1. A Linear-Space Top-down Algorithm for Tree Inclusion Problem
Prof. 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. Motivation
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 v1 , ..., vk of nodes
such that for
T0 = T and
Ti+1 = delete(Ti, vi +1) for i = 0, ..., k - 1,
we have Tk = P. If this is the case, we say, P is included in T, T contains P,
or say, T covers P. a b d e f T: c
delete(T, c) b d e f T: a
By “ordered trees”, we mean those trees, in which the order of siblings is significant.

4. Motivation Linguistic analysis s vp v n adv “reads” “book” s np vp det
“The”
“student”
“reads”
adj
“the”
“interesting”
“book”
“again and again”

5. Tree inclusion algorithm
Definition
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) G: F: a a b b d b e b b
By “ordered trees”, we mean those trees, in which the order of siblings is significant.

6. Tree inclusion algorithm
Let T = <t; T1, ..., Tk> (k  1) be a tree and G = <P1 , ..., Pl>
(l  1) be a forest. We handle G as a tree P = <pv; P1, ..., Pl>,
where pv represents a virtual node, matching any node in T.
Consider a node v in P with children v1, ..., vj. We use a pair <i, v>
(i  j) to represent an ordered forest containing the first i subtrees
of v: <P[v1], ..., P[vi]>. Then, <j, pv> represents the first j trees
in G. P: v v1 vi vk … …
<i, v>
referred to as a left corner

7. Tree inclusion algorithm
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]. P:
(v1)
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(v3) = {v1, v2, v3} v1
(v2) v2 v5 v3 v4

8. Tree inclusion algorithm
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. G: … P1 P2 pv Pl v
T = <t; T1, ..., Tk>
T’ = <T1’, ..., Tk’>
G = <P1, ..., PL>
Each of the two functions returns a pair <i, v> with v being pv or a node on
the left-most path in P1.
Each node in T will be associated with a data structure, (t).
- Initially, (t) is set .
- By each call of top-down(<t; T1, ..., Tk>, G), (t) is changed to <i, v>,
where <i, v> is the return value of top-down(<t; T1, ..., Tk>, G).
((t) is mainly used in the execution of bottom-up(T’, G) to avoid
repeated computation.)

9. Tree inclusion algorithm
Function: top-down(T, G)
In top-down(T, G), two cases will be handled.
Case 1: G = <P1>; or G = <P1, ..., Pl> (l > 1), but |T|  |P1| + |P2|.
In this case, we try to find a pair <i, v> such that T contains the first i
subtrees of v, where v = pv , or v  -1(v’) and v’ is the leaf node on the
left-most path in P1. T: t G: pv p1 P1 G: pv T: t p1 P2 … Pl
|T|  |P1| + |P2|. P1

10. Tree inclusion algorithm
Function: top-down(T, G)
case 1:
i) If t is a leaf node, we will check whether label(t) = label((p1)), where p1
is the root of P1. If it is the case, return <1, parent of (p1)>.
Otherwise, return <0, parent of (p1)>.
T = <t; T1, ..., Tk>: G: pv t P1 G: pv
T = <t; T1, ..., Tk>: t … Pl
|T|  |P1| + |P2|. P1 P2

11. Tree inclusion algorithm
Function: top-down(T, G)
case 1:
If |T| < |P1| or height(t) < height(p1), we will make a recursive call
top-down(T , <P11, ..., P1j>), where <P11, ..., P1j> be a forest of
the subtrees of p1. The return value of top-down(T , <P11, ..., P1j>)
is used as the return value of top-down(T, G) pv T: G: t
|T| < |P1| p1 …
P11
P1j
P1i … Pl

12. Tree inclusion algorithm
Function: top-down(T, G)
case 1:
If |T|  |P1| (but |T |  |P1| + |P2|) and height(t)  height(p1), two cases
need to be considered:
label(t) = label(p1). Call bottom-up(<T1, ..., Tk>, <P11, ..., P1j>). p1 …
P11
P1j
P1i t T1 Tk Ti
label(t) = label(p1)
label(t)  label(p1). Call bottom-up(<T1, ..., Tk>, <P1>). t p1
label(t)  label(p1) T1 Ti Tk
P11
P1i
P1j … … … …

13. Tree inclusion algorithm
Function: top-down(T, G)
case 1:
In both sub-cases, assume that the return value of the corresponding
bottom-up function is <i, v>. A further checking needs to be conducted:
If label(t) = label(v) and i = the outdegree of v, the return value should
be <1, v’s parent>.
Otherwise, the return value is the same as <i, v>.
P1: p1 v T:
label(t) = label(v) t
or label(t)  label(v)

14. Tree inclusion algorithm
Function: top-down(T, G)
Case 1: G = <P1>; or G = <P1, ..., Pl> (l > 1), but |T|  |P1| + |P2|.
Case 2: G = <P1, ..., Pl> (l > 1), and |T| > |P1| + |P2|. In this case, we
will call bottom-up(<T1, ..., Tk>, G). Assume that the return value is <i, v>.
The following checkings will be continually conducted. T: t G: pv
|T| > |P1| + |P2| Tk … Pl … T1 T2 P1 P2

15. Tree inclusion algorithm
Function: top-down(T, G)
Case 2: G = <P1, ..., Pl> (l > 1), and |T| > |P1| + |P2|. In this case, we
will call bottom-up(<T1, ..., Tk>, G).
Assume that the return value is <i, v>. The following checkings will be
continually conducted.
iv) If v = p1’s parent, the return value is the same as <i, v>.
v) If v  p1’s parent, check whether label(t) = label(v)) and
i = the outdegree of v. If so, the return value will be changed to
<1, v’s parent>. Otherwise, the return value remains <i, v>.
v = p1’s parent = pv
v  p1’s parent G: pv pv … … Pl v … Pl P1 P2 Pi P1 P2

16. Tree inclusion algorithm
Function: bottom-up(T’, G)
bottom-up(T’, G) is designed to handle the case that both T’ and G are
forests. Let T’ = <T1, ..., Tk> and G = <P1, ..., Pq>. In bottom-up(T’, G),
we will make a series of calls top-down(Tl, <Pjl, ..., Pq>), where l = 1, ..., k,
j1 = 0, and j1  j2  ...  jh  q (for some h  k), controlled as follows.
T’: G: T1 T2 Ti Tk P1 Pi Pq … … … …
Before each call of top-down(Tl, <Pjl, ..., Pq>),
(tl) will be checked to determine whether the call
will be conducted.
top-down(Tl, <Pjl, ..., Pq>)

17. Tree inclusion algorithm
Function: bottom-up(T’, G)
bottom-up(T’, G) is designed to handle the case that both T’ and G are
forests. Let T’ = <T1, ..., Tk> and G = <P1, ..., Pq>. In bottom-up(T’, G),
we will make a series of calls top-down(Tl, <Pjl, ..., Pq>), where l = 1, ..., k,
j1 = 0, and j1  j2  ...  jh  q (for some h  k), controlled as follows.
Two index variables l, j are used to scan T1, ..., Tk and P1, ..., Pq,
respectively.
Let <il, vl> be the return value of top-down(Tl, <Pj, ..., Pq>). If vl = pj’s
parent, set j to be j + il - 1. Otherwise, j is not changed. Set l to be l + 1.
Go to (2).
The loop terminates when all Tl’s or all Pj’s are examined.
In the above process, (tl) will be checked before each top-down(Tl, <Pj, ..., Pq>)
to determine whether the call will be conducted

18. Tree inclusion algorithm
Function: bottom-up(T’, G)
If j > 0 when the loop terminates, bottom-up(T’, G) returns
<j, p1’s parent>. T1 T2 Ti Tk P1 Pi Pj Pq … … … …

19. Tree inclusion algorithm
Function: bottom-up(T’, G)
If j > 0 when the loop terminates, bottom-up(T’, G) returns
<j, p1’s parent>.
Otherwise, j = 0. In this case, we will continue to search for a pair
<i, v> 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 P1, as described below.
Let <i1, v1>, <i2, v2>, ..., <ik, vk> be the respective return values of
top-down(T1, <P1, ..., Pq>),
top-down(T2, <P1, ..., Pq>),
top-down(Tk, <P1, ..., Pq>).
Since j = 0, each vl  -1(v’) (l = 1, ..., k). P1 P2 v2 … v1 … vk

20. Tree inclusion algorithm
Function: bottom-up(T’, G)
i) Let <i1, v1>, ..., <ik, vk> be the return values of top-down(T1, <P1, ..., Pq>),
..., top-down(Tk, <P1, ..., Pq>), respectively. Since j = 0, each vl  -1(v’)
(l = 1, ..., k).
ii) If each il = 0, return <0, >, where  is considered to be a descendant of
any node in G. Otherwise, find the first vg with children w1, ..., wh such that
vg is not a descendant of any other vj, and ig > 0. P1 T1 T2 Tg
Tg+1 Tk vg … … … … v1 vk ig

21. Tree inclusion algorithm
Function: bottom-up(T’, G)
iii) Check <P[wig+1], ..., P[wh]>) against <Tg+1, ..., Tk>. This can be done by a
series of calls top-down(Tl, <P[wjl] ..., P[wh]>), where l = g + 1, …, k, and
j1  j2  ...  jh  h (for some h  k), as illustrated below.
<Tg+1, ..., Tk>:
<P[wig+1], ..., P[wh]>:
Tg+1
Tg+2
Tg+i Tk
P[wig+1]
P[wig+i]
P[wh] … … … …
Let <x, y> be its return value. If y = vg, then the return value of
bottom-up(T’, G) is set to be <ig + x, vg>.
Otherwise, the return value is <ig, vg>.

22. Tree inclusion algorithm
Further improvements
In the case j = 0:
Let <i1, v1>, ..., <ik, vk> be the return values of top-down(T1, <P1, ..., Pq>),
..., top-down(Tk, <P1, ..., Pq>). We will find the first vg such that it is not a
descendant of any other vj and ig > 0. Then,
<Tg+1, ..., Tk> will be checked against <P[wig+1], ..., P[wh]>.
This shows that all the return values except <ig, vg> are not used
in the subsequent computation. Thus, the work for looking for such values
should be avoided. P1 T1 T2 Tg
Tg+1 Tk vg … … … … v1 vk
Any of these subtrees should not be checked against <P1, ..., Pq>).

23. Tree inclusion algorithm
Further improvements
Let <ij, vj> be the return value of top-down(Tj, <P1, ..., Pq>) such that ij > 0 and vj is p1 or a
descendant of p1. Then, during the execution of top-down(Tj+1, <P1, ..., Pq>), once we have
detected that it can only produce a return value <ij+1, vj+1> with vj+1 being a descendant of vj, 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(Tj+1, <P1, ..., Pq>) to
top-down(Tj+1, <P1, ..., Pq>, vj) with vj being used to transfer information, called a
controlling-node.
Assume that in the execution of top-down(Tj+1, <P1, ..., Pq>, vj), we have the following
function calls:
top-down(Tj+1, <P1, ..., Pq>, u1) returns <a1, u1>,
top-down(Tj+2, <P1, ..., Pq>, u2) returns <a1, u2>,
… …
with all uj’s being a proper descendant of vj. Then the bottom-up function call with
some ui as a controlling node should not be conducted.
bottom-up(<Tj+i , ... >, <… …>, ui ).

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

25. Thank you.