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Covering Indexes for XML Queries by Prakash Ramanan presented by Dilek Demirel

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Contents XML query languages Some definitions and concepts Bisimulation and simulation relations Results

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The paper is about Minimizing the search tree, trying to build similar but smaller graphs equivalent the original XML document graph.

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An XML document can be represented as a graph D=(N, E, E ref ), where N is the set of nodes, E is the set of edges and E ref is a set of idref edges. Idref edges denotes an element– subelement relationship. The subgraph T=(N, E) is a tree.

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Example

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XML query languages Some XML query languages XPath XQuery They allow navigation in an XML document along different axes, to locate the desired element.

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Axes XPath provides 13 different axes Self Child Descendant/Descendant or self Parent Ancestor/ Ancestor or self Preceding/Preceding sibling Following/Following sibling Attribute Namespace

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Subset languages of XPath Core Xpath (CXPath) Branching Path Queries (BPQ) Tree Pattern Queries (TPQ) TPQ = TPQ + subsetof BPQ + subsetof CXPath + subsetof Xpath Where C + denotes query language C without the operator NOT

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Core XPath Does not contain arithmetic and string operations Has the full navigational power of XPath Consists all queries involving the thirteen axes and three boolean operators and, or and not

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Branching Path Queries A subset of CXPath CXPath queries that ignore the order of sibling elements Allows nine axes, excluding the order respecting axes

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Tree Pattern Queries Involve four axes Self Child Descendant Descendant or self The only operator and Do not involve idref edges

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Definitions and concepts An index for an XML document Obtained by merging “equivalent” nodes into a single node. “equivalent” according to what, coming soon…

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Index of an XML doc.

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Definitions cont’d A query Q distinguishes between two nodes in an XML document D, if exactly one of the two nodes is in the result of evaluating query Q on D.

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Definitions cont’d An index D I is a covering index for a class C of queries, if the following holds: No query in C can distinguish between two nodes of D that are in the same extend in D I. The important point about the covering index is: A covering index D I can be used to evaluate the queries in C, without using D.

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Focus of the paper The paper have studied the evaluation of CXPath queries and covering indexes for the above mentioned subclasses of CXPath.

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Definitions cont’d CXPath + is complete, in the sense that, For any node n in an XML document D, one can always construct a query, which starts from the root, Q in CXPath +, that distinguishes n from all the other nodes. The paper presented a method to build this query.

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We, till now, Described some classes of XML queries Give some definitions and concepts Will describe the equivalence relations that are mentioned in the beginning: Define the simulation relation on vertices of an ordinary graph Define simulation and bisimulation relations on an XML document

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Simulation and bisimulation

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Question Why do people deal with these simulation quotients? Because, for an XML document, if its simulation quotient is small, then a set of queries can be evaluated faster by using this index instead of the bigger XML document graph.

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Simulation for Ordinary Graphs Directed graphs G1=(V1, A1), G2=(V2,A2), each vertex v has a type t(v) Simulation is a binary relation between the vertex sets V1 and V2 of two graphs. It provides a possible notion of dominance/equivalence between the vertices of the two graphs.

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Forward simulation Fsimulation of G1 by G2 is the largest binary relation subset of V1 * V2, such that Preserves vertex types t(v1)=t(v2) Preserve outgoing arcs: for each v1’ elementOf post(v1), there exists v2’ elementOf post(v2) such that v1’ is Fsimulated by v2’ Fsimilarity is an equivalence relation

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Backward Simulation Analogous to Fsimulation Deal with the incoming arcs at a vertex, as opposed to forward simulation which deals with outgoing arcs.

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Forward and Backward Simulation Fbsimulation Preserves vertex types Preserves outgoing arcs Preserves incoming arcs

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Simulation for an XML Document Fsimulation of D is the largest binary relation on N (node set of D), such that Preserves node types If n1=root(D) then n2=root(D) Else t(n2)=t(n1) Preserve outgoing tree edges For each tree edge (n1,n1’), there exists a tree edge (n2, n2’) such that n1’ is fsimulated by n2’. Preserve outgoing idref edges For each idref edge (n1,n1’), there exists an idref edge (n2, n2’) such that n1’ is fsimulated by n2’.

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FBsimulation of D Deals with both incoming and outgoing arcs Preserves node types Preserve outgoing tree edges Preserve outgoing idref edges Preserve incoming tree edges Preserve incoming idref edges

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Bisimulation Relation Forward bisimulation of D is the largest binary relation on N (node set of D), such that Preserves node types If n1=root(D) then n2=root(D) and vice versa Else t(n2)=t(n1) Preserve outgoing tree edges For each tree edge (n1,n1’), there exists a tree edge (n2, n2’) such that n1’ is fsimulated by n2’ and vice versa. Preserve outgoing idref edges For each idref edge (n1,n1’), there exists an idref edge (n2, n2’) such that n1’ is fsimulated by n2’ and vice versa.

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The Quotients An equivalence relation on N partitions N into equivalence classes. Any two nodes in the same class are related, any two nodes in different classes are not. The quotient graph D~ is obtained from D by merging the nodes of each equivalence class into a single node.

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Example

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Results A CXPath+ query Q can be evaluated on an XML document D by computing the simulation of Q by D. For an XML document, its simulation quotient is the smallest covering index for BPQ+. For an XML document, its simulation quotient, with idref edges ignored throughout, is the smallest covering index for TPQ.

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Questions?

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