1. Mapping Data in Peer-to-Peer Systems: Semantics and Algorithmic Issues By A. Kementsietsidis, M. Arenas and R.J. Miller Presented by Md. Anisur Rahman: 3558643 Anahit Martirosyan: 100628480 LianXiang Qiu: 3603336 University Of Ottawa Winter 2004

2. Outline  P2P Data-Sharing-System  Mapping Table  Alternative Semantics for Mapping Tables  Mapping Tables as Constraints  An algorithm for checking consistency of the existing mappings and inferring new mappings from them  Conclusion and Future work

3. Peer-to-Peer Data-Sharing System

4. What is a Mapping Table? GDB_idSwissProt_id G1 G2 G3 P9 Q62 P40 P38 GDB_idGene_Name G1 G2 G3 NF1 NF2 NGFB SwissProt_idProtein_ name P9 P40 NF1 MERL Relation GDB Relation SwissProt Mapping Table  A mapping table m from a set of attributes X to a set of attributes Y is a finite set of mappings over X  Y

5. Alternative Semantics for Mapping Tables  Closed-Closed-World Semantics  Closed-Open-World Semantics GDB_idSwissProt_id G2P40 GDB_idSwissProt_id G2 v - {G2} P40 v’ - {P40}

6. Valuation over a mapping table  A valuation p over mapping table m is a function that maps  each constant value in m to itself and  each variable v of m to a value of the domain of the attribute where v appears  If v appears in the expression of the form v-S, then p(v)  S Attr1Attr2 a3 b2 v-{a,b}1 dom(Attr1)={a, b, c, d} dom(Attr2)={1, 2, 3} p(a) = a p(3) = 3 p(v) = c p(v) = d Mapping table m

7. Mapping Constraint GDB_idGene_Name G1 G2 G3 NF1 NF2 NGFB SwissProt_idProtein_ name P9 P40 NF1 MERL GDB_idSwissProt_id G2 v - {G2} P40 v’ - {P40} Relation GDB Relation SwissProt GDB_idGENE_NameSwissprot_idProtein_ Name G1 G2 G3 G2 NF1 NF2 NGFB NF2 P9 P40 P9 NF1 MERL NF1 Mapping table m A relation having attributes from both GDB and SwissProt  Mapping Constraint

8. Extension of a mapping constraint  Given a mapping constraint ext (  ) = {  (t) | t  m and  is a valuation over m } Attr1Attr2 a3 b2 v-{a,b}1 Mapping table m dom(Attr1)={a, b, c, d} dom(Attr2)={1, 2, 3} Attr1Attr2 a3 b2 c1 d1 ext(µ)

9.  A mapping constraint is called the cover of a set of mapping constraints  if   is consistent if and only if there exists t  ext(  )  For every mapping constraint,  ╞  ’ if and only if ext(  )  ext(  ’) Cover of a set of mapping constraints

10. Example of Cover B1B1 B2B2 px1 qy2 rz3 rx4 A1A1 A2A2 B1B1 pxpx qyqy v-{p,q}v’ v’’-{px,qy} C1C1 C2C2 ai bj ck A1A1 A2A2 px qy rz B1B1 C1C1 C2C2 pxai qybj v-{px,qy} v’v’’-{I,j} A1A1 A2A2 C1C1 C2C2 pxai qybj Mapping table m 1 Mapping table m 2 Mapping table m Relation r 1 Relation r 2 Relation r 3  ={  1,  2 }

11. The Algorithm  Input  A path  = P 1, P 2,…., P n of peers  A set  of mapping constraints over path   Two sets of attributes X and Y in peers P 1 and P n  Output:  A mapping constraint that is a cover of 

12. How is the Algorithm useful?  To check whether  ╞  ’  Run the algorithm to find the cover   Check whether ext(  )  ext(  ’).  To check whether  is consistent  Run the algorithm to find the cover   Check whether ext(  ) is nonempty

13. An Example P1P1 P3P3  =P 1, P 2, P 3, P 4  = {µ 1, µ 2,…, µ 11 } {A 1, A 2,.., A 6 } P2P2 {B 1, B 2,.., B 6 } {C 1,C 2,C 3,C 4 } P4P4 {D 3, D 4 }

14. Partitions µ2µ2 µ1µ1 µ3µ3 µ5µ5 µ4µ4 µ6µ6 11 22 33 44

15. Inferred Partitions Peer P 1 Peer P 2 11 22 33 44 55 66 77 11 55 22 66 33 77 44 Inferred partition over P 1 and P 2

16. Advantages of Partitioning  While computing the cover, partitioning reduces computational cost as fewer constraints are considered at a time.  Different partitions can be processed in parallel.

17. Description of the Algorithm  The algorithm has two phases  The Information gathering Phase  The Computation Phase

18. Information Gathering Phase P1P1 P2P2 P3P3 P4P4  Compute partitions  For each partition send to P 2 the set of attributes in the partition  Compute own partitions  Compute inferred partitions using the information of partitions of P 1  Compute own partitions  Compute inferred partitions using the information of propagated inferred partitions from P 2

19. Computation Phase P1P1 P2P2 P3P3 P4P4  Using the local constraints of the inferred partition, computes a cover between P 3 and P 4  The mappings belonging to the cover are streamed to peer P 2.  Determines with which of its own partitions the incoming stream of mapping should be associated  With this information it generates a cover between itself and P 4  Uses the incoming stream of mappings to generate a cover between its own attributes and those of peer P 4

20. Conclusion and Future Scope  This paper showed that by treating mapping tables as constraints on the exchange of information between peers it is possible to reason about them and check their consistency.  There is scope for investigating the use of mapping tables in support of query answering.

21. Thank You