1. Introduction to XML Algebra
CS561

2. Data Model
data model ~ core data structures and data types supported by DBMS
relational database is a table (set-oriented) data model
XML format is a tree-structured hierarchical model

3. Why Query Algebra (for XML) ?
It is common to translate a query language into an algebra.
First, the algebra is used to give a semantics for the query language.
Second, the algebra is used to support query optimization.

4. XML Algebra History Lore Algebra (August 1999) -- Stanford University
IBM Algebra (September 1999)
--Oracle; IBM; Microsoft Corp
YAT Algebra (May 2000)
AT&T Algebra (June 2000)
--AT&T; Bell Labs
Niagara Algebra (2001)
-- University of Wisconsin -Madison

5. NIAGARA
Title : Following the paths of XML Data: An algebraic framework for XML query evaluation
By : Leonidas Galanis, Efstratios Viglas, David J. DeWitt, Jeffrey. F. Naughton, and David Maier.
Univ. of Wisconsin

6. Outline
Concepts of Niagara Algebra
Operations
Optimization

7. Goals of Niagara Algebra
Be independent of schema information
Query on both structure and content
Generate simple, flexible, yet powerful algebraic expressions
Allow re-use of traditional optimization techniques

8. Example: XML Source Documents
Invoice.xml
<Invoice_Document>
<invoice No = 1>
<account_number>2 </account_number>
<carrier>AT&T</carrier>
<total>$0.25</total>
</invoice>
<invoice>
<account_number>1 </account_number>
<carrier>Sprint</carrier>
<total>$1.20</total>
<total>$0.75</total>
</Invoice_Document>
Customer.xml
<Customer_Document>
<customer>
<account>1 </account>
<name>Tom </name>
</customer >
<account>2 </account>
<name>George </name>
</Customer _Document>

9. XML Data Model and Tree Graph
Example:
Invoice_Document
<Invoice_Document>
<invoice>
<number>2</number>
<carrier>Sprint</carrier>
<total>$0.25</total>
</invoice>
<number>1</number> <carrier>Sprint</carrier> <total>$1.20</total>
</Invoice_Document> …
Invoice
Invoice
number
carrier
number
total
total
carrier 2
AT&T
$0.25 1
$1.20
Sprint
Ordered Tree Graph,
Semi structured Data

10. XML Data Model (for Querying)
SQL: relations in, relation out.
Relational Algebra: relations in, relation out.
XQuery: XML doc in, XML docs out
XML Algebra: ??

11. XML Data Model [GVDNM01] Collection of bags of vertices.
Vertices in a bag have no order.
Example:
Root invoice.xml invoice invoice.account_number
< account_number >
element-content
</ account_number >
<invoice>
Invoice-element-content
</invoice>
[Root“invoice.xml”, invoice, invoice. account_number ]

12. Data Model Bag elements are reachable by path expressions.
Path expression consists of two parts:
An entry point
A relative forward part
Example: account_number:invoice

13. Outline
Concepts of Niagara Algebra
Operations
Optimization

14. Operators Source S , Follow , Expose , Vertex ,
Source S , Select , Join , Rename , Group , Union , Intersection , Difference - , Cartesian Product .

15. Source Operator S Input : a list of documents
Output :a collection of singleton bags
Examples :
S (*) All known XML documents
S (invoice*.xml) All XML documents whose filename match
“invoice*.xml
S (*,schema.dtd) All known XML documents that conform
to schema.dtd

16. Follow operator  Input : a path expression in entry point notation
Functionality : extracts vertices reachable by path expression
Output : a new bag that consists of the extracted vertex + all contents of original bag (in case of unnesting follow)

17. Follow operator (Example*)
{[Root invoice.xml , invoice, invoice.carrier]}
Root invoice.xml invoice invoice.carrier
<carrier>
carrier -element-content
</carrier >
<invoice>
Invoice-element-content
</invoice>
*Unnesting Follow
(carrier:invoice)
Root invoice.xml invoice
<invoice>
Invoice-element-content
</invoice>
{[Root invoice.xml , invoice]}

18. Select operator  Input : a set of bags
Functionality : filters the bags of a collection using a predicate
Output : a set of bags that conform to the predicate
Predicate : Logical operator (,,), or simple qualifications (,,,,,)

19. Select operator (Example)
{[Root invoice.xml , invoice],… }
Root invoice.xml invoice
<invoice>
Invoice-element-content
</invoice>
 invoice.carrier =Sprint
Root invoice.xml invoice
Root invoice.xml invoice
<invoice>
Invoice-element-content
</invoice>
<invoice>
Invoice-element-content
</invoice>
{[Root invoice.xml , invoice], [Root invoice.xml , invoice], ……………}

20. Join operator Input: two collections of bags
Functionality: Joins the two collections based on a predicate
Output: the concatenation of pairs of pages that satisfy the predicate

21. Join operator (Example)
{[Root invoice.xml , invoice, Root customer.xml , customer]}
Root invoice.xml invoice Root customer.xml customer
<invoice>
Invoice-element-content
</invoice>
<customer>
customer-element-content
</customer>
account_number: invoice =number:customer
Root invoice.xml invoice
Root customer.xml customer
<invoice>
Invoice-element-content
</invoice>
<customer>
customer-element-content
</customer>
{[Root invoice.xml , invoice]}
{[Root customer.xml , customer]}

22. Expose operator 
Input: a list of path expressions of vertices to be exposed
Output: a set of bags that contains vertices in the parameter list with the same order

23. Expose operator (Example)
{[Root invoice.xml , invoice.bill_period, invoice.carrier]}
Root invoice.xml invoice. bill_period invoice.carrier
<carrier>
bill_period -element-content
</carrier >
<invoice>
carrier-element-content
</invoice>
(bill_period,carrier)
Root invoice.xml invoice invoice.carrier invoice.bill_period
<invoice>
Invoice-element-content
</invoice>
<invoice>
carrier-element-content
</invoice>
<carrier>
bill_period -element-content
</carrier >
{[Root invoice.xml , invoice, invoice.carrier, invoice.bill_period]}

24. Vertex operator 
Creates the actual XML vertex that will encompass everything created by an expose operator
Example :
 (Customer_invoice)[((account)[invoice.account_number], (inv_total)[invoice.total])]

25. Other operators
Group  : is used for arbitrary grouping of elements based on their values
Aggregate functions can be used with the group operator (i.e. average)
Rename  : Changes entry point annotation of elements of a bag.
Example: (invoice.bill_period,date)

26. Example: XML Source Documents
Invoice.xml
<Invoice_Document>
<invoice>
<account_number>2 </account_number>
<carrier>AT&T</carrier>
<total>$0.25</total>
</invoice>
<account_number>1 </account_number>
<carrier>Sprint</carrier>
<total>$1.20</total>
<total>$0.75</total>
<auditor> maria </auditor>
</Invoice_Document>
Customer.xml
<Customer_Document>
<customer>
<account>1 </account>
<name>Tom </name>
</customer >
<account>2 </account>
<name>George </name>
</Customer _Document>

27. Xquery Example
List account number, customer name, and invoice total for all invoices that have carrier = “Sprint”.
FOR $i in (invoices.xml)//invoice,
$c in (customers.xml)//customer
WHERE $i/carrier = “Sprint” and
$i/account_number= $c/account
RETURN
<Sprint_invoices>
$i/account_number,
$c/name,
$i/total
</Sprint_invoices>

28. Example: Xquery output
<Sprint_Invoice>
<account_number>1 </account_number>
<name>Tom </name>
<total>$1.20</total>
</Sprint_Invoice >

29. Algebra Tree Execution
Account_number name total
Expose (*.account_number , *.name, *.total )
invoice(2) customer(1)
Join (*.invoice.account_number=*.customer.account)
invoice (2)
Select (carrier= “Sprint” )
Invoice (1)
invoice (2)
invoice (3)
customer(1)
customer (2)
Follow (*.invoice)
Follow (*.customer)
Source (Invoices.xml)
Source (cutomers.xml)

30. Outline
Concepts of Niagara Algebra
Operations
Optimization

31. Optimization with Niagara
Optimizer based on Niagara algebra:
Use the operation more efficiently
Produce simpler expressions by combining operations

32. Language Convention A and B are path expressions
A< B -- Path Expression A is prefix of B
AnB  Common prefix of path A and B
AńB  Greatest common prefix
of path A and B
┴  Null path Expression

33. Heuristics using Rewrite Rules
Allow optimization based on path selectivity
When applying un-nesting with operation Φμ

34. Interchangeability of Follow operation
Φμ(A) [Φμ(B)]=Φμ (B)[Φμ (A)]
TRUE or FALSE?
TRUE when
exists C such that C < A && C < B and C = AńB
Or AnB = ┴

35. Application of Rule on Invoice
Φμ(acc_Num:invoice)[Φμ(carrier:invoice)] ==
Φμ(carrier:invoice)[Φμ(acc_Num:invoice)] ?
TRUE or FALSE?

36. Application of Rule on Invoice
Φμ(acc_Num:invoice)[Φμ(carrier:invoice)] =
Φμ(carrier:invoice)[Φμ(acc_Num:invoice)]
TRUE because both share common prefix “invoice”.
Case AńB = invoice

37. Benefit of Rule Application
NOTE: Assume acc_Num is required for each invoice element, while carrier is not
THEN:
Φμ(acc_Num:invoice)[Φμ(carrier:invoice)] ==
Φμ(carrier:invoice)[Φμ(acc_Num:invoice)]
Then what algebra tree do we prefer?

38. Discussion Reduction of Input Size on first Sub-operation:
Φμ(carrier:invoice)  vs
Φμ(acc_Num:invoice) (:

39. Can we apply the rule below?
Φμ(acc_Num:invoice)[Φμ(acc_Num:Customer)]

40. Example “acc_Num:invoice” and “acc_Num:customer”
are two totally different paths
Case is: AnB = ┴
So yes, rule is valid.

41. Summary
XML Algebra
Operations
Optimization