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Comparing Index Structuresfor Completeness ReasoningFariz Darari*, Werner Nutt†, Simon Razniewski‡*Universitas Indonesia, Indonesia†Free University of Bozen-Bolzano, Italy‡Max Planck Institute for Informatics, GermanyIWBIS 2018 - Jakarta, Indonesia

Imagine you are a Spielberg movie fan

Imagine you are a Spielberg movie fanwho happens to be a knowledge graph fan too

Now you want to query to the knowledge graph:Give movies written by Spielberg!Movies written by Spielberg?

The knowledge graph gives some answers..Movies written by Spielberg?AnswersPoltergeistET.....

The knowledge graph gives some answers..Movies written by Spielberg?AnswersPoltergeistET.....But are these answers complete?

The knowledge graph gives some answers..Movies written by Spielberg?AnswersPoltergeistET.....But are these answers complete? Maybe.Maybe yes,maybe no..

Imagine you are a Spielberg movie fanwho happens to be a knowledge graph fan too*Darari, et al. Completeness Statements about RDF Data Sources and Their Use for Query Answering. ISWC 2013.*But now the knowledge graph has been augmented with completeness statements, as in:This knowledge graph is completefor all movies written by Spielberg

Now you want to query to the knowledge graph:Give movies written by Spielberg!Movies written by Spielberg?This knowledge graph is completefor all movies written by Spielberg

The knowledge graph gives some answers..Movies written by Spielberg?AnswersPoltergeistET.....But are these answers complete?This knowledge graph is completefor all movies written by Spielberg

The knowledge graph gives some answers..Movies written by Spielberg?AnswersPoltergeistET.....But are these answers complete? Yes!This knowledge graph is completefor all movies written by Spielberg

Completeness reasoning:Movies written by Spielberg?This knowledge graph is completefor all movies written by SpielbergChecking if completeness statements guarantee query completenessQuery: (?x, type, Movie), (?x, writtenBy, Spielberg)Completeness statement:(?x, type, Movie), (?x, writtenBy, Spielberg)

Completeness reasoning:Movies written by Spielberg?This knowledge graph is completefor all movies written by SpielbergChecking if completeness statements guarantee query completenessQuery: (?x, type, Movie), (?x, writtenBy, Spielberg)Completeness statement:(?x, type, Movie), (?x, writtenBy, Spielberg)Can the statement(s) cover all query components?In general, multiple completeness statements may be requiredto cover all components of the query!

Completeness reasoning scalability challenge:What if there were a million completeness statements?**We'd need this many because a large KG (= knowledge graph) naturallywould require many completeness statements!QueryX 1 million!

X 1 million!QueryUnoptimized completeness reasoningtakes minutes!Completeness reasoning scalability challenge:What if there were a million completeness statements?**We'd need this many because a large KG (= knowledge graph) naturallywould require many completeness statements!

X 1 million!QueryUnoptimized completeness reasoningtakes minutes!Completeness reasoning scalability challenge:What if there were a million completeness statements?*Can we do any better?*We'd need this many because a large KG (= knowledge graph) naturallywould require many completeness statements!

We propose techniquesfor scalable completeness reasoningOur contributions:1. Principle for filtering out irrelevant completeness statements2. Indexing methods for completeness statements3. Experimental evaluations ofnaive completeness reasoning vs. indexed completeness reasoning

Constant-relevance principleA statement contributes to query completeness only ifits constants are among the query’sMovies written by Spielberg?Query: (?x, type, Movie), (?x, writtenBy, Spielberg)Completeness statements:(?x, type, Movie), (?x, writtenBy, Spielberg)(?x, type, Movie), (?x, writtenBy, Bob)(?x, type, Song, (?x, writtenBy, Mary)(Spielberg, child, ?y)

Constant-relevance principleA statement contributes to query completeness only ifits constants are among the query’sMovies written by Spielberg?Query: (?x, type, Movie), (?x, writtenBy, Spielberg)Completeness statements:(?x, type, Movie), (?x, writtenBy, Spielberg)(?x, type, Movie), (?x, writtenBy, Bob)(?x, type, Song, (?x, writtenBy, Mary)(Spielberg, child, ?y)Constants:type, Movie, writtenBy, Spielbergtype, Movie, writtenBy, Bobtype, Song, writtenBy, MarySpielberg, child

Constant-relevance principleA statement contributes to query completeness only ifits constants are among the query’sMovies written by Spielberg?Query: (?x, type, Movie), (?x, writtenBy, Spielberg)Completeness statements:(?x, type, Movie), (?x, writtenBy, Spielberg)(?x, type, Movie), (?x, writtenBy, Bob)(?x, type, Song, (?x, writtenBy, Mary)(Spielberg, child, ?y)Constants: type, Movie, writtenBy, SpielbergConstants:type, Movie, writtenBy, Spielbergtype, Movie, writtenBy, Bobtype, Song, writtenBy, MarySpielberg, child

Constant-relevance principle: subset queryingA statement contributes to query completeness only ifits constants are among the query’sMovies written by Spielberg?Query: (?x, type, Movie), (?x, writtenBy, Spielberg)Completeness statements:(?x, type, Movie), (?x, writtenBy, Spielberg)(?x, type, Movie), (?x, writtenBy, Bob)(?x, type, Song, (?x, writtenBy, Mary)(Spielberg, child, ?y)Constants: type, Movie, writtenBy, SpielbergConstants:type, Movie, writtenBy, Spielbergtype, Movie, writtenBy, Bobtype, Song, writtenBy, MarySpielberg, childReduce the problem into:Subset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constants

Index structure for subset querying:Standard hashing, inverted index, trieSubset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsHash indexes map keys to values: Set-equality queriesHow do we perform subset querying using hash indexes?

Index structure for subset querying:Subset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsHash indexes map keys to values: Set-equality queriesHow do we perform subset querying using hash indexes?By enumerating all possible non-empty subsets of query constants!Standard hashing, inverted index, trie

Index structure for subset querying:Subset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsconst(C1) = a, bconst(C2) = a, b, cconst(C3) = a, b, cconst(C4) = dconst(Q) = a, bStandard hashing, inverted index, trie

Index structure for subset querying:Subset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsconst(C1) = a, bconst(C2) = a, b, cconst(C3) = a, b, cconst(C4) = dconst(Q) = a, bM(a, b) = C1M(a, b, c) = C2, C3M(d) = C4Standard hashing, inverted index, trie

Index structure for subset querying:Subset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsconst(C1) = a, bconst(C2) = a, b, cconst(C3) = a, b, cconst(C4) = dconst(Q) = a, bM(a, b) = C1M(a, b, c) = C2, C3M(d) = C4Relevant completeness statements:M(a) U M(b) U M(a, b) = C1Standard hashing, inverted index, trie

Index structure for subset querying:Standard hashing, inverted index, trieconst(C1) = a, bconst(C2) = a, b, cconst(C3) = a, b, cconst(C4) = dconst(Q) = a, bSubset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constants

Index structure for subset querying:const(C1) = a, bconst(C2) = a, b, cconst(C3) = a, b, cconst(C4) = dconst(Q) = a, bSubset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsStandard hashing, inverted index, trie

Index structure for subset querying:const(C1) = a, bconst(C2) = a, b, cconst(C3) = a, b, cconst(C4) = dconst(Q) = a, bSubset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsBag(Q) = Inv(a) U Inv(b)= C1, C2, C3, C1, C2, C3Constant-relevant statements:Those statements appearing as many times in Bag(Q)as the statements' constantsStandard hashing, inverted index, trie

Index structure for subset querying:Standard hashing, inverted index, trieconst(C1) = a, bconst(C2) = a, b, cconst(C3) = a, b, cconst(C4) = dconst(Q) = a, bSubset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsTraverse the above trie usingquery constant sequence (a, b)

Index structure for subset querying:const(C1) = a, bconst(C2) = a, b, cconst(C3) = a, b, cconst(C4) = dconst(Q) = a, bSubset queryingRetrieve completeness statementswhose constants are subsets ofthe query's constantsTraverse the above trie usingquery constant sequence (a, b)visited node = constant-relevantStandard hashing, inverted index, trie

Experimental evaluation:SetupSynthetic dataset with realistic parametersfrom DBpedia, a large real-world knowledge graphParameters:- Number of completeness statements, default value: 1 million- Max length of completeness statements, default value: 6- Query length, two default values, short = 3 and long = 6

Experimental evaluation:Results – Varying number of statementsShort queries Long queries

Experimental evaluation:Results – Varying max length of statementsShort queries Long queries

Experimental evaluation:Results – Varying query length

Experimental evaluation:Results – Unoptimized vs Indexed Reasoning

Experimental evaluation:Results – Unoptimized vs Indexed ReasoningConclusion: We fast forward completeness reasoning.

Comparing Index Structures for Completeness Reasoning

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