Multivalued dependencies and a 4NF for XML

1
Multivalued dependencies and a 4NF for XML

• Millist W. Vincent and Jixue Liu
• School of Computer and Information Science
• University of South Australia

2
Overview

• Motivation.
• Basic Definitions.
• MVDs in XML.
• Redundancy in XML.
• Hierarchical XMVDs.
• A 4NF for XML.
• Conclusions and future research.

3
Motivation

• Large amounts of data stored in XML format.
• How to normalize XML documents to avoid
  redundancy?
• FDs and MVDs are the basis for normalization in
  relations.
• Recent research has defined functional
  dependencies and a redundancy free normal form
  for XML.
• How to extend the approach to multivalued
  dependencies in XML?

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Multivalued Dependencies in XML

• Modeling XML Documents
• ltrootgt
• ltDeptgt
• ltSection Empe1 Projectj1/gt
• lt/Deptgt
• ltDeptgt
• ltSection Empe3 Projectj2/gt
• lt/Deptgt
• lt/rootgt

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Basic Definitions

• Path is a sequence of labels from the root, e.g.,
• root.Dept.Section
• Path intersection e.g.,
• root.Dept.Section.Emp ? root.Dept.Section.Project
  root.Dept.Section

E
root
vr
E
E
Dept
v2
Dept
v1
E
E
Section
v3
Section
v7
A
Emp
v4
A
A
A
Project
v6
Emp
v8
Project
v10
e1
j1
e3
j2
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Basic Definitions (cont.)

• Path instances of a path. For example, a path
  instance of
• root.Dept.Section is vr. v1. v3
• Function Paths(q) is the set of all path
  instances of path q.
• Function N(root.Dept)v1, v2
• Function Nodes(v1, root.Dept.Section)v3
• Function val(v3)v3, val(v4)e1.

E
root
vr
E
E
Dept
v2
Dept
v1
E
E
Section
v3
Section
v7
A
Emp
v4
A
A
A
Project
v6
Emp
v8
Project
v10
e1
j1
e3
j2
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XMVD Definition

• Let P be a consistent set of paths and let T
  be an XML tree that conforms to P and is
  complete. An XMVD is a statement of the form p
  \rightarrow\rightarrow
• q r where p, q and r are paths in P.
  T satisfies p \rightarrow\rightarrow
• q r if whenever there exists two distinct
  paths path instances v _1. \cdots.
• v _n and
• w_1. \cdots. w _n in Paths(q) such that
• (i) val(v_n) \neq val(w _n)
• (ii) there exists two nodes z_1, z_2, where
  z_1 \in
• Nodes(x_1_1, r) and z_2 \in Nodes(y_1_1,
  r) such that val(z_1)
• \neq val(z_2)
• (iii) there exists two nodes z_3 and z_4,
  where
• z_3 \in Nodes(x_1_1_1, p) and z_4 \in
  Nodes(y_1_1_1, p), such that val(z_3)
  val(z_4)
• then
• (a) there exists a path v'_1. \cdots. v'_n in
  Paths(q) such that
• val(v'_n) val(v_n) and there exists a node
  z'_1 in

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XMVD Example

• root.Id.Course ??
• root.Id.Id.Teacher.S root.Id.Id.Text.S
• is satisfied.

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XMVD Definition

• How do we know the definition of an XMVD is
  correct?
• NO definite answer, but in another paper we show
  that for a very wide class of mappings from a
  relation to XML a relation satisfies an MVD if
  the XML document satisfies the XMVD.
• Hence there is a natural correspondence between
  MVDs in relations and XMVDs in XML.

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XMVD Example (contd.)

• root.Project.P??root.Project.Person.Nameroot.Pr
  oject.Part.Pid
• is violated.

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XMVD Example

• root.Id.Course ?? root.Id.Id.Teacher.Sroot.Id.Id.
  Text.S is satisfied.

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XMVD Example (contd.)

• root.Project.Person.Name ??Root.Project.Person.Ski
  llroot.Project.P
• is violated.

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Redundancy in XML

• An XML tree has redundancy if there exists a node
  such that every change of the value of the node
  to a new value results in violating the MVDs.
• Example Consider the tree below and XMVD
  root.Id.Course? ?root.Id.Id.Teacher
  root.Id.Id.Id.Text. All Teacher and Text nodes
  are redundant.

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Hieararchical XMVDs and Keys

• A set of XMVDs is said to be hierarchical if for
  every XMVD p \rightarrow\rightarrow qr
  Parnt(p) is a strict prefix of both q
• and r. A path p is a key if N(p) contains no
  duplicate values.

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4NF for XML

• A set of hierarchical XMVDs and keys is in 4XNF
  if for every p ?? qr if either of the following
  conditions hold(i)q and r are both
  keys(ii)p is a key and q ? r p.
• Theorem If a set of XMVDs and keys is in 4XNF
  then no tree which satisfies the XMVDs has
  redundancy.

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4NF Example
4NF Example

• Example Consider the set of XMVDs
  root.Id.Course? ?root.Id.Id.Teacher
  root.Id.Id.Id.Text. Not in 4XNF.

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4NF Example (contd.)

• Consider the set of XMVDs root.Id.Course ??
• root.Id.Id.Teacher.S root.Id.Id.Text.S and
• root.Id.Id.Teacher is a key and
• root.Id.Id.Text is a key. 4XNF is satisfied.

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Future Work

• Need to extend main theorem to the case of
  arbitrary XMVDs.
• Is 4XNF a necessary condition for no redundancy.
• XFD implication is decidable. Is XMVD implication
  decidable?
• Axiom system for XMVD implication?
• Polynomial time algorithm for XMVD implication?
• Interaction between XFDs and XMVDs.