1. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20021 XAL - An XML ALgebra for Query Optimization Flavius Frasincar Geert-Jan Houben Cristian Pau Databases & Hypermedia Group Division of Computer Science

2. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20022 Contents 1.Motivation 2.XML Query Algebra Goals 3.XML Query Algebras 4.XAL 5.XAL Optimization Laws 6.XAL Heuristic Optimization Algorithm 7.XAL Query Example 8.Conclusion and Future Work

3. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20023 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20024 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20025 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) … 3. XML Query Algebras

6. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20026 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20027 4.1. XAL Data Model Rooted connected directed graph with a partial order relation on edges –Acyclic (lexical view) –Cyclic (semantic view) Formally,

8. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20028 Properties for Vertex Basic Property Element Vertex Result Simple Vertex Result valueidentifiervalue (e.g.“Dali”) typeelementtype of value (e.g.string) Derived Property Result namename of the incoming E edge parentparent vertex (via E edge) parentedgeincoming E edge childelementsoutgoing E edges attributesoutgoing A edges referencesoutgoing R edges

9. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 20029 Properties for Edge Basic Property Result nameelement name (E) attribute name (A) ID attribute name (R) “Data” (D) typeE, A, R, D parentsource vertex of the edge childtarget vertex of the edge Derived Property Result nextfollowing sibling edge previouspreceding sibling edge Note: Derived Property apply to E, D edges

10. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200210 4.2. XAL Operators All operators have the following form o[f](x 1, x 2, … x n : 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200211 Operator Semantics o[f](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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200212 Collection Generalization of list and set (collections have a boolean order property) Similar to the mathematician’s monad and functional programmer’s (list) comprehension Monad, where M is a type is a triplet of functions (map, unit, join ) 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200213 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200214 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)  (y:expr) Difference (x: expr)  (y:expr) Intersection (x: expr)  (y:expr) Note: Flexibility, x and y do not have to be “union compatible” like in relational algebra

15. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200215 Projection  [type, name](e: expression) type = E, A, R, D or disjunctions (|) of these name = regular expression over strings Example.  [E, (P|p)ainter[s]#)](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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200216 Meta-operators & Construction Operators Map map[f](e: expression) Kleene Star *[f](e: expression) Note: e is included in the result Create vertex vertex[type](value) Note: for element vertices the value (identifier) is given by the system Create edge edge[type, name, parent](child)

17. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200217 An Example Copy a complete graph starting from the vertex v map[edge[type(e), name(e), vertex[type(parent(e))](value(parent(e))) ](vertex[type(child(e))](value(child(e)))) ](e) where e = *[parentedge(  [E|A|D, #](child(x))) ](x: parentedge(  [E|A|D, #](v)))

18. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200218 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200219 Law 1 (Left unit) If e 1 is of unit type (singleton collection), then e 2 (e 1 ) = e 2 (v := e 1 ) Law 2 (Right unit) If e 2 is the identity function, i.e. e 2 (v) = v, then e 2 (e 1 ) = e 1 Law 3 (Associativity) (e 1 o e 2 ) o e 3 = e 1 o ( e 2 o e 3 ) Law 4 (Empty collection) If e 2 is the empty function, i.e. e 2 (v) = (), then e 2 (e 1 ) = () Law 5 (Decomposition of join) e 1 ⋈ [condition] e 2 =  [condition](e 1  e 2 )

20. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200220 Law 6 (Decomposition of projection) If name is a regular expression that can be decomposed in several regular expressions n 1, n 2, … n n and e is an unordered collection, then  [name](e) =  [n 1 ](e)   [n 2 ](e)  …  [n n ](e) Law 7 (Cascading of selection)  [c 1 ∧ c 2 ∧ … c n ](e) =  [c 1 ](  [ c 2 ]( … (  [ c n ](e)) … )) Law 8 (Commutativity of selection)  [c 1 ](  [c 2 ](e)) =  [c 2 ](  [c 1 ](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 e 1, then  [c](e 1  e 2 ) =  [c](e 1 )  e 2

21. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200221 Law 11 (Commutativity of selection with binary operators) If  is one of the set operators: , , or , then  [c](e 1  e 2 ) =  [c](e 1 )   [c](e 2 ) Law 12 (Commutativity of binary operators) If  is one of the set operators: , , or  and e 1 and e 2 are unordered collections, then e 1  e 2 = e 2  e 1 Law 13 (Commutativity of projection with cartesian product) If name is a regular expression that can decomposed in two regular expressions name 1 and name 2, name 1 involves solely vertices in e 1 and name 2 involves solely vertices in e 2, then  [name](e 1  e 2 ) =  [name 1 ](e 1 )   [name 2 ](e 2 ) Law 14 (Commutativity of projection with union)  [name](e 1  e 2 ) =  [name](e 1 )   [name](e 2 )

22. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200222 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200223 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200224 7. XAL Query Example XML repository with three documents: painters.xml Rembrandt Dutch painter … catalogue.xml Painting_ID01 1500000 … paintings.xml Painting_ID01 The Stone Bridge Rembrandt …

25. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200225 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 element as many times as the number of their paintings that fulfill the above condition) XQuery 1.0: { FOR $i IN document(“painters.xml”)/painters/painter, $j IN document(“paintings.xml”)/paintings/painting[author = $i/name], $k IN document(“catalogue.xml”)/items/item[paintingid = $j/id] WHERE $k/price/data() > 1000000 RETURN $i/name SORTBY./data() }

26. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200226 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200227 Initial Query Tree –Output is alphabetically ordered!  Cartesian Product: 3 x 350 x 350 = 367 500 elements XQUERYXAL FOR , ,  WHERE  SORTBY 

28. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200228 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200229 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. / department of mathematics and computer science TU/e eindhoven university of technology ADC 2002January 29, 200230 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