Converting Disjunctive Data to Disjunctive Graphs

1
Converting Disjunctive Data to Disjunctive Graphs

• Lars Olson
• Data Extraction Group
• Funded by NSF

2
Introduction

• Disjunctive databases
• Needed to represent disjunctive data
• Queries are CoNP-complete in general Imielinski
  and Vadaparty, 1989
• Transitive closure in disjunctive graphs
• CoNP-complete in general
• Polynomial time, under certain circumstances
  Lobo et. al, 1995

3
The Problem

• How do we convert the data into a disjunctive
  graph?
• What is the complexity of the conversion?
• Time
• Space / Memory

4
Implementation

• XML data repository
• Shore / Niagara (Univ. of Wisconsin)
• Xerces XML parser (Apache.org)
• How do we represent a disjunctive database in
  storage?
• Needs to be easy to convert to disjunctive graph
• Needs to minimize the changes to the DTD and
  thus, the existing data

5
XML ? Graph Conversion
doc

• XML ? DOM tree

Node
ltdocgt ltNode nameAgt ltEdgeTo
refB/gt lt/Nodegt ltNode nameBgtlt/Nodegt ... lt/
docgt
Node
EdgeTo
A
B
B

• Use primary key to distinguish doc?Node edges
• Use foreign key to perform join (EdgeTo.ref
  Node.name)

6
Disjunctions in XML, 1st Case
ltNode nameAgt ltEdgeTo refB/gt ltDisjgt ltEdge
To refC/gt ltEdgeTo refD/gt lt/Disjgt lt/Nodegt
...
B
A
C
D
but how do we represent a disjunctive tail?
7
Disjunctions in XML, 1st Case
ltNode nameAgt ltEdgeTo refB/gt ltDisjgt ltEdge
To refC/gt ltEdgeTo refD/gt lt/Disjgt lt/Nodegt
ltDisjgt ltNode nameEgt ltEdgeTo
refG/gt ltEdgeTo refH/gt lt/Nodegt ltNode
nameFgt ltEdgeTo refG/gt ltEdgeTo
refH/gt lt/Nodegt lt/Disjgt ...
or
8
Disjunctions in XML, 2nd Case
ltDisjgt ltTailgt ltNode nameE/gt ltNode
nameF/gt lt/Tailgt ltHeadgt ltEdgeTo
refG/gt ltEdgeTo refH/gt lt/Headgt lt/Disjgt ...
E
G
F
H
What if the disjunction isnt the full
cross-product?
9
Disjunctions in XML, 3rd Case
ltDisjgt ltTailgt ltNode nameI/gt lt/Tailgt ltHeadgt
ltEdgeTo refK/gt lt/Headgt ltTailgt ltNode
nameJ/gt lt/Tailgt ltHeadgt ltEdgeTo
refK/gt ltEdgeTo refL/gt lt/Headgt lt/Disjgt ...
10
Time and Space Complexity

• n of nodes in DOM tree
• counts edges as well
• not necessarily proportional to of values in
  the database
• Ordinary XML traverse tree, add edges.
  Distinguish records with primary keys, add edges
  for foreign keys. O(n) time, O(n) space.

11
Time and Space Complexity

• ltDisjgt same, except only one edge to all
  children. O(n), O(n).
• ltDisjgt with ltTailgt and ltHeadgt traverse tree,
  add ltTailgt and ltHeadgt elements to a list, add one
  edge, repeat for each Tail/Head pair. O(n), O(n).

12
Summary

• We need to introduce new XML constructs
• ltDisjgt
• Helper constructs ltTailgt and ltHeadgt
• Three cases
• simple tail, compound head
• full cross-product
• partial cross-product
• Time and space requirements consistent with the
  transitive closure algorithm

13
Future Work

• Solving path queries
• Adding XML constructs for more complicated
  disjunctions
• e.g. Tail (A or B), Head ((C and D) or E)
• Determining frequency of disjunctive data in
  real-world data
• Developing a normal form for disjunctive XML
• Minimize redundancy
• Minimize disjunctive tails