XAL An XML ALgebra for Query Optimization

1
XAL - An XML ALgebra for Query Optimization

• Flavius Frasincar
• Geert-Jan Houben
• Cristian Pau

Databases Hypermedia Group Division of Computer
Science
2
Contents

• Motivation
• XML Query Algebra Goals
• XML Query Algebras
• XAL
• XAL Optimization Laws
• XAL Heuristic Optimization Algorithm
• XAL Query Example
• Conclusion and Future Work

3
1. Motivation

• Hera project automatic hypermedia presentation
  of data residing in the heterogeneous deep web
• Use XML technologies for querying, transforming,
  and integrating large amounts of Web data
• Optimization of XML queries is important need of
  an XML algebra for query optimization

4
2. XML Query Algebra Goals

• Based on W3C XML Query Data Model
• Genericity logical operators independent of the
  underlying storage representation
• Optimizability support query optimizations
• Expressivity express a large class of queries
• Composability operators are closed on the same
  data type
• Flexibility support various data types

5
3. XML Query Algebras

• Lore (Stanford)
• specific set of logical operators
• Beech et al. (industry)
• logical model, no optimization strategies
• YATL (INRIA)
• specific data model, focus on data
  integration

• XOM (Zhang Dong)
• complete and closed, no optimization support
• SAL (Beeri Tzaban)
• focus on semistructured data,
• limited optimization support
• XQuery (W3C)
• weak support for optimization (unordered
  forests)

6
4. XAL

• Based on W3C XML Query Data Model
• Reduces the impedance mismatch between databases
  and XML (query languages) by allowing a mix of
  ordered/unordered operators
• Support for optimization (reuse the query
  optimization heuristics from relational systems)
  
• Fine grained algebra of vertices and edges
  (Genericity)
• Composability, Flexibility, XQuery Compatibility

7
4.1. XAL Data Model

• Rooted connected directed graph with a partial
  order relation on edges
• Acyclic (lexical view)
• Cyclic (semantic view)
• Formally,

8
Properties for Vertex
9
Properties for Edge
Note Derived Property apply to E, D edges
10
4.2. XAL Operators

• All operators have the following form
• of(x1, x2, xn expression)
• Unary operators evaluate the input to a
  collection of vertices and use the implicit map
  operation to evaluate the result
• Closedness all operators are closed on
  collections (support composability)

11
Operator Semantics

• of(x expression)
• Variable x is bound to each vertex in the
• input collection. For each such binding f(x) is
• evaluated
• The semantics of the operator o defines how
• the partial result (resulting from one variable
• binding) is computed from f(x)
• The operator result is built by concatenating
• all the partial results

12
Collection

• Generalization of list and set (collections have
  a boolean order property)
• Similar to the mathematicians monad and
  functional programmers (list) comprehension
• MonadltMgt, where M is a type is a triplet of
  functions
• (mapltMgt, unitltMgt, join ltMgt)
• XAL has map and join (called union) but no unit
  operator
• (the singleton collection is written as the
  singleton itself)
• Collections have elements of arbitrary types

13
Operators Type

• Extraction operators retrieve the needed
  information from XML documents
• Meta-operators control the evaluation of
  expressions
• Construction operators build new XML documents
  from the extracted data
• Note two vertices are equal if they have the
  same value

14
Extraction Operators

• Projection ?type, name(e expr)
• Selection ?condition(e expr)
• Unorder ?(e expr)
• Join (x expr) ?condition
  (y expr)
• Cartesian Product (x expr) ? (y expr)
• Union (x expr) ? (yexpr)
• Difference (x expr) ? (yexpr)
• Intersection (x expr) ? (yexpr)

Note Flexibility, x and y do not have to be
union compatible like in relational algebra
15
Projection

• ?type, name(e expression)
• type E, A, R, D or disjunctions () of these
• name regular expression over strings
• Example. ?E, (Pp)ainters)(e) produces all
  the target vertices of element containment (E)
  edges that have names starting with Painter,
  painter, Painters, or painters, and that
  originate from the vertices in e

16
Meta-operators Construction Operators

• Map
• mapf(e expression)
• Kleene Star
• f(e expression)
• Note e is included in the result
• Create vertex
• vertextype(value)
• Note for element vertices the value
  (identifier) is given by the system
• Create edge
• edgetype, name, parent(child)

17
An Example

• Copy a complete graph starting from the vertex v
• mapedgetype(e), name(e),
• vertextype(parent(e))(value(par
  ent(e)))
• (vertextype(child(e))(value(chi
  ld(e))))
• (e)
• where e parentedge(?EAD, (child(x)))
• (x parentedge(?EAD,
  (v)))

18
5. XAL Optimization Laws

• The main factor in the execution cost of algebra
  expressions is the iteration (explicit or
  implicit map operator) over collections
• The proposed set of optimization laws aims at
  reducing iteration size for the data extraction
  expressions
• The laws are inspired by monad laws and
  relational algebraic optimization rules

19

• Law 1 (Left unit)
• If e1 is of unit type (singleton
  collection), then
• e2(e1) e2 (v e1)
• Law 2 (Right unit)
• If e2 is the identity function, i.e. e2 (v)
  v, then
• e2(e1) e1
• Law 3 (Associativity)
• (e1 o e2) o e3 e1 o ( e2 o e3 )
• Law 4 (Empty collection)
• If e2 is the empty function, i.e. e2(v)
  (), then
• e2(e1) ()
• Law 5 (Decomposition of join)
• e1 ?condition e2 ?condition(e1 ? e2)

20

• Law 6 (Decomposition of projection)
• If name is a regular expression that can be
  decomposed in several regular expressions n1, n2
  , nn and e is an unordered collection, then
• ?name(e) ?n1(e) ? ?n2(e) ? ?nn(e)
• Law 7 (Cascading of selection)
• ?c1?c2? cn(e) ?c1(? c2( (? cn (e))
  ))
• Law 8 (Commutativity of selection)
• ?c1(?c2(e)) ?c2(?c1(e))
• Law 9 (Commutativity of selection with
  projection)
• If the condition c involves solely vertices
  that have incoming edges named by the regular
  expression name, then
• ?name(?c(?name)(e)) ?c(?name(e))
• Law 10 (Commutativity of selection with cartesian
  product)
• If the condition c involves solely vertices
  from e1 , then
• ?c(e1 ? e2) ?c(e1 ) ? e2

21

• Law 11 (Commutativity of selection with binary
  operators)
• If ? is one of the set operators ?, ?, or
  ?, then
• ?c(e1 ? e2) ?c(e1) ? ?c(e2)
• Law 12 (Commutativity of binary operators)
• If ? is one of the set operators ?, ?, or
  ? and e1 and e2 are unordered collections, then
• e1 ? e2 e2 ? e1
• Law 13 (Commutativity of projection with
  cartesian product)
• If name is a regular expression that can
  decomposed in two regular expressions name1 and
  name2, name1 involves solely vertices in e1 and
  name2 involves solely vertices in e2 , then
• ?name(e1 ? e2) ?name1(e1) ? ?name2(e2)
• Law 14 (Commutativity of projection with union)
• ?name(e1 ? e2) ?name(e1) ? ?name(e2)

22
6. XAL Heuristic Optimization Algorithm

• S1. Eliminate unnecessary iterations (use Laws 1,
  2, and 4). After each following step, S1 is
  applied again.
• S2. Unorder collections (use unorder operator).
  Collections for which order is not relevant are
  unordered.
• S3. Decompose joins (use Law 5).
• S4. Decompose selections (use Law 7). Break down
  selections into a cascade of selections. It
  enables moving select operations down in the
  query tree.
• S5. Move selections down as far as possible (use
  Laws 8, 9, 10, and 11). Based on the
  commutativity of selection with other operators
  move selections down in the query tree as far as
  it is permitted by the selection condition.

23

• S6. Apply the most restrictive selections first
  (use Laws 3 and 12). Based on the commutativity
  and associativity of binary operators rearrange
  the leaf vertices so that the most restrictive
  selections apply first.
• Note As a selectivity criterion one can use
  the size of the collection.
• The most restrictive selections are the
  selections that produce collections with the
  fewest elements.
• S7. Decompose projections (use Law 6). Break down
  projections into a union of projections. It
  enables moving the project operations down in the
  query tree.
• S8. Move projections down as far as possible (use
  Laws 1, 2, and 4). Based on the commutativity of
  projection with other operators, move projections
  down in the query tree as far as possible.
• S9. Identify combined operations (use composition
  laws). Identify subtrees that group operations
  that can be executed by a single program.

24
7. XAL Query Example

• XML repository with three documents

painters.xml ltpaintersgt ltpaintergt ltnamegtRembrandt
lt/namegt ltdescriptiongtDutch painterlt/descriptiongt lt
/paintergt lt/paintersgt
catalogue.xml ltitemsgt ltitemgt ltpaintingidgtPainting
_ID01lt/paintingidgt ltpricegt1500000lt/pricegt lt/itemgt
lt/itemsgt
paintings.xml ltpaintingsgt ltpaintinggt ltidgtPainting
_ID01lt/idgt ltnamegtThe Stone Bridgelt/namegt ltauthorgtR
embrandtlt/authorgt lt/paintinggt lt/paintingsgt
25

• Query
• Return in alphabetical order the name of the
  painters that have a painting over 1 000 000
• (the name of the painters will appear in the
  ltresultgt element as many times as the number of
  their paintings that fulfill the above condition)
• XQuery 1.0
• ltresultgt
• FOR i IN document(painters.xml)/painters/paint
  er,
• j IN document(paintings.xml)/painting
  s/paintingauthor i/name,
• k IN document(catalogue.xml)/items/ite
  mpaintingid j/id
• WHERE k/price/data() gt 1000000
• RETURN i/name
• SORTBY ./data()
• lt/resultgt

26

• Input
• painters.xml 3 painters (1,2,3)
• paintings.xml 100 paintings for painter 1
• 150 paintings for
  painter 2
• 100 paintings for
  painter 3
• catalogue.xml Only painter 1 has 20 paintings
  more expensive than 1 000 000, all the other
  paintings are below 1 000 000

27

• Initial Query Tree
• Output is alphabetically ordered!
• Cartesian Product
• 3 x 350 x 350 367 500
• elements

28

• I Optimization
• Step 2 Unorder collections
• (commutativity of XAL binary operators)
• Step 4 Decompose selections
• Step 5 Move selections down as far as possible
• Cartesian Product
• 3 x 350
• 350 x 20 8 050 elements

29

• II Optimization
• Step 6 Apply the most restrictive selections
  first
• (switch positions of painter and item)
• Cartesian Product
• 20 x 350
• 20 x 3 7 060 elements

30
8. Conclusion and Future Work

• XAL provides an elegant way (by applying the
  unorder ? operator) to reuse the heuristic
  optimization algorithm from relational queries
• Investigate new optimization laws that take
  advantage of the XML specific features (e.g. tree
  structure, internal references)
• Build a translation scheme from XQuery to XAL,
  exploring the power of expression of XAL