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Defining relations on graphs: how hard is it in the presence of node partitions?
- M. Praveen and B. Srivathsan
CMI
Defining relations on graphs: how hard is it in the presence of node - - PowerPoint PPT Presentation
Defining relations on graphs: how hard is it in the presence of node - - PowerPoint PPT Presentation
Defining relations on graphs: how hard is it in the presence of node partitions? M. Praveen and B. Srivathsan CMI 9 February 2015 Regular Path Queries on Graphs a b u v c a b u v Regular Path Queries on Graphs a b u v c a
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Regular Path Queries on Graphs
u v u′ v′ a b c a b Regular path query Q : x
ab
− − → y
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Regular Path Queries on Graphs
u v u′ v′ a b c a b Regular path query Q : x
ab
− − → y Q(G) = {u, v, u′, v′}
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RPQ-definability
◮
u v u′ v′ a b c a b
◮ {u, v}
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RPQ-definability
◮
u v u′ v′ a b c a b
◮ {u, v} ◮ Is there a RPQ Q such that Q(G) = {u, v}?
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RPQ-definability
◮
u v u′ v′ a b c a b
◮ {u, v} ◮ Is there a RPQ Q such that Q(G) = {u, v}? ◮ [Antonopoulos, Neven, Servais 2013] RPQ-definability is
Pspace-complete.
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Node partitions
u 1 2 v 1 u′ 1 2 v′ 2 a b c a b
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Node partitions
u 1 2 v 1 u′ 1 2 v′ 2 a b c a b Regular data path query Q : x
↓r1.ab[r=
1 ]
− − − − − − − → y
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Node partitions
u 1 2 v 1 u′ 1 2 v′ 2 a b c a b Regular data path query Q : x
↓r1.ab[r=
1 ]
− − − − − − − → y Q(G) = {u, v}
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Node partitions
u 1 2 v 1 u′ 1 2 v′ 2 a b c a b Regular data path query Q : x
↓r1.ab[r=
1 ]
− − − − − − − → y Q(G) = {u, v} e ::= ε | a | e + e | e+ | ↓ r.e | e[c] c ::= r= | r= | c ∧ c | c ∨ c | ¬c
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RDPQ-definability
◮
u 1 2 v 1 u′ 1 2 v′ 2 a b c a b
◮ {u, v}
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RDPQ-definability
◮
u 1 2 v 1 u′ 1 2 v′ 2 a b c a b
◮ {u, v} ◮ Is there a RDPQ Q such that Q(G) = {u, v}? ◮ We study the complexity of RDPQ-definability.
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Motivation - schema mappings
Chennai Bordeaux Chennai Stuttgart Bordeaux friend colleague friend colleague
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Motivation - schema mappings
Chennai Bordeaux Chennai Stuttgart Bordeaux friend colleague friend colleague
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Motivation - schema mappings
Chennai Bordeaux Chennai Stuttgart Bordeaux friend colleague friend colleague ↓ r1.(friend + collegue)∗[r=
1 ]
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Related work
G.H.L. Fletcher, M. Gyssens, J. Paredaens, and D. V. Gucht. On the expressive power of the relational algebra on finite sets
- f relation pairs.
IEEE Trans. Knowledge and Data Engg., 21(6):939–942, 2009.
- G. Gottlob and P. Senellart.
Schema mapping discovery from data instances.
- J. ACM, 57(2):6:1–6:37, 2010.
- A. Das Sarma, A. Parameswaran, H. Garcia-Molina, and
- J. Widom.
Synthesizing view definitions from data. In Proceedings, ICDT, pages 89–103, 2010.
- B. Alexe, B. T. Cate, P. G. Kolaitis, and W. Tan.
Designing and refining schema mappings via data examples. In SIGMOD, pages 133–144, 2011.
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Related work . . .
- D. Calvanese, G. De Giacomo, M. Lenzerini, and M. Y. Vardi.
Simplifying schema mappings. In ICDT, pages 114–125, 2011.
- P. Barcel´
- , J. P´
erez, and J. Reutter. Schema mappings and data exchange for graph databases. In ICDT, pages 189–200, 2013.
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Regular Expressions with Equality
u1 1 v1 1 a a a u2 3 v2 1 a a a u3 1 2 3 v3 1 a a a Q = (a · (a)= · a)=
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Regular Expressions with Equality
u1 1 v1 1 a a a u2 3 v2 1 a a a u3 1 2 3 v3 1 a a a Q = (a · (a)= · a)= Q(G) = {u1, v1}
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Regular Expressions with Equality
u1 1 v1 1 a a a u2 3 v2 1 a a a u3 1 2 3 v3 1 a a a Q = (a · (a)= · a)= Q(G) = {u1, v1} e ::= ε | a | e + e | e+ | e= | e=
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Number of registers
u1 2 3 2 v1 3 a a a u2 1 v2 2 a a a u3 1 2 3 v3 2 a a a
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Number of registers
u1 2 3 2 v1 3 a a a u2 1 v2 2 a a a u3 1 2 3 v3 2 a a a Q =↓ r1 · a· ↓ r2 · a[r=
1 ] · a[r= 2 ]
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Number of registers
u1 2 3 2 v1 3 a a a u2 1 v2 2 a a a u3 1 2 3 v3 2 a a a Q =↓ r1 · a· ↓ r2 · a[r=
1 ] · a[r= 2 ]
Q(G) = {u1, v1}
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Results
◮ RDPQmem-definability is Expspace-complete. ◮ k − RDPQmem-definability is in NSpace(O(n2δk)). ◮ RDPQ=-definability is Pspace-complete. ◮ UCRDPQ-definability is coNP-complete.
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Witnesses for RPQ-definability
u v u′ v′ a a c a b
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Witnesses for RDPQ-definability
u v u′ v′ 1 2 1 1 2 2 a a a a c
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Witnesses for RDPQ-definability
u v u′ v′ u′ v′ c
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Witnesses for RDPQ-definability
u v u′ v′ u′ v′ c
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Witnesses for RDPQ-definability
u v u′ v′ u′ v′ c
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Witnesses for RDPQ-definability
u v u′ v′ u′ v′ c
- ↓ r
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Witnesses for RDPQ-definability
u v u′ v′ u′ v′ c
- ↓ r
[r=]
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Witnesses for RDPQ-definability
u v u′ v′ u′ v′ c
- ↓ r
[r=] ↓ r
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Witnesses for RDPQ-definability
u v u′ v′ u′ v′ c
- ↓ r
[r=] ↓ r [r=]
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Expspace Lower Bound
ti t2 t3 . . . tf 2n
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Expspace Lower Bound
ti t2 t3 . . . tf 2n p1 illegal tilings q1 p2 all tilings q2
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Expspace Lower Bound
ti t2 t3 . . . tf 2n p1 illegal tilings q1 p2 all tilings q2 There exists a legal tiling iff {p2, q2} is definable.
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Expspace Lower Bound
ti t2 t3 . . . tf 2n
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Expspace Lower Bound
ti t2 t3 . . . tf 2n ti t2 . . .
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Expspace Lower Bound
ti t2 t3 . . . tf 2n ti t2 . . . t3 . . .
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Expspace Lower Bound
ti t2 t3 . . . tf 2n ti t2 . . . t3 . . . ↓ rn · · · ↓ r2 ↓ r2
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Expspace Lower Bound
ti t2 t3 . . . tf 2n ti t2 . . . t3 . . . ↓ rn · · · ↓ r2 ↓ r2 r=
n
· · · r=
2
r=
1
r=
n
· · · r=
2
r=
1
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Conclusion
◮ RDPQmem-definability is Expspace-complete. ◮ k − RDPQmem-definability is in NSpace(O(n2δk)). ◮ RDPQ=-definability is Pspace-complete. ◮ UCRDPQ-definability is coNP-complete.
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Conclusion
◮ RDPQmem-definability is Expspace-complete. ◮ k − RDPQmem-definability is in NSpace(O(n2δk)). ◮ RDPQ=-definability is Pspace-complete. ◮ UCRDPQ-definability is coNP-complete.