Repairing Entities using Star Constraints in Multi-relational Graphs - PowerPoint PPT Presentation

Repairing Entities using Star Constraints in Multi-relational Graphs Peng Lin 1 Qi Song 1 Yinghui Wu 2,3 Jiaxing Pi 4 1 2 4 3

Erroneous entities: how to capture? § Multi-relational graphs: a labeled graph with attributes on nodes 𝒘 𝟏 Player name: VanPersie playsFor teammate playsFor coachedBy Coach Club Club Player name: Wenger name: AFC name: MU name: Rooney operates trainsAt operates trainsAt worksAt Stadium Facility Stadium Facility name: ATC name: EM name: OT name: AON owner: AHP owner: AHP owner: MUP owner: MUP city: LDN city: BZ city: MAN city: LD 𝒘 𝟐 𝒘 𝟑 𝒘 𝟓 𝒘 𝟒 Graph G: a football database 1

Erroneous entities: how to capture? § Multi-relational graphs: a labeled graph with attributes on nodes § Entity errors: incorrect node attributes 𝒘 𝟏 Player name: VanPersie playsFor teammate playsFor coachedBy Coach Club Club Player name: Wenger name: AFC name: MU name: Rooney operates trainsAt operates trainsAt worksAt Stadium Facility Stadium Facility name: ATC name: EM name: OT name: AON owner: AHP owner: AHP owner: MUP owner: MUP city: LDN city: BZ city: MAN city: LD 𝒘 𝟐 𝒘 𝟑 𝒘 𝟓 𝒘 𝟒 Graph G: a football database 1

Erroneous entities: how to capture? § Multi-relational graphs: a labeled graph with attributes on nodes § Entity errors: incorrect node attributes § Semantics: relevant paths from a center node “For stadium and facility relevant to player ( 𝒘 𝟏 ) 𝒘 𝟏 Player from Premier League, if they have the same name: VanPersie owner, then they should locate at the same city.” playsFor teammate playsFor coachedBy Coach Club Club Player name: Wenger name: AFC name: MU name: Rooney operates trainsAt operates trainsAt worksAt Stadium Facility Stadium Facility name: ATC name: EM name: OT name: AON owner: AHP owner: AHP owner: MUP owner: MUP city: LDN city: BZ city: MAN city: LD 𝒘 𝟐 𝒘 𝟑 𝒘 𝟓 𝒘 𝟒 Graph G: a football database 1

Regular path queries Regular expressions: 𝑆 = 𝑚 𝑚 &' 𝑆 % 𝑆|𝑆 ∪ 𝑆 § 𝒘 𝟏 Player name: VanPersie playsFor teammate playsFor coachedBy Coach Club Club Player name: Wenger name: AFC name: MU name: Rooney operates trainsAt operates trainsAt worksAt Facility Stadium Stadium Facility name: ATC name: EM name: OT name: AON owner: AHP owner: AHP owner: MUP owner: MUP city: LDN city: BZ city: MAN city: LD 𝒘 𝟐 𝒘 𝟑 𝒘 𝟓 𝒘 𝟒 Graph G: a football database 2

Regular path queries Regular expressions: 𝑆 = 𝑚 𝑚 &' 𝑆 % 𝑆|𝑆 ∪ 𝑆 § § Paths from Player to Stadium 𝑆 ! = (playsFor , operates) ∪ (coachedBy , worksAt) § 𝒘 𝟏 Player name: VanPersie playsFor teammate playsFor coachedBy Coach Club Club Player name: Wenger name: AFC name: MU name: Rooney operates trainsAt operates trainsAt worksAt Facility Stadium Stadium Facility name: ATC name: EM name: OT name: AON owner: AHP owner: AHP owner: MUP owner: MUP city: LDN city: BZ city: MAN city: LD 𝒘 𝟐 𝒘 𝟑 𝒘 𝟓 𝒘 𝟒 Graph G: a football database 2

Regular path queries Regular expressions: 𝑆 = 𝑚 𝑚 &' 𝑆 % 𝑆|𝑆 ∪ 𝑆 § § Paths from Player to Stadium 𝑆 ! = (playsFor , operates) ∪ (coachedBy , worksAt) § 𝒘 𝟏 Player § Paths from Player to Facility 𝑆 " = (playsFor , operates) ∪ (teammate #! , trainsAt) name: VanPersie § playsFor teammate playsFor coachedBy Coach Club Club Player name: Wenger name: AFC name: MU name: Rooney operates trainsAt operates trainsAt worksAt Facility Stadium Stadium Facility name: ATC name: EM name: OT name: AON owner: AHP owner: AHP owner: MUP owner: MUP city: LDN city: BZ city: MAN city: LD 𝒘 𝟐 𝒘 𝟑 𝒘 𝟓 𝒘 𝟒 Graph G: a football database 2

Contributions StarRepair framework Repair 𝐻’ Graph 𝐻 , StarFDs Σ Error detection Repair ( 𝐻 does not satisfy Σ ) ( 𝐻’ satisfies Σ ) 3

Contributions StarFDs: star functional dependencies Entity repair problem: minimum new constraints for graphs editing cost, NP-hard and APX-hard StarRepair framework Repair 𝐻’ Graph 𝐻 , StarFDs Σ Error detection Repair ( 𝐻 does not satisfy Σ ) ( 𝐻’ satisfies Σ ) Feasible framework with provable guarantees whenever possible 3

Contributions StarFDs: star functional dependencies Entity repair problem: minimum new constraints for graphs editing cost, NP-hard and APX-hard StarRepair framework Repair 𝐻’ Graph 𝐻 , StarFDs Σ Error detection Repair ( 𝐻 does not satisfy Σ ) ( 𝐻’ satisfies Σ ) Repair workflow Is approximable? Feasible framework with provable guarantees whenever possible No Yes Is optimal repairable? Heuristic solution Yes No Optimal solution Approximation solution 3

Star constraints StarFDs: 𝜒 = (𝑄(𝑣 ( ), 𝑌 → 𝑍) § Star pattern 𝑄(𝑣 ( ) : § Value constraints: 𝑌 → 𝑍 § 4

Star constraints StarFDs: 𝜒 = (𝑄(𝑣 ( ), 𝑌 → 𝑍) § Star pattern 𝑄(𝑣 ( ) : § Value constraints: 𝑌 → 𝑍 § - A two-level tree with center node 𝑣 ( - Each branch is a regular expression 𝒗 𝟏 Player 𝑺 𝟐 𝑺 𝟑 Stadium Facility 𝒗 𝟐 𝒗 𝟑 𝑆 % = (playsFor 0 operates) ∪ (coachedBy 0 worksAt) 𝑆 # = (playsFor 0 operates) ∪ (teammate $% 0 trainsAt) 4

Star constraints StarFDs: 𝜒 = (𝑄(𝑣 ( ), 𝑌 → 𝑍) § Star pattern 𝑄(𝑣 ( ) : § Value constraints: 𝑌 → 𝑍 § - A two-level tree with center node 𝑣 ( - 𝑌 and 𝑍 are two sets of literals Literals: 𝑣. 𝐵 = 𝑑 , or 𝑣. 𝐵 = 𝑣 ) . 𝐵′ - Each branch is a regular expression - 𝒗 𝟏 Player 𝑌 : 𝑣 $ . league = EPL, 𝑣 ! . owner = 𝑣 " . owner 𝑺 𝟐 𝑺 𝟑 𝑍 : 𝑣 ! . city = 𝑣 " . city Stadium Facility 𝒗 𝟐 𝒗 𝟑 𝑆 % = (playsFor 0 operates) ∪ (coachedBy 0 worksAt) 𝑆 # = (playsFor 0 operates) ∪ (teammate $% 0 trainsAt) 4

Star constraints § Matching semantics: maximum set matched by star pattern 𝒗 𝟏 Player 𝑺 𝟑 𝑺 𝟐 Facility Stadium 𝒗 𝟐 𝒗 𝟑 Star pattern 𝑄(𝑣 $ ) 𝑌 : 𝑣 & . league = EPL, 𝑣 % . owner = 𝑣 # . owner 𝑍 : 𝑣 % . city = 𝑣 # . city 5

Star constraints 𝒗 𝟏 matches 𝒘 𝟏 § Matching semantics: maximum set matched by star pattern 𝒗 𝟐 matches 𝒘 𝟐 and 𝒘 𝟓 𝒗 𝟑 matches 𝒘 𝟑 and 𝒘 𝟒 𝒘 𝟏 Player name: VanPersie 𝒗 𝟏 Player playsFor teammate playsFor coachedBy 𝑺 𝟑 𝑺 𝟐 Coach Club Club Player name: Wenger name: AFC name: MU name: Rooney Facility Stadium operates trainsAt operates trainsAt worksAt 𝒗 𝟐 𝒗 𝟑 Facility Stadium Stadium Facility Star pattern 𝑄(𝑣 $ ) name: ATC name: EM name: OT name: AON owner: AHP owner: AHP owner: MUP owner: MUP 𝑌 : 𝑣 & . league = EPL, 𝑣 % . owner = 𝑣 # . owner city: LDN city: BZ city: MAN city: LD 𝑍 : 𝑣 % . city = 𝑣 # . city 𝒘 𝟐 𝒘 𝟑 𝒘 𝟓 𝒘 𝟒 5

Star constraints 𝒗 𝟏 matches 𝒘 𝟏 § Matching semantics: maximum set matched by star pattern 𝒗 𝟐 matches 𝒘 𝟐 and 𝒘 𝟓 Inconsistencies 𝑱 : matches that 𝑌 holds but 𝑍 does not hold § 𝒗 𝟑 matches 𝒘 𝟑 and 𝒘 𝟒 𝒘 𝟏 Player name: VanPersie 𝒗 𝟏 Player playsFor teammate playsFor coachedBy 𝑺 𝟑 𝑺 𝟐 Coach Club Club Player name: Wenger name: AFC name: MU name: Rooney Facility Stadium operates trainsAt operates trainsAt worksAt 𝒗 𝟐 𝒗 𝟑 Facility Stadium Stadium Facility Star pattern 𝑄(𝑣 $ ) name: ATC name: EM name: OT name: AON owner: AHP owner: AHP owner: MUP owner: MUP 𝑌 : 𝑣 & . league = EPL, 𝑣 % . owner = 𝑣 # . owner city: LDN city: BZ city: MAN city: LD 𝑍 : 𝑣 % . city = 𝑣 # . city 𝒘 𝟐 𝒘 𝟑 𝒘 𝟓 𝒘 𝟒 5

Summary of results Problem Description Hardness Solution Input: Σ Satisfiability NP-complete decide whether there exists 𝐻 that satisfies Σ Input: Σ and 𝜒 Implication coNP-hard decide whether for all 𝐻 satisfy Σ , they satisfy 𝜒 Input: 𝐻 and Σ Error detection PTIME Evaluate regular path queries and validate values Output: all inconsistencies 𝑱 time complexity: 𝑃( Σ V + |𝑊|( 𝑊 + |𝐹|)) (validation) - Input: Σ and 𝐻 that does not satisfy Σ Repair NP-hard Approximable cases (PTIME checkable) time complexity 𝑃( 𝑱 Σ ! + 𝑱 ( 𝑱 Σ ! + |𝑱| Σ )) Ouput: 𝐻′ that satisfies Σ with least repair cost APX-hard - approximation ratio: 𝑱 Σ ! - Optimal cases time complexity 𝑃( 𝑱 Σ )) - Heuristic cases time complexity 𝑃( 𝑱 Σ ! + 𝑱 ( 𝑱 Σ ! + |𝑱| Σ )) - bounded repairable: cost ≤ 𝑱 - Notations 𝐻 : graph 𝑊 : nodes 𝐹 : edges § Σ : a set of StarFDs 𝜒 : a single StarFD 𝑱 : all inconsistencies. 6

Updates and repairs Updates 𝑃 : operators 𝑝 = (𝑤. 𝐵, 𝑏, 𝑑) with editing cost cost 𝑃 = ∑ (∈+ cost 𝑝 § Repair 𝑃 : applying 𝑃 to 𝐻 , such that obtain 𝐻′ that satisfies Σ § 7