p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Discovering relative importance of skyline attributes D. Mindolin & J. Chomicki Department of Computer Science and Engineering University at Buffalo, SUNY VLDB 2009
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Main contributions 1. generalizing skylines to p-skylines to capture relative attribute importance 2. discovering p-skylines on the basis of user feedback: algorithms and complexity
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Skylines [B¨ orzs¨ onyi et al., ICDE’01] Skyline preferences ◮ Atomic preferences ( H ): total orders over attribute values ◮ Skyline preference relation ( sky H ): t 1 preferred to t 2 if ◮ t 1 equal or better than t 2 in every attribute, and ◮ t 1 strictly better than t 2 in at least one attribute ◮ Skyline: the set w sky H ( O ) of best tuples (according to sky H ) in a set of tuples O
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Skylines [B¨ orzs¨ onyi et al., ICDE’01] Skyline preferences ◮ Atomic preferences ( H ): total orders over attribute values ◮ Skyline preference relation ( sky H ): t 1 preferred to t 2 if ◮ t 1 equal or better than t 2 in every attribute, and ◮ t 1 strictly better than t 2 in at least one attribute ◮ Skyline: the set w sky H ( O ) of best tuples (according to sky H ) in a set of tuples O Example Y X
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Skylines [B¨ orzs¨ onyi et al., ICDE’01] Skyline preferences ◮ Atomic preferences ( H ): total orders over attribute values ◮ Skyline preference relation ( sky H ): t 1 preferred to t 2 if ◮ t 1 equal or better than t 2 in every attribute, and ◮ t 1 strictly better than t 2 in at least one attribute ◮ Skyline: the set w sky H ( O ) of best tuples (according to sky H ) in a set of tuples O Example Skyline properties ◮ Simple, unique way of Y composing atomic preferences ◮ Equal attribute importance ◮ Skyline of exponential size X
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work p-skylines p-skyline relation ≻ ◮ Induced by an atomic preference relation > A ∈ H ≻ = { ( t , t ′ ) | t . A > A t ′ . A } ◮ Pareto accumulation (“ ≻ 1 equally important as ≻ 2 “) ≻ = ≻ 1 ⊗ ≻ 2 ◮ Prioritized accumulation (“ ≻ 1 more important than ≻ 2 “) ≻ = ≻ 1 & ≻ 2
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work p-skylines p-skyline relation ≻ ◮ Induced by an atomic preference relation > A ∈ H ≻ = { ( t , t ′ ) | t . A > A t ′ . A } ◮ Pareto accumulation (“ ≻ 1 equally important as ≻ 2 “) ≻ = ≻ 1 ⊗ ≻ 2 ◮ Prioritized accumulation (“ ≻ 1 more important than ≻ 2 “) ≻ = ≻ 1 & ≻ 2 Each atomic preference must be used exactly once in ≻
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Pareto accumulation [Kießling’02] Definitions Var ( ≻ ) - set of attributes used in definition of ≻ E S = { ( t . A , t ′ . A ) | A ∈ S ∧ t . A = t ′ . A } - pairs of tuples equal in every attribute in S Pareto accumulation: ≻ 1 as important as ≻ 2 ≻ 1 ⊗ ≻ 2 = ( ≻ 1 ∩ E Var ( ≻ 2 ) ) ∪ ( ≻ 2 ∩ E Var ( ≻ 1 ) ) ∪ ( ≻ 1 ∩ ≻ 2 )
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Pareto accumulation [Kießling’02] Definitions Var ( ≻ ) - set of attributes used in definition of ≻ E S = { ( t . A , t ′ . A ) | A ∈ S ∧ t . A = t ′ . A } - pairs of tuples equal in every attribute in S Pareto accumulation: ≻ 1 as important as ≻ 2 ≻ 1 ⊗ ≻ 2 = ( ≻ 1 ∩ E Var ( ≻ 2 ) ) ∪ ( ≻ 2 ∩ E Var ( ≻ 1 ) ) ∪ ( ≻ 1 ∩ ≻ 2 ) ≻ X ⊗ ≻ Y Y X
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Prioritized accumulation [Kießling’02] Definitions Var ( ≻ ) - set of attributes used in definition of ≻ E S = { ( t . A , t ′ . A ) | A ∈ S ∧ t . A = t ′ . A } - pairs of tuples equal in every attribute in S Prioritized accumulation: ≻ 1 more important than ≻ 2 ≻ 1 & ≻ 2 = ≻ 1 ∪ ( ≻ 2 ∩ E Var ( ≻ 2 ) )
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Prioritized accumulation [Kießling’02] Definitions Var ( ≻ ) - set of attributes used in definition of ≻ E S = { ( t . A , t ′ . A ) | A ∈ S ∧ t . A = t ′ . A } - pairs of tuples equal in every attribute in S Prioritized accumulation: ≻ 1 more important than ≻ 2 ≻ 1 & ≻ 2 = ≻ 1 ∪ ( ≻ 2 ∩ E Var ( ≻ 2 ) ) ≻ X & ≻ Y Y X
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work p-skyline properties p-skyline properties ◮ Many different ways of composing atomic preferences (different combinations of ⊗ and & ) ◮ Reduction in query result size w.r.t. skylines ◮ Differences in attribute importance
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Representing attribute importance with p-graphs p-graph Γ ≻ represents attribute importance induced by a p-skyline relation ≻ ◮ Nodes: attributes Var ( ≻ ) ◮ Edges: from more important to less important attributes
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Representing attribute importance with p-graphs p-graph Γ ≻ represents attribute importance induced by a p-skyline relation ≻ ◮ Nodes: attributes Var ( ≻ ) ◮ Edges: from more important to less important attributes ≻ ′ = ≻ A ⊗ ≻ B ⊗ ≻ C A B C
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Representing attribute importance with p-graphs p-graph Γ ≻ represents attribute importance induced by a p-skyline relation ≻ ◮ Nodes: attributes Var ( ≻ ) ◮ Edges: from more important to less important attributes ≻ ′ = ≻ A ⊗ ≻ B ⊗ ≻ C ≻ ′′ = ≻ A & ( ≻ B ⊗ ≻ C ) A A B C B C
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Properties of p-graphs Necessary and sufficient conditions for p-graphs Γ is a p-graph of a p-skyline relation iff Γ is ◮ SPO ◮ satisfies Envelope property Envelope ∀ A , B , C , D ∈ A , all different ( A , B ) ∈ Γ ∧ ( C , D ) ∈ Γ ∧ ( C , B ) ∈ Γ ⇒ ( C , A ) ∈ Γ ∨ ( A , D ) ∈ Γ ∨ ( D , B ) ∈ Γ A C B D
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Dominance testing using p-graphs Is o preferred to o ′ by ≻ ? o ≻ o ′ iff ◮ o � = o ′ , and ◮ for every attribute B in which o is worse than o ′ , there is a parent A of B in which o is better than o ′
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Dominance testing using p-graphs Is o preferred to o ′ by ≻ ? o ≻ o ′ iff ◮ o � = o ′ , and ◮ for every attribute B in which o is worse than o ′ , there is a parent A of B in which o is better than o ′ Example ≻ = ≻ A & ( ≻ B ⊗ ≻ C ) A : b better than w B : b better than w A C : b better than w Then B C (b , w , b) ≻ (w , b , b) (b , w , b) �≻ (b , b , w)
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Containment of p-skyline relations Containment hierarchy Using p-graphs for checking containment A B C ≻ ⊂ ≻ ′ ⇔ E (Γ ≻ ) ⊂ E (Γ ≻ ′ ) A C A C B C B C A B A B B B A A C C A C A B C B A B C B C B A A C C A B A A B B C C C B A C A B B C C A B A
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Containment of p-skyline relations Containment hierarchy Using p-graphs for checking containment A B C ≻ ⊂ ≻ ′ ⇔ E (Γ ≻ ) ⊂ E (Γ ≻ ′ ) A C A C B C B C A B A B Minimal extensions of ≻ B B A A C C ◮ Correspond to immediate children of Γ ≻ in the hierarchy A C A B C B A B C ◮ Obtained using rewriting rules B C B A A C C A B applied to syntax trees of p-skyline formulas A A B B C C C B A C A B B C C A B A
p-skylines Properties of p-skyline relations p-skyline relation elicitation Related & future work Minimal extension rewriting rules Rules to compute minimal extensions of p-skyline relation ◮ Applied to syntax trees of p-skyline formulas ◮ Every minimal extension computed by a single rule application in PTIME ◮ Full set consists of four rule templates ◮ All minimal extensions of p-skyline relation can be computed in PTIME
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