An Algebra forBasic Graph Patterns

George FletcherWashington State University, Vancouver

LID 2008, Rome19-20 May 2008

(web) data:

‣ social

‣ semantic

‣ structured, semi-structured, unstructured

semantic web, data spaces, web 3.0, ...

data model?

requirements

1. “things” and their relationships

2. don’t force artificial data/metadata distinctions

3. support both structure and no structure

Resource Description Framework

• W3C standard

• data model for social semantic web

Resource Description Framework

• W3C standard

• data model for social semantic web

• triples

‣ dataspace = finite ternary relation over “things”

‣ things = URI’s, literals

• fulfills 1, 2, and 3

An Algebra For Basic Graph Patterns

George H.L. Fletcher

School of Engineering and Computer ScienceWashington State University, Vancouver

fletcher@vancouver.wsu.edu

Abstract. Motivated by recent developments in the dataspaces, web,and personal information management communities, we outline researchdirections on query processing for SPARQL, the W3C recommendationlanguage for querying RDF triple stores. The core of each SPARQL queryis a basic graph pattern (BGP). BGP is a little logic for extracting sub-sets of related nodes in an RDF graph. In this paper we undertake aformal study of BGP with an eye towards e!cient SPARQL query evalu-ation. Our main contributions are (1) an algebraization of BGP, and (2)first steps towards a framework for the design of structural indexes toaccelerate processing of queries in this algebra.

1 Introduction

The flexibility and fundamental character of ternary relations for data modelingwas recognized very early in the development of modern logic [34]. Althoughthere were some initial e!orts towards realizing this idea in information systems(e.g., [31]), “triples” have only recently gained traction with the developmentof the W3C Resource Description Framework (RDF) [24, 29]. As a data modelfor capturing “things” and their “relationships,” treating each as co-equal firstclass objects, RDF successfully blurs the somewhat arbitrary distinction one istypically forced to make between “data” and “metadata.”1

Example 1 Consider the following set of atoms, denoting a small subset of theobjects in a particular individual’s dataspace of information:

{Yamada, McShea, Herzog, doc1, doc2, doc3, knows, performed, authored,type, social action, is a kind of, past action, created on, 14.5.08, PDF, MP3}

A sample of some of the (triple) relationships holding between these objects isgiven in Figure 1(a). For example, the first triple asserts that Yamada authoredDocument 1; the second triple asserts that Yamada knows McShea; and the thirdtriple asserts that ‘knows’ is a kind of social action. This “graph” of relationshipscan be visualized as in Figure 1(b).

1 Note that RDF has a!nities to data models previously proposed in the context ofgraph databases (e.g., [5, 18]) and data integration systems (e.g., [4, 30, 47]).

RDF

˘!Yamada, authored, doc1",!Yamada, knows, McShea",!knows, is a kind of, social action",!Herzog, authored, doc2",!Herzog, authored, doc3",!McShea, performed, doc3",!McShea, past action, authored",!doc1, type, PDF",!doc2, type, MP3",!doc3, type, MP3",!doc3, created on, 14.5.08"

¯

(a) A triple graph

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

(b) A visualization of this graph

Fig. 1. A small subset of a personal dataspace graph.

The flexibility of the RDF data model has made it a natural choice for tech-nologies being developed in the social and semantic web research communities[2, 40]. In addition to the central role RDF plays in these areas, recent e!ortstowards the development of dataspaces [13, 20] of personal information [26] havealso highlighted the natural fit of triples in modeling modern data scenarios. Asthe use of RDF grows, the need for RDF data management becomes increasinglycrucial.

There has been an explosion of proposals for RDF query languages [16]. TheW3C recommendation query language SPARQL [36], however, is emerging asthe most mature and accepted. SPARQL is a declarative language with a syntaxvery much like SQL.

Example 2 Consider the query “Retrieve authors and the types of their docu-ments.” In SPARQL, where variables are identified by a preceding ?, this querycan be posed against the graph in Figure 1(a) as follows:

SELECT ?a ?tWHERE { ?a authored ?d . ?d type ?t . }

The WHERE clause of a SPARQL query specifies a basic graph pattern (BGP).Such patterns, which are at the heart of all SPARQL queries, identify a subsetof related atoms to be extracted from the RDF graph, which is then returned asa set of variable mappings.

Example 3 The BGP of Example 2 is (?a authored ?d) ; (?d type ?t). Thesemantics of this pattern with respect to the graph of Figure 1(a) is a set ofvariable mappings, each of which makes the pattern hold in the graph, i.e.,

1 2 3Yamada doc1 PDFHerzog doc2 MP3Herzog doc3 MP3

RDF

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

˘!Yamada, authored, doc1",!Yamada, knows, McShea",!knows, is a kind of, social action",!Herzog, authored, doc2",!Herzog, authored, doc3",!McShea, performed, doc3",!McShea, past action, authored",!doc1, type, PDF",!doc2, type, MP3",!doc3, type, MP3",!doc3, created on, 14.5.08"

¯

(a) A triple graph

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

(b) A visualization of this graph

Fig. 1. A small subset of a personal dataspace graph.

The flexibility of the RDF data model has made it a natural choice for tech-nologies being developed in the social and semantic web research communities[2, 40]. In addition to the central role RDF plays in these areas, recent e!ortstowards the development of dataspaces [13, 20] of personal information [26] havealso highlighted the natural fit of triples in modeling modern data scenarios. Asthe use of RDF grows, the need for RDF data management becomes increasinglycrucial.

There has been an explosion of proposals for RDF query languages [16]. TheW3C recommendation query language SPARQL [36], however, is emerging asthe most mature and accepted. SPARQL is a declarative language with a syntaxvery much like SQL.

Example 2 Consider the query “Retrieve authors and the types of their docu-ments.” In SPARQL, where variables are identified by a preceding ?, this querycan be posed against the graph in Figure 1(a) as follows:

SELECT ?a ?tWHERE { ?a authored ?d . ?d type ?t . }

The WHERE clause of a SPARQL query specifies a basic graph pattern (BGP).Such patterns, which are at the heart of all SPARQL queries, identify a subsetof related atoms to be extracted from the RDF graph, which is then returned asa set of variable mappings.

Example 3 The BGP of Example 2 is (?a authored ?d) ; (?d type ?t). Thesemantics of this pattern with respect to the graph of Figure 1(a) is a set ofvariable mappings, each of which makes the pattern hold in the graph, i.e.,

1 2 3Yamada doc1 PDFHerzog doc2 MP3Herzog doc3 MP3

RDF

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

˘!Yamada, authored, doc1",!Yamada, knows, McShea",!knows, is a kind of, social action",!Herzog, authored, doc2",!Herzog, authored, doc3",!McShea, performed, doc3",!McShea, past action, authored",!doc1, type, PDF",!doc2, type, MP3",!doc3, type, MP3",!doc3, created on, 14.5.08"

¯

(a) A triple graph

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

(b) A visualization of this graph

Fig. 1. A small subset of a personal dataspace graph.

The flexibility of the RDF data model has made it a natural choice for tech-nologies being developed in the social and semantic web research communities[2, 40]. In addition to the central role RDF plays in these areas, recent e!ortstowards the development of dataspaces [13, 20] of personal information [26] havealso highlighted the natural fit of triples in modeling modern data scenarios. Asthe use of RDF grows, the need for RDF data management becomes increasinglycrucial.

There has been an explosion of proposals for RDF query languages [16]. TheW3C recommendation query language SPARQL [36], however, is emerging asthe most mature and accepted. SPARQL is a declarative language with a syntaxvery much like SQL.

Example 2 Consider the query “Retrieve authors and the types of their docu-ments.” In SPARQL, where variables are identified by a preceding ?, this querycan be posed against the graph in Figure 1(a) as follows:

SELECT ?a ?tWHERE { ?a authored ?d . ?d type ?t . }

The WHERE clause of a SPARQL query specifies a basic graph pattern (BGP).Such patterns, which are at the heart of all SPARQL queries, identify a subsetof related atoms to be extracted from the RDF graph, which is then returned asa set of variable mappings.

Example 3 The BGP of Example 2 is (?a authored ?d) ; (?d type ?t). Thesemantics of this pattern with respect to the graph of Figure 1(a) is a set ofvariable mappings, each of which makes the pattern hold in the graph, i.e.,

1 2 3Yamada doc1 PDFHerzog doc2 MP3Herzog doc3 MP3

RDF

subjects predicates objects

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

˘!Yamada, authored, doc1",!Yamada, knows, McShea",!knows, is a kind of, social action",!Herzog, authored, doc2",!Herzog, authored, doc3",!McShea, performed, doc3",!McShea, past action, authored",!doc1, type, PDF",!doc2, type, MP3",!doc3, type, MP3",!doc3, created on, 14.5.08"

¯

(a) A triple graph

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

(b) A visualization of this graph

Fig. 1. A small subset of a personal dataspace graph.

The flexibility of the RDF data model has made it a natural choice for tech-nologies being developed in the social and semantic web research communities[2, 40]. In addition to the central role RDF plays in these areas, recent e!ortstowards the development of dataspaces [13, 20] of personal information [26] havealso highlighted the natural fit of triples in modeling modern data scenarios. Asthe use of RDF grows, the need for RDF data management becomes increasinglycrucial.

There has been an explosion of proposals for RDF query languages [16]. TheW3C recommendation query language SPARQL [36], however, is emerging asthe most mature and accepted. SPARQL is a declarative language with a syntaxvery much like SQL.

Example 2 Consider the query “Retrieve authors and the types of their docu-ments.” In SPARQL, where variables are identified by a preceding ?, this querycan be posed against the graph in Figure 1(a) as follows:

SELECT ?a ?tWHERE { ?a authored ?d . ?d type ?t . }

The WHERE clause of a SPARQL query specifies a basic graph pattern (BGP).Such patterns, which are at the heart of all SPARQL queries, identify a subsetof related atoms to be extracted from the RDF graph, which is then returned asa set of variable mappings.

Example 3 The BGP of Example 2 is (?a authored ?d) ; (?d type ?t). Thesemantics of this pattern with respect to the graph of Figure 1(a) is a set ofvariable mappings, each of which makes the pattern hold in the graph, i.e.,

1 2 3Yamada doc1 PDFHerzog doc2 MP3Herzog doc3 MP3

RDF

RDF/XML, N3, NTriples, ...

... an old idea

Charles S. Peirce (1839-1914)

SPARQL

SPARQL

• W3C standard RDF query language

SPARQL

• W3C standard RDF query language

• SQL-like syntax

SELECT ?a ?tWHERE{

?a authored ?d .?d type ?t

}

SPARQL

• W3C standard RDF query language

• SQL-like syntax

SELECT ?a ?tWHERE{

?a authored ?d .?d type ?t

}

• query processing?

SPARQL

• core of expression is a basic graph pattern

SELECT ?a ?tWHERE{

?a authored ?d .?d type ?t

}

Basic graph patterns

(v1, v2, v3); (v4, v5, v6); v2 = authored; v3 = v4; v5 = type

SELECT ?a ?tWHERE{

?a authored ?d .?d type ?t

}

Basic graph patterns

(v1, v2, v3); (v4, v5, v6); v2 = authored; v3 = v4; v5 = type

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

Basic graph patterns

(v1, v2, v3); (v4, v5, v6); v2 = authored; v3 = v4; v5 = type

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

1 2 3 4 5 6Yamada authored doc1 doc1 type PDFHerzog authored doc2 doc2 type MP3Herzog authored doc2 doc2 type MP3

Basic graph patterns

1 6Yamada PDFHerzog MP3

(v1, v2, v3); (v4, v5, v6); v2 = authored; v3 = v4; v5 = type

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

Basic graph patterns

(v1, v2, v3); · · · ; (v3n!2, v3n!1, v3n); c1; · · · ; cm

Basic triple algebra

E ::= ! | E · E | !c(E)

Basic triple algebra

Given RDF graph G

E ::= ! | E · E | !c(E)

[[!]]G = G

[[E1 · E2]]G = [[E1]]G " [[E2]]G[[!c(E)]]G = {m # [[E]]G | m satisfies c}

(v1, v2, v3); (v4, v5, v6); v2 = authored; v3 = v4; v5 = type

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

Basic triple algebra

PDF

doc3doc2

MP3

Yamada Herzog McShea

socialaction

14.5.08

doc1

type

knows

authored performed

created on

past action

is a kind of

1 2 3 4 5 6Yamada authored doc1 doc1 type PDFHerzog authored doc2 doc2 type MP3Herzog authored doc2 doc2 type MP3

Basic triple algebra!3=4(!2=authored(!) · !2=type(!))

FactFor every BGP there is a semantically equivalent BTA expression, and vice versa.

Basic triple algebra

Basic triple algebra

• little conjunctive algebra

• can leverage ~40 years of research on relational algebra evaluation

‣ e.g., Stocker et al. WWW 2008

• our interest: design space of index data structures for acceleration of SPARQL query processing

Indexing RDF

• state of the art: value-based indexing

Indexing RDF

• state of the art: value-based indexing

• missing: structure-based indexing

‣ e.g., dataguide and A(k) indexes for XML

‣ e.g., join indexes for relational and OO data

Indexing RDF

• methodology for coupling query languages and structural indexes (PODS 2006 & DBPL 2007)

• language indistinguishability

• build data structure over the partition induced by language indistinguishability of objects

Indexing RDF

Let G be a graph, s, s! ! subjects(G), and L be a subset ofBTA.

We say s and s! are L-equivalent, denoted s "L s!, if forany E ! L and n ! N, it is the case that there exists amapping m ! [[E]]G with m[3n + 1] = s if and only if thereexists a mapping m! ! [[E]]G with m![3n + 1] = s!.

The partition induced by "L on subjects(G) is called theL-partition of subjects(G).

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L0 = BTA

• partition is too fine

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L0 = BTA

• partition is too fine

• !1=s(!) " BTA

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L0 = BTA

{[Yamada], [Herzog], [McShea],[doc1], [doc2], [doc3]}

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L1

• disallow conditions involving atoms

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L1

• disallow conditions involving atoms

•

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L1

!3=4(! ·! )

{[Yamada, Herzog, McShea],[doc1, doc2, doc3]}

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L2

• only allow atoms in selections on predicates

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L2

• only allow atoms in selections on predicates

• partition is too fine

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L2

• only allow atoms in selections on predicates

• partition is too fine

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

L2

{[Yamada], [Herzog], [McShea],[doc1], [doc2], [doc3]}

Lk3 ! L2

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

• only allow atoms in selections on predicates

Lk3 ! L2

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

• only allow atoms in selections on predicates

• only allow join paths of length at most k

Lk3 ! L2

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

• only allow atoms in selections on predicates

• only allow join paths of length at most k

Lk3 ! L2

PDF

doc3doc2

MP3

Yamada Herzog McShea

14.5.08

doc1

type

authored performed

created on

{[Yamada, Herzog], [McShea],[doc1], [doc2, doc3]}

Lki ! BTA

• localized fragments of BTA

Lki ! BTA

• localized fragments of BTA

• building blocks of large classes of queries

Lki ! BTA

• design and implementation of index data structures built over partitions induced by localized fragments of BTA

• query processing using these indexes

‣ query decomposition

ongoing work

• BP expressiveness results for SPARQL

• “ontology” discovery for RDF

future work

Thank you!

Questions?