T = : PlPl case 1:

11. Function: top-down(T, G) Tree inclusion algorithm ii)If |T| < |P 1 | or height(t) < height(p 1 ), we will make a recursive call top-down(T, ), where be a forest of the subtrees of p 1. The return value of top-down(T, ) is used as the return value of top-down(T, G) |T | < |P 1 | G:G: …… pvpv p1p1 … … P 11 P1jP1j P1iP1i T:T: t PlPl case 1:

12. Function: top-down(T, G) Tree inclusion algorithm iii)If |T|  |P 1 | (but |T |  |P1| + |P2|) and height(t)  height(p 1 ), two cases need to be considered: label(t) = label(p 1 ). Call bottom-up(, ). label(t)  label(p 1 ). Call bottom-up(, ). p1p1 … … P 11 P1jP1j P1iP1i t … … T1T1 TkTk TiTi label(t) = label(p 1 ) p1p1 … … P 11 P1jP1j P1iP1i t … … T1T1 TkTk TiTi label(t)  label(p 1 ) case 1:

13. In both sub-cases, assume that the return value is. A further checking needs to be conducted: Function: top-down(T, G) Tree inclusion algorithm If label(t) = label(v) and i = the outdegree of v, the return value should be. Otherwise, the return value is the same as. T:T: t P1:P1: p1p1 v or label(t)  label(v) label(t) = label(v) case 1:

14. Function: top-down(T, G) Tree inclusion algorithm Case 2: G = (l > 1), and |T| > |P 1 | + |P 2 |. In this case, we will call bottom-up(, G). Assume that the return value is. The following checkings will be continually conducted. Case 1: G = ; or G = (l > 1), but |T |  |P 1 | + |P 2 |. G:G: …… P1P1 P2P2 pvpv |T | > |P 1 | + |P 2 | PlPl T:T: …… T1T1 T2T2 t TkTk

15. Function: top-down(T, G) Tree inclusion algorithm iv)If v = p 1 ’s parent, the return value is the same as. v)If v  p 1 ’s parent, check whether label(t) = label(v)) and i = the outdegree of v. If so, the return value will be changed to. Otherwise, the return value remains. Case 2: G = (l > 1), and |T | > |P 1 | + |P 2 |. In this case, we will call bottom-up(, G). Assume that the return value is. The following checkings will be continually conducted. G:G: … … P1P1 P2P2 pvpv v = p 1 ’s parent = p v …… P1P1 P2P2 pvpv v  p 1 ’s parent v PiPi PlPl PlPl

16. Function: bottom-up(T’, G) Tree inclusion algorithm bottom-up(T’, G) is designed to handle the case that both T’ and G are forests. Let T’ = and G =. In bottom-up(T’, G), we will make a series of calls top-down(T l, ), where l = 1,..., k, j 1 = 0, and j 1  j 2 ...  j h  q (for some h  k), controlled as follows. … … PiPi … … TkTk T1T1 TiTi P1P1 PqPq T2T2 … top-down(T l, ) T’: G:G:

17. Function: bottom-up(T’, G) Tree inclusion algorithm 1.Two index variables l, j are used to scan T 1,..., T k and P 1,..., P q, respectively. 2.Let be the return value of top-down(T l, ). If v l = p j ’s parent, set j to be j + i l - 1. Otherwise, j is not changed. Set l to be l + 1. Go to (2). 3.The loop terminates when all T l ’s or all P j ’s are examined. bottom-up(T’, G) is designed to handle the case that both T’ and G are forests. Let T’ = and G =. In bottom-up(T’, G), we will make a series of calls top-down(T l, ), where l = 1,..., k, j 1 = 0, and j 1  j 2 ...  j h  q (for some h  k), controlled as follows.

18. Function: bottom-up(T’, G) Tree inclusion algorithm If j > 0 when the loop terminates, bottom-up(T’, G) returns. … … PiPi … … TkTk T1T1 TiTi P1P1 PqPq T2T2 … PjPj

19. Function: bottom-up(T’, G) Tree inclusion algorithm i)Let,,..., be the respective return values of top-down(T 1, ), top-down(T 2, ),...... top-down(T k, ). Since j = 0, each v l   -1 (v’) (l = 1,..., k). Otherwise, j = 0. In this case, we will continue to searching for a pair such that T’ contains the first i subtrees of v, where v   -1 (v’) and v’ is the leaf node on the left-most path in P 1, as described below. If j > 0 when the loop terminates, bottom-up(T’, G) returns. P1P1 v1v1 v2v2 vkvk …

20. ii)If each i l = 0, return, where  is considered to be a descendant of any node in G. Otherwise, find the first v g with children w 1,..., w h such that v g is not a descendant of any other v j, and i g > 0. Call bottom-up(, ). Function: bottom-up(T’, G) Tree inclusion algorithm i)Let,..., be the return values of top-down(T 1, ),..., top-down(T k, ), respectively. Since j = 0, each v l   -1 (v’) (l = 1,..., k). Let be its return value. If y = v g, then the return value of bottom-up(T’, G) is set to be. Otherwise, the return value is. … … T g+1 T1T1 TgTg T2T2 P1P1 v1v1 vgvg vkvk TkTk … … igig

21. Further improvements Tree inclusion algorithm In the case j = 0: Let,..., be the return values of top-down(T 1, ),..., top-down(T k, ). We will find the first v g such that it is not a descendant of any other v j and i g > 0. Then, bottom-up(, ). is invoked. This shows that all the return values except are not used in the subsequent computation. Thus, the work for looking for such values should be avoided. … … T g+1 T1T1 TgTg T2T2 P1P1 v1v1 vgvg vkvk TkTk … …

22. Let be the return value of top-down(T j, ) such that i j > 0 and v j is p 1 or a descendant of p 1. Then, during the execution of top-down(T j+1, ), once we have detected that it can only produce a return value with v j+1 being a descendant of v j, we should stop the corresponding computation immediately since this return value will not be used in the subsequent searching. For this purpose, we rearrange top-down(T j+1, ) to top-down(T j+1,, v j ) with v j being used to transfer information, called a controlling-node. Further improvements Tree inclusion algorithm Assume that in the execution of top-down(T j+1,, v j ), we have the following function calls: top-down(T j+1,1,, u 1 ) returns, top-down(T j+1,2,, u 2 ) returns, With all u j ’s being a proper descendant of v j. Then the bottom-up function call with some u i as a controlling node should not be conducted. … bottom-up(,, u i ).

23. Summary An efficient method for tree inclusion problem -O|T|min{D P, |leaves(P)|}) time and -O(|T| + |P|) space where D P – the height of P, and Future work -adapt the algorithm to a data stream environment -adapt the algorithm to an indexing environment leaves(P) - set of the leaf nodes of P.

24. Thank you.