Tree Tuples II

1
Tree Tuples II

• Lecturer Dr. Pavle Mogin

2
Plan for the Tree Tuples II Topic

• Maximal tree tuples
• Tree subsumption
• Constructing a document tree using a set of
  maximal tree tuples
• Reading
• M. Arenas, L. Lipkin A Normal Form for XML, in
  Proceedings of ACM PODS 02

3
An XML Tree as A Set of Tuples

• To represent an XML tree as a set of tree tuples,
  we need to introduce the notion of subsumption
  that disregards the tree ordering
• Given two XML trees T1 (V1, lab1, ele1, att1,
  root1) and T2 (V2, lab2, ele2, att2, root2), T1
  is subsumed by T2, denoted T1 ? T2 if
• V1 ? V2
• root1 root2
• lab2V1 lab1
• att2V1?Att att1
• For all v?V1, ele1(v) is a sublist of a
  permutation of ele2(v)

4
Comments on Subsumption

• lab is a set of pairs (v, e)
• Projection of lab onto the set of vertices V ,
  denoted labV is a subset of lab pairs having
  elements from V as the first component
• In the definition we request that each vertex v
  from V1 maps into the same element e as does the
  vertex v from V2
• att is a set of pairs ((v, a), s)
• Projection of att onto V?Att, denoted attV?Att
  is a subset of att pairs having (v,a) from V?Att
  as the first component
• In the definition we request that each pair (v,a)
  from V1?Att maps into the same string s as does
  the pair (v,a) from V2?Att
• Also, we request that if v is in V1, and (v, a)
  is defined in T2, it is also defined in T1

5
XML Tree as a Set of Tuples (tuplesD(T))

• Given a DTD D and an XML tree T such that T ? D,
  tuplesD(T ) is defined to be the set of tree
  tuples t such that for each t, treeD(t) is
  subsumed by T
• t??(D) ? treeD(t) ? T

6
A Class Document Tree
CLASS
1
_at_name
COMP442
LECTURER
2
_at_lecId
STUDENT
STUDENT
4
Pavle
8
ASSIGNMENT
111
NAME
_at_no_of_pts
_at_origin
_at_origin
_at_no_of_pts
9
6
5
7
NAME
Bad Student
Domestic
_at_no
20.0
International
Ahmed
_at_no
10.0
0.0
10.0
1
2
7
t1 Tuple as An XML Tree
CLASS
1
_at_name
COMP442
LECTURER
STUDENT
4
2
ASSIGNMENT
Pavle
_at_lecid
_at_no_of_pts
_at_origin
6
5
NAME
111
20.0
International
Ahmed
_at_no
10.0
1
8
t2 Tuple as An XML Tree
CLASS
1
_at_name
_at_no_of_pts
COMP442
31
AUDITOR
LECTURER
STUDENT
4
3
2
ASSIGNMENT
Pavle
_at_lecid
Sharon
_at_no_of_pts
_at_origin
6
5
NAME
111
20.0
International
Ahmed
_at_no
10.0
2
9
Functions lab of the Class Tree

• V 1, 2, 4, 5, 6, 7, 8, 9
• root 1
• lab (1, CLASS), (2, LECTURER), (4,
  STUDENT), (5, NAME), (6, ASSIGNMENT),
• (7, ASSIGNMENT), (8, STUDENT), (9, NAME)

10
Functions ele and att of the Class Tree

• ele(1) (2, 3, 4, 8), ele(2) Pavle,
  ele(4) (5, 6, 7), ele(5) Ahmed, ele(6)
  10, ele(7) 10, ele(8) (9), ele(9) Bad
  Student
• att ((1, _at_name), COMP442), ((2, _at_lecid),
  111), ((4, _at_no_of_pts), 20.0), ((4, _at_origin),
  International), ((6, _at_no), 1), ((7, _at_no), 2),
  ((8, _at_no_of_pts), 0.0), ((8, _at_origin),
  Domestic)

11
The Class Tree and Tuples t1, t2

• Now, we check whether T1 ? T2 holds for
• T1?treeD(t1), treeD(t2) and the Class tree as
  T2
• treeD(t1) is subsumed by the Class tree
• treeD(t2) is not subsumed by the Class tree,
  since
• The condition V1 ? V2 is violated (the vertex 3
  is in V1 but not in V2)
• The condition for all v?V1, ele1(v) is a sublist
  of a permutation of ele2(v) is violated
  (ele1(1) (2, 3, 4), but ele2(1) (2, 4,
  8), hence ele1(1) is not a sublist of ele2(1))
• The condition att2V1?Att att1 is violated (the
  element
• ((1, _at_no_of_std), 31) is in att1 but not in
  att2)

12
t3 Tuple as An XML Tree
CLASS
1
_at_name
COMP442
LECTURER
STUDENT
8
2
Pondy
_at_lecid
_at_no_of_pts
_at_origin
222
0.0
Domestic
13
The Class Tree and the Tuple t3

• treeD(t3) is not subsumed by the Class tree,
  since
• The condition for all v?V1, ele1(v) is a sublist
  of a permutation of ele2(v) is violated
• Namely, both trees contain the vertex 2, but
  ele2(2) Pavle and ele1(2) Pondy, and Pondy
  is not a sublist of Pavle
• The condition att2V1?Att att1 is also violated
  
• Namely, (2, lecid) is defined in both T1 and T2,
  but (2, lecid) 222 in T2 and (2, lecid) 111
  in T1

14
Subsumption Example

• Consider the XML tree given on the following
  slide and check whether the Class tree T2
  subsumes the next tree T1

15
XML Tree T1
CLASS
1
_at_name
COMP442

• V1 1
• root1 1
• lab1 (1, CLASS)
• att1 ((1, _at_name), COMP442)
• ele(1) ? (empty list)

V1 ? V2 root1 root2 lab1 lab2 V1 att1 att2
V1xAtt ? (empty list) is a sublist of each list
16
A Question for You

• How many tree tuples can be inferred from the
  Class XML tree T2
• 3 tree tuples,
• 33 tree tuples,
• 125 tuples

17
Maximal Tree Tuples

• Many tree tuples in a ?(D) associate a null value
  to a not null path in a tree T that is compatible
  with a DTD D (T ? D)
• Also, many tree tuples with nulls are subsumed by
  such tree tuples from ?(D), which are also
  subsumed by T
• We are interested only for those tree tuples
  that
• Belong to ?(D),
• Are subsumed by the tree T, and
• Are not subsumed by any other tree tuple that is
  subsumed by T
• Such tree tuples carry maximal information about
  the tree T (since they contain minimal number of
  null values)
• The set of tree tuples that satisfy the three
  conditions above will be denoted by tuplesD(T )

18
Defining tuplesD(T)

• Given a DTD D, an XML treeT, and a set of tuples
  tuplesD(T ), where T ? D
• All tuples in tuplesD(T ) satisfy the following
  conditions
• t??(D),
• treeD(t ) ? T,
• ?t1, t2 ?tuplesD(T )(treeD(t1 ) ? treeD(t2 ) ? t1
  t2 )

19
tuplesD(T) Example

• Consider the Class DTD, the Class XML tree, and
  the following tuples
• tx (1, COMP442, 31, ?, ?, ?, ?, ?, ?, ?, ?, ?,
  ?, ?, ?, ?),
• ty (1, COMP442, 31, 2, 111, Pavle, ?, ?, 4,
  20.0, International, 5, Ahmed, 6, 1, 10.0),
• tz (1, COMP442, 31, ?, ?, ?, ?, ?, 4, 20.0,
  International, 5, Ahmed, 6, 1, 10.0)
• ty , tx, tz ? ?(D )
• ty ?tuplesD (T )
• tx?tuplesD (T ), since treeD(tx ) ? treeD(ty )
• tz?tuplesD (T ), since treeD(tz ) ? treeD(ty )

20
Relationship Between T and tuplesD(T)

• A tree made of a tree tuple t in tuplesD(T ) can
  be viewed as being extracted from T by taking a
  branch from the root to a leaf in the tree T for
  each path from the root to a leaf in paths(T )
  that ends with a S or an attribute
• A tuple t in tuplesD(T ) is allowed to have null
  values only for paths in paths(D) \ paths(T )

21
A Question for You

• How many maximal tree tuples can be inferred from
  the Class XML tree T2
• 3 tree tuples,
• 6 tree tuples,
• 9 tree tuples,
• other

22
Summary

• A tree tuple t is a function that assigns a value
  in Vert ? Str ? ? to each path in D
• A tree tuple t can be represented as an XML tree
  T
• An XML tree T2 can subsume another XML tree T1
• Tree tuples subsumed by the document tree only
  are called maximal tree
• A document tree can be viewed as constructed
  using its maximal tuples from tuplesD(T ) (the
  set of maximal tree tuples)