Introduction to XML Algebra

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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

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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.

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XML Algebra History

• Lore Algebra (August 1999)
• -- Stanford University
• IBM Algebra (September 1999)
• --Oracle IBM Microsoft Corp
• YAT Algebra (May 2000)
• ATT Algebra (June 2000)
• --ATT Bell Labs
• Niagara Algebra (2001)
• -- University of Wisconsin -Madison

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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

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Outline

• Concepts of Niagara Algebra
• Operations
• Optimization

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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

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Example XML Source Documents
Invoice.xml ltInvoice_Documentgt ltinvoice No
1gt ltaccount_numbergt2 lt/account_numbergt
ltcarriergtATTlt/carriergt lttotalgt0.25lt/totalgt
lt/invoicegt ltinvoicegt ltaccount_numbergt1
lt/account_numbergt ltcarriergtSprintlt/carriergt
lttotalgt1.20lt/totalgt lt/invoicegt
ltinvoicegt ltaccount_numbergt1 lt/account_numbergt
ltcarriergtATTlt/carriergt lttotalgt0.75lt/totalgt
lt/invoicegt lt/Invoice_Documentgt

• Customer.xml
• ltCustomer_Documentgt
• ltcustomergt
• ltaccountgt1 lt/accountgt
• ltnamegtTom lt/namegt
• lt/customer gt
• ltcustomergt
• ltaccountgt2 lt/accountgt
• ltnamegtGeorge lt/namegt
• lt/customer gt
• lt/Customer _Documentgt

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XML Data Model and Tree Graph

• Example

Invoice_Document
ltInvoice_Documentgt ltinvoicegt
ltnumbergt2lt/numbergt ltcarriergtSprintlt/carriergt
lttotalgt0.25lt/totalgt lt/invoicegt
ltinvoicegt ltnumbergt1lt/numbergt ltcarriergtSprintlt/car
riergt lttotalgt1.20lt/totalgt lt/invoicegt lt/Invoice
_Documentgt

Invoice
Invoice
number
carrier
number
total
total
carrier
2
ATT
0.25
1
1.20
Sprint
Ordered Tree Graph, Semi structured Data
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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 ??

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XML Data Model GVDNM01

• Collection of bags of vertices.
• Vertices in a bag have no order.
• Example

Root invoice.xml invoice
invoice.account_number
lt account_number gt element-content lt/
account_number gt
ltinvoicegt Invoice-element-content lt/invoicegt
Rootinvoice.xml, invoice, invoice.
account_number
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Data Model

• Bag elements are reachable by path expressions.
• Path expression consists of two parts
• An entry point
• A relative forward part
• Example account_numberinvoice

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Outline

• Concepts of Niagara Algebra
• Operations
• Optimization

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Operators

• Source S , Follow ?, Expose ?, Vertex ?,
• Source S , Select ?, Join , Rename ?, Group
  ?, Union ?, Intersection ?, Difference - ,
  Cartesian Product ?.

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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

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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)

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Follow operator (Example)
Root invoice.xml , invoice, invoice.carrier
Root invoice.xml invoice
invoice.carrier
ltcarriergt carrier -element-content lt/carrier gt
ltinvoicegt Invoice-element-content lt/invoicegt
Unnesting Follow
?(carrierinvoice)
Root invoice.xml invoice
ltinvoicegt Invoice-element-content lt/invoicegt
Root invoice.xml , invoice
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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 (?,?,?,?,?,?)

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Select operator (Example)
Root invoice.xml , invoice,
Root invoice.xml invoice
ltinvoicegt Invoice-element-content lt/invoicegt
? invoice.carrier Sprint
Root invoice.xml invoice
Root invoice.xml invoice
ltinvoicegt Invoice-element-content lt/invoicegt
ltinvoicegt Invoice-element-content lt/invoicegt
Root invoice.xml , invoice, Root invoice.xml
, invoice,
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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

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Join operator (Example)
Root invoice.xml , invoice, Root customer.xml ,
customer
Root invoice.xml invoice
Root customer.xml customer
ltinvoicegt Invoice-element-content lt/invoicegt
ltcustomergt customer-element-content lt/customergt
account_number invoice numbercustomer
Root invoice.xml invoice
Root customer.xml customer
ltinvoicegt Invoice-element-content lt/invoicegt
ltcustomergt customer-element-content lt/customergt
Root invoice.xml , invoice
Root customer.xml , customer
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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

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Expose operator (Example)
Root invoice.xml , invoice.bill_period,
invoice.carrier
Root invoice.xml invoice.
bill_period invoice.carrier
ltcarriergt bill_period -element-content lt/carrier gt
ltinvoicegt carrier-element-content lt/invoicegt
?(bill_period,carrier)
Root invoice.xml invoice
invoice.carrier invoice.bill_period
ltcarriergt bill_period -element-content lt/carrier gt
ltinvoicegt Invoice-element-content lt/invoicegt
ltinvoicegt carrier-element-content lt/invoicegt
Root invoice.xml , invoice, invoice.carrier,
invoice.bill_period
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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)
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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)

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Example XML Source Documents
Invoice.xml ltInvoice_Documentgt
ltinvoicegt ltaccount_numbergt2 lt/account_numbergt
ltcarriergtATTlt/carriergt lttotalgt0.25lt/totalgt
lt/invoicegt ltinvoicegt ltaccount_numbergt1
lt/account_numbergt ltcarriergtSprintlt/carriergt
lttotalgt1.20lt/totalgt lt/invoicegt
ltinvoicegt ltaccount_numbergt1 lt/account_numbergt
lttotalgt0.75lt/totalgt lt/invoicegt ltauditorgt
maria lt/auditorgt lt/Invoice_Documentgt
Customer.xml ltCustomer_Documentgt
ltcustomergt ltaccountgt1 lt/accountgt ltnamegtTom
lt/namegt lt/customer gt ltcustomergt ltaccountgt
2 lt/accountgt ltnamegtGeorge lt/namegt
lt/customer gt lt/Customer _Documentgt
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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
• ltSprint_invoicesgt
• i/account_number,
• c/name,
• i/total
• lt/Sprint_invoicesgt

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Example Xquery output

• ltSprint_Invoicegt
• ltaccount_numbergt1 lt/account_numbergt
• ltnamegtTom lt/namegt
• lttotalgt1.20lt/totalgt
• lt/Sprint_Invoice gt

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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 )
customer (2)
customer(1)
Invoice (1)
invoice (2)
invoice (3)
Follow (.invoice)
Follow (.customer)
Source (Invoices.xml)
Source (cutomers.xml)
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Outline

• Concepts of Niagara Algebra
• Operations
• Optimization

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Optimization with Niagara

• Optimizer based on Niagara algebra
• Use the operation more efficiently
• Produce simpler expressions by combining
  operations

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Language Convention

• A and B are path expressions
• Alt B --? Path Expression A is prefix of B
• AnB ---? Common prefix of path A and B
• AnB ---? Greatest common prefix
• of path A and B
• - ---? Null path Expression

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Heuristics using Rewrite Rules

• Allow optimization based on path selectivity
• When applying un-nesting with operation Fµ

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Interchangeability of Follow operation

• Fµ(A) Fµ(B)Fµ (B)Fµ (A)
• TRUE or FALSE?
• TRUE when
• exists C such that C lt A C lt B and C AnB
• Or AnB -

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Application of Rule on Invoice

• Fµ(acc_Numinvoice)Fµ(carrierinvoice)
• Fµ(carrierinvoice)Fµ(acc_Numinvoice) ?
• TRUE or FALSE?

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Application of Rule on Invoice

• Fµ(acc_Numinvoice)Fµ(carrierinvoice)
• Fµ(carrierinvoice)Fµ(acc_Numinvoice)
• TRUE because both share common prefix invoice.
• Case AnB invoice

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Benefit of Rule Application

• NOTE Assume acc_Num is required for each invoice
  element, while carrier is not
• THEN
• Fµ(acc_Numinvoice)Fµ(carrierinvoice)
• Fµ(carrierinvoice)Fµ(acc_Numinvoice)
• Then what algebra tree do we prefer?

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Discussion

• Reduction of Input Size on first
• Sub-operation
• Fµ(carrierinvoice) ?
• vs
• Fµ(acc_Numinvoice) (

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Can we apply the rule below?

• Fµ(acc_Numinvoice)Fµ(acc_NumCustomer)

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Example

• acc_Numinvoice and
• acc_Numcustomer
• are two totally different paths
• Case is AnB -
• So yes, rule is valid.

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Summary

• XML Algebra
• Operations
• Optimization