XML Algebra

1
XML Algebra

• Comparison between
• XPERANTO
• NIAGARA

2
Part I

• NIAGARA
• XML Query Optimization
• XML Algebra
• Data Model
• Operator
• Query Plan
• Equivalent Rules

• XPERANTO
• XML Query to SQL
• XML Algebra
• Data Model
• Operator
• Query Plan
• Composition Rules
• Translation Example

3
Example of Telephone Bill
lt?xml version1.0 encodingUS-ASCII ?gt
lt!DOCTYPE invoice lt!ELEMENT invoice
(account_number, bill_period, carrier,
itemized_call, total)gt lt!ELEMENT
account_number (PCDATA)gt lt!ELEMENT bill_period
(PCDATA)gt lt!ELEMENT carrier (PCDATA)gt lt!ELEMENT
itemized_call EMPTYgt lt!ATTLIST itemized_call no
ID REQUIRED date CDATA REQUIRED number_called
CDATA REQUIRED time CDATA REQUIRED rate
(NIGHTDAY) REQUIRED min CDATA REQUIRED amount
CDATA REQUIREDgt lt!ELEMENT total (PCDATA)gt gt
ltinvoicegt ltaccount_numbergt555 777-3158 573 234
3lt/account_numbergt ltbill_periodgtJun 9 - Jul 8,
2000lt/bill_periodgt ltcarriergtSprintlt/carriergt
ltitemized_call no1 dateJUN 10
number_called973 555-8888 time1017pm
rateNIGHT min1 amount0.05 /gt
ltitemized_call no2 dateJUN 13
number_called973 650-2222 time1019pm
rateDAY min1 amount0.15 /gt
ltitemized_call no3 dateJUN 15
number_called206 365-9999 time1025pm
rateNIGHT min3 amount0.15 /gt
lttotalgt0.35lt/totalgt lt/invoicegt
4
Example XQuery

• User XQuery
• ltsummarygt
• FOR
• rate IN distinct(document(invoice)/invoice/item
  ized_call_at_rate)
• LET
• itemized_call document(invoice)/invoice/item
  ized_call_at_raterate
• WHERE
• itemized_call/_at_number_called LIKE 973
• RETURN
• ltrategtratelt/rategt
• ltnumber_of_callsgtcount(itemized_call)lt/number_of_
  callsgt
• lt/summarygt

Count number of itemized_calls in calling area
973 grouped by the calling rate.
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.
  

6
Goals

• 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

7
Data Model

• A collection of bags of vertices.
• The vertices in the bag have no order.
• Example

Root invoice.xml invoice
invoice.account_number
lt account_number gt carrier -element-content lt/
account_number gt
ltinvoicegt Invoice-element-content lt/invoicegt
Rootinvoice.xml, invoice, invoice.
account_number
8
Data Model

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

9
Operators

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

10
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 matches
• invoice.xml
• S (,schema.dtd) All known XML documents that
  conform to schema.dtd

11
Follow operator ?

• Input a path expression in entry point notation
  
• Functionality extracts vertices reachable by
  path expression
• Output a new bag that consist of the extracted
  vertex all the contents of the original bag (in
  care of unnesting follow)

12
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
13
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 (?,?,?,?,?,?)

14
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,
15
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

16
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
17
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

18
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
19
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)
20
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 the entry point annotation of
  the elements of a bag.
• Example ?(invoice.bill_period,date)

21
Example XQuery

• User XQuery
• ltsummarygt
• FOR
• rate IN distinct(document(invoice)/invoice/item
  ized_call_at_rate)
• LET
• itemized_call document(invoice)/invoice/item
  ized_call_at_raterate
• WHERE
• itemized_call/_at_number_called LIKE 973
• RETURN
• ltrategtratelt/rategt
• ltnumber_of_callsgtcount(itemized_call)lt/number_of_
  callsgt
• lt/summarygt

Count number of itemized_calls in calling area
973 grouped by the calling rate.
22
Query Plan Algebra

• ?(summary)
• e(?(rate)rate
• ?(number_of_calls)number)
• ?(rateinvoice.itemized_call, rate),
• ?(count(invoice.itemized_call), number)
• ?(rateinvoice.itemized_call,
  count(invoice.itemized_call))
• s number calledinvoice.itemized_call ?973
  
• Fµ(invoice.itemized_call)
• s(invoice.xml)

23
Equivalent Rules

• 14 equivalent rules so far.
• Definition of Auxiliary Operators for Equiv.
• A gt B Path expression A is a prefix of B
• - The null path expression
• A?B The greatest common prefix of path
  expressions A and B
• A?B The common prefix of path expressions A and
  B

.
24
Equivalent Rules Examples

• Rule applications
• Follow ordering
• Fµ(A) Fµ(B) Fµ (B)Fµ (A)
• iff C lt A, C lt B C A?B, or A?B -

.
C
X
X
A
B
B
A
...
25
Equivalent Rules Examples

• Rule applications
• Join commutability and associability
• (A B) C (C B) A

26
Equivalent Rules Examples

• Rule applications
• Selection distribution and interchangeability
• scA B sc1A sc2B
• where c is a conjoin of the conditions c1 and c2,
  each of which only refers to one of the join
  inputs

27
Equivalent Rules Examples

• Rule applications
• Elimination of unused bag elements
• e(P)(JA) J(e(PA))
• iff J uses only elements exposed by P

28
XPERANTO

• Goal
• XQuery ? SQL
• References
• J. Shanmugasundaram, et. Al. Querying XML Views
  of Relational Data, VLDB 2001.
• J. Shanmugasundaram, et. Al. Efficiently
  Publishing Relational Data as XML Documents, VLDB
  2000.
• J. Shanmugasundaram, Ph.D. Dissertation. July,
  2001.

29
Query Processing Architecture
User
XQuery
User XML View
XQuery
RDB
30
Data Model

• Tables of A List of XML Fragments

ltcarriergt carrier lt/carrier ltcarriergt
carrier lt/carriergt .
carriers
carriers


Groupby carrier aggXMLFrags(carrier_entry)
carrier_entry
ltcarriergt carrier lt/carrier
carrier_entry


Project carrier_entry ltcarriergtcarrierlt/carri
ergt
carrier
carrier


Select invoice_id id
invoice_id carrier


invoice_id
carrier
Table Carrier
31
Operators

• Table, Project, Select, Join, Groupby, Orderby,
  Union, Unnest, View, Function
• Select, Project, join, groupby, orderby and union
  have the same semantics as their relational
  counterparts.
• Project to invoke various function defined
• Table/View to refer to relational table or XML
  view
• Unnest to unnest XML list
• Function to invoke XQuery valued functions
• Groupby to create XML Fragments

32
XML Functions Operators
XML Function Description Operators
1 cr8Elem(Tag, Atts, Clist) Creates an element with tag name Tag, attribute list Atts, and contents Clist Project
2 cr8AttList(A1,..An) Creates a list of attributes from the attributes passed as parameters Project
3 cr8Att(Name, Val) Creates an attribute with name Name and value Val Project
4 cr8XMLFragList(C1,Cn) Creates an XML fragment list from the content parameters Project
5 aggXMLFrags Aggregate XML function that creates an XML fragment list Groupby
6 getTagName(Elem) Returns the element name of the Elem Project, Select
7 getAttributes(Elem) Returns the list of attributes of Elem Project, Select
8 getContents(Elem) Returns the XML fragment list of contents of Elem Project, Select
9 getAttName(Att) Returns the name of attribute Att Project, Select
10 getAttValue Returns the value of the attribute Att Project, Select
11 isElement(E) Returns true if E is an element, returns false otherwise Select
12 isText(T) Returns true if T is text, returns false otherwise Select
13 Unnest(List) Superscalar function that unnests a list Unnest
33
Operators - Examples
elems
ltaccount_numbergt508-753-2352lt/account_numbergt ltbill_periodgt24 july 23 august, 2001lt/bill_periodgt .. .. ..

count
3
elems
Project elems getContents(invoice)
count
Groupby count count(itemized_call)
invoice
ltinvoicegt ltaccount_numbergt508-753-2352lt/account_numbergt ltbill_periodgt24 july 23 august, 2001lt/bill_periodgt .. .. lt/invoicegt
itemized_call
ltitemized_call gt lt/itemized_callgt
ltitemized_call gt lt/itemized_callgt
ltitemized_call gt lt/itemized_callgt
34
Operators - Examples
entries
ltrategt DAY lt/rategt ltnumber_of_callsgt 20 lt/number_of_callsgt ltrategt NIGHT lt/rategt ltnumber_of_callsgt 23 lt/number_of_callsgt
result
ltsummarygt ltrategt DAY lt/rategt ltnumber_of_callsgt 20 lt/number_of_callsgt ltrategt NIGHT lt/rategt ltnumber_of_callsgt 23 lt/number_of_callsgt lt/summarygt
entries
result
Groupby entries aggXMLFrags(entry)
Project result cr8Elem(summary, Att, entries)
entry
ltrategt DAY lt/rategt ltnumber_of_callsgt 20 lt/number_of_callsgt
ltrategt NIGHT lt/rategt ltnumber_of_callsgt 23 lt/number_of_callsgt

entries
ltrategt DAY lt/rategt ltnumber_of_callsgt 20 lt/number_of_callsgt ltrategt NIGHT lt/rategt ltnumber_of_callsgt 23 lt/number_of_callsgt
35
Operator - Examples
elem
ltrategt DAY lt/rategt ltnumber_of_callsgt 20 lt/number_of_callsgt ltrategt NIGHT lt/rategt ltnumber_of_callsgt 23 lt/number_of_callsgt
elem
Unnest elem unnest(elems)
elems
ltrategt DAY lt/rategt ltnumber_of_callsgt 20 lt/number_of_callsgt ltrategt NIGHT lt/rategt ltnumber_of_callsgt 23 lt/number_of_callsgt
36
XML Query
result
Project result ltsummarygt entries lt/summarygt
entries
Groupby entries aggXMLFrags(entry)
entry

• User XQuery
• ltsummarygt
• FOR
• rate IN distinct(document(invoice)/invoice/item
  ized_call_at_rate)
• LET
• itemized_call document(invoice)/invoice/item
  ized_call_at_raterate
• WHERE
• itemized_call/_at_number_called LIKE 973
• RETURN
• ltrategtratelt/rategt
• ltnumber_of_callsgtcount(itemized_call)lt/number_of_
  callsgt
• lt/summarygt

XQGM
Project entry ltrategt rate lt/rategt
ltnumber_of_callsgt count lt/number_of_callsgt
rate
count
count
Join (Correlated)
Groupby count count(itemized_call)
itemized_call
Selection number LIKE 973
rate
itemized_call
Select distinct(rate)
Select rate irate
rate
itemized_call
number
irate
Navigate doc/invoice/itemized_call_at_rate
Navigate irate doc/invoice/itemized_call_at_rate
number doc/invoice/itemized_call_at_number_calle
d
doc
View document(invoice.xml)
37
Navigation in XQGM
XQGM
account_number
account_number
Select getTagName(elem)account_number
Navigate invoice/account_number
elem
invoice
Unnest elem unnest(elems)
elems
Project elems getContents(invoice)
invoice
38
Default XML View

• ltinvoicegt
• ltrowgt
• ltidgt 1 lt/idgt
• ltaccount_numbergt555 777-3158 573 234
  3lt/account_numbergt
• ltbill_periodgt Jun 9 Jun 8, 2000 lt/bill_periodgt
• lttotalgt0.35lt/totalgt
• lt/rowgt
• lt/invoicegt
• ltcarriergt
• ltrowgt
• ltinvoice_idgt 1 lt/invoice_idgt
• ltcarriergtSprintlt/carriergt
• lt/rowgt
• lt/carriergt
• ...

invoice
id account_number bill_period total
1 555 777-3158 573 234 3 Jun 9 Jun 8, 2000 0.35
carrier
invoice_id carrier
1 Sprint
itemized_call
invoice_id no date number_called time rate min amount
1 1 JUN 10 973 555-8888 1017pm NIGHT 1 0.05
1 2 JUN 13 973 650-2222 1019am DAY 1 0.15
1 3 JUN 15 206 365-9999 1025pm NIGHT 3 0.15
39
User Defined XML View
ltinvoicegt ltaccount_numbergt555 777-3158 573 234
3lt/account_numbergt ltbill_periodgtJun 9 - Jul 8,
2000lt/bill_periodgt ltcarriergtSprintlt/carriergt
ltitemized_call no1 dateJUN 10
number_called973 555-8888 time1017pm
rateNIGHT min1 amount0.05 /gt
ltitemized_call no2 dateJUN 13
number_called973 650-2222 time1019pm
rateDAY min1 amount0.15 /gt
ltitemized_call no3 dateJUN 15
number_called206 365-9999 time1025pm
rateNIGHT min3 amount0.15 /gt
lttotalgt0.35lt/totalgt lt/invoicegt
Invoice
Id account_number bill_period total
1 555 777-3158 573 234 3 Jun 9 Jun 8, 2000 0.35
Carrier
Invoice_id Carrier
1 Sprint
Itemized_call
Invoice_id No Date Number_called Time Rate Min Amount
1 1 JUN 10 973 555-8888 1017pm NIGHT 1 0.05
1 2 JUN 13 973 650-2222 1019am DAY 1 0.15
1 3 JUN 15 206 365-9999 1025pm NIGHT 3 0.15
40
User Defined XML View Cont.

• Create view invoice as (
• FOR
• invoice IN view(default)/invoice/row
• RETURN
• ltinvoicegt
• ltaccount_numbergtinvoice/account_numberlt/account_n
  umbergt
• ltbill_periodgtinvoice/bill_periodlt/bill_periodgt
• FOR
• carrier in view(default)/carrier/row
• WHERE
• carrier/invoice_id invoice/id
• RETURN
• ltcarriergtcarrierlt/carriergt
• FOR
• itemized_call in view(default)/itemized_call/ro
  w
• WHERE
• itemized_call/invoice_id invoice/id
• RETURN
• ltitemized_call noitemized_call/no
  dateitemized_call/date number_calleditemized_c
  all/number_called timeitemized_call/time
  rateitemized_call/rate minitemized_call/min
  amountitemized_call/amount /gt

41
XML View XQGM
doc
Project doc ltinvoicegt ltaccount_numbergt
account_number lt/account_numbergt
ltbill_periodgtbill_periodlt/bill_periodgt carriers
itemized_calls lttotalgttotallt/totalgt lt/invoicegt

• Create view invoice as (
• FOR
• invoice IN view(default)/invoice/row
• RETURN
• ltinvoicegt
• ltaccount_numbergtinvoice/account_numberlt/account_n
  umbergt
• ltbill_periodgtinvoice/bill_periodlt/bill_periodgt
• FOR
• carrier in view(default)/carrier/row
• WHERE
• carrier/invoice_id invoice/id
• RETURN
• ltcarriergtcarrierlt/carriergt
• FOR
• itemized_call in view(default)/itemized_call/ro
  w
• WHERE
• itemized_call/invoice_id invoice/id
• RETURN
• ltitemized_call noitemized_call/no
  dateitemized_call/date number_calleditemized_c
  all/number_called timeitemized_call/time
  rateitemized_call/rate minitemized_call/min
  amountitemized_call/amount /gt

account_number
bill_period
total
items
carriers
Join (Correlated)
carriers
Groupby carrier aggXMLFrags(carrier_entry)
items
Subquery.
carrier_entry
Project carrier_entry ltcarriergtcarrierlt/carri
ergt
invoice_id
carrier
Table Carrier
carrier
Select invoice_id id
invoice_id
carrier
Table Carrier
id
account_number
bill_period
total
Table Invoice
42
View Composition

• User Query XQGM User View XQGM
• To cancel out the Navigation operators
• By using the composition rules

cr8Elem(invoice, cr8AttList(), cr8XMLFragList(
cr8Elem(account_number, cr8AttList(), cr8XMLFra
gList(account_number)), cr8Elem(bill_period,
cr8AttList(), cr8XMLFragList(bill_period)),
carriers, items, cr8Elem(total,
cr8AttList(), cr8XMLFragList(total)) ) )
account_number
Select getTagName(elem)account_number
elem
Unnest elem unnest(elems)
elems
Project elems getContents(invoice)
invoice
43
12 Composition Rules
Function COMPOSES WITH REDUCTION
1 getTagName cr8Elem(Tag, Atts, Clist) Tag
2 getAttributes Cr8Elem(Tag, Atts, Clist) Atts
3 getContents cr8Element(Tag, Atts, Clist) Clist
4 getAttName cr8Att(Name, Val) Name
5 getAttValue cr8Att(Name, Val) Val
6 isElement cr8Element(Tag, Atts, Clist) True
7 isElement Other than cr8Eleme False
8 isText PCDATA True
9 isText Other than PCDATA False
10 Unnest aggXMLFrags(C) C
11 Unnest cr8XMLFragList(C1, ..., Cn) C1 U ... U Cn
12 Unnest cr8AttList(A1, ..., An) A1 U ... U An
44
View Composition Example
account_number
Select getTagName(elem)account_number
elem
account_number
Unnest elem unnest(elems)
Join (Correlated)
elems
Project elems getContents(invoice)
invoice
Project invoice ltinvoicegt ltaccount_numbergt
account_number lt/account_numbergt ltbill_periodgt
bill_period lt/bill_periodgt carriers
itemized_calls lttotalgt total lt/totalgt lt/invoicegt
account_number
bill_period
total
items
carriers
Join (Correlated)
45
Computation Pushdown

• Goal XQGM ? SQLs Tagger Graph
• Step1 Query Decorrelation
• Correlated Join ? Out Unions
• Reference P. Seshadri, et. Al. Complex Query
  Decorrelation, ICDE 1996.
• Step2 Tagger Pull-Up
• XQGM ? Tagger Run-Time Graph
• Use Sorted Outer Union
• Reference J. Shanmugasundaram, et. Al.
  Efficiently Publishing Relational Data as XML
  Documents.
• Separation of SQL and Tagger Operations
• Semantically equivalent fragment by pattern.

46
Comparison
XPERANTO NIAGARA
Goal XQuery ? SQL XQuery ? Algebra
Algebra XQGM and Tagger Graph XML Algebra
Data Model Tables of a list of XML Fragments A collection of bags of vertices
Operators 10 operators with 13 functions 12 operators
Variable Binding Lot of temporary variables No variables.
Order Sensitive Semi-sensitive (missing orderby)
Regular Expression No Support at operator level Support at operator level
Text-in-context No Support Support
Level of abstraction Function level (lower) Logical level (higher)
Transition rules Composition rules (ad-hoc) 1 Semantically equivalent pattern (ad-hoc) Equivalent rules
Operation History Not maintained Maintained
47
Conclusions and Future Work

• WE NEED OUR OWN ALGEBRA.
• More Reading
• David Beech, et. Al. A Formal Data Model and
  Algebra for XML.
• Mary Fernandez, et. Al. An Algebra for XML Query.