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XAL - An X ML AL gebra for Query Optimization

XAL - An X ML AL gebra for Query Optimization. Flavius Frasincar Geert-Jan Houben Cristian Pau. Databases & Hypermedia Group Division of Computer Science. Contents. Motivation XML Query Algebra Goals XML Query Algebras XAL XAL Optimization Laws

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XAL - An X ML AL gebra for Query Optimization

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  1. XAL - An XML ALgebra for Query Optimization Flavius Frasincar Geert-Jan Houben Cristian Pau Databases & Hypermedia Group Division of Computer Science ADC 2002

  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 ADC 2002

  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 ADC 2002

  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 ADC 2002

  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) • … ADC 2002

  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 ADC 2002

  7. 4.1. XAL Data Model • Rooted connected directed graph with a partial order relation on edges • Acyclic (lexical view) • Cyclic (semantic view) • Formally, ADC 2002

  8. Properties for Vertex ADC 2002

  9. Properties for Edge Note: Derived Property apply to E, D edges ADC 2002

  10. 4.2. XAL Operators • All operators have the following form o[f](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) ADC 2002

  11. 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 ADC 2002

  12. 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<M>, where M is a type is a triplet of functions (map<M>, unit<M>, join <M>) 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 ADC 2002

  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 ADC 2002

  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)  (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 ADC 2002

  15. 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 ADC 2002

  16. 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) ADC 2002

  17. 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))) ADC 2002

  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 ADC 2002

  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 oe2) oe3 = e1 o(e2oe3 ) • Law 4 (Empty collection) Ife2 is the empty function, i.e. e2(v) = (), then e2(e1) = () • Law 5 (Decomposition of join) e1 ⋈[condition] e2 = [condition](e1  e2) ADC 2002

  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 ADC 2002

  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 e1and 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) ADC 2002

  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. ADC 2002

  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. ADC 2002

  24. 7. XAL Query Example • XML repository with three documents: painters.xml <painters> <painter> <name>Rembrandt</name> <description>Dutch painter</description> </painter> … </painters> catalogue.xml <items> <item> <paintingid>Painting_ID01</paintingid> <price>1500000</price> </item> … </items> paintings.xml <paintings> <painting> <id>Painting_ID01</id> <name>The Stone Bridge</name> <author>Rembrandt</author> </painting> … </paintings> ADC 2002

  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 <result> element as many times as the number of their paintings that fulfill the above condition) • XQuery 1.0: <result> { 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() } </result> ADC 2002

  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 ADC 2002

  27. Initial Query Tree • Output is alphabetically ordered! • Cartesian Product: 3 x 350 x 350 = 367 500 elements ADC 2002

  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 ADC 2002

  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 ADC 2002

  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 ADC 2002

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