Covering Indexes for XML Queriesby Prakash Ramanan

presented by

Dilek Demirel

Contents

XML query languages

Some definitions and concepts

Bisimulation and simulation relations

Results

The paper is about

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

An XML document can be represented as a graph D=(N, E, Eref), where N is the set of nodes, E is the set of edges and Eref is a set of idref edges.

Idref edges denotes an element–subelement relationship.

The subgraph T=(N, E) is a tree.

Example

XML query languages

Some XML query languagesXPathXQuery

They allow navigation in an XML document along different axes, to locate the desired element.

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

Subset languages of XPath

Core Xpath (CXPath)Branching Path Queries (BPQ)Tree Pattern Queries (TPQ)

TPQ = TPQ+ subsetof BPQ+ subsetof CXPath+ subsetof XpathWhere C+ denotes query language C without

the operator NOT

Core XPath

Does not contain arithmetic and string operations

Has the full navigational power of XPathConsists all queries involving the thirteen axes and three boolean operators and, or and not

Branching Path Queries

A subset of CXPath

CXPath queries that ignore the order of sibling elements

Allows nine axes, excluding the order respecting axes

Tree Pattern Queries

Involve four axesSelfChildDescendantDescendant or self

The only operator and

Do not involve idref edges

Definitions and concepts

An index for an XML documentObtained by merging “equivalent” nodes

into a single node. “equivalent” according to what, coming

soon…

Index of an XML doc.

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.

Definitions cont’d

An index DI 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 DI.

The important point about the covering index is:A covering index DI can be used to

evaluate the queries in C, without using D.

Focus of the paper

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

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.

We, till now,Described some classes of XML queriesGive some definitions and concepts

Will describe the equivalence relations that are mentioned in the beginning:Define the simulation relation on vertices of

an ordinary graphDefine simulation and bisimulation

relations on an XML document

Simulation and bisimulation

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.

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.

Forward simulation

Fsimulation of G1 by G2 is the largest binary relation subset of V1 * V2, such thatPreserves 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

Backward Simulation

Analogous to Fsimulation

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

Forward and Backward Simulation

FbsimulationPreserves vertex typesPreserves outgoing arcsPreserves incoming arcs

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’.

FBsimulation of D

Deals with both incoming and outgoing arcsPreserves node typesPreserve outgoing tree edgesPreserve outgoing idref edgesPreserve incoming tree edgesPreserve incoming idref edges

Bisimulation RelationForward 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.

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.

Example

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.

Questions?