1. Aggregate Queries in Peer-to-Peer OLAP Mauricio Minuto Espil Faculty of Engineering Universidad Católica Argentina Alejandro A. Vaisman Computer Science Department Universidad de Buenos Aires 7 th International Workshop on Data Warehousing & OLAP

2. OUTLINE: CHARACTERIZATION PROBLEM AND PROPOSAL FACT INTEGRATION DIMENSION INTEGRATION AGGREGATE QUERIES CONCLUSIONS Aggregate Queries in Peer-to-Peer OLAP

3. Peer-to-Peer Systems  Involves a network of interconnected peer systems;  The network topology is not relevant;  Each peer maintains full autonomy over its own data resources;  Each peer may assume the role of local. The rest become acquaintances of the local peer;  The roles of local and acquaintance among peers are not static; they are functional and are determined with respect to an operation. MAIN CHARACTERISTICS:

4. Peer-to-Peer Data Management  No global schema is assumed to exist for data;  Each peer must manage its data according its own perspective;  A query may be posed on any peer, the responsive peer becomes local with respect to the query;  Answers to queries must conform the best attempt to gather data from all peers;  Answers to queries posed by local peer users must conform the view those users have of their data;  Peers must cooperate in maintaining the local views of data; MAIN CHARACTERISTICS:

5. OUTLINE: CHARACTERIZATION PROBLEM AND PROPOSAL FACT INTEGRATION DIMENSION INTEGRATION AGGREGATE QUERIES CONCLUSIONS Aggregate Queries in Peer-to-Peer OLAP

6. OLAP Data in a Peer-to-Peer System OLAP data is essentially multidimensional; Multidimensional data consists in a collection of views of base and derived aggregated data, describing fact indicators by dimensions of analysis; Concepts for aggregation within dimensions are obtained from finer grain concepts through hierarchies; Different peers may have affine fact indicators described by different dimension hierarchies; Integration is needed: Any summary concept that appears in a hierarchy of a peer acquaintance must be transformed into a summary concept meaningful to the local peer. > THE PROBLEM:

7. OLAP Data in a Peer-to-Peer System The expected integration is not always possible; Users may pose OLAP queries in a local peer expecting results involving all relevant data stored in all peers. Local queries must be propagated among the acquaintances; A rewriting of the propagated queries is needed to conform the view of the local user. The rewriting technique must accomplish the data integration on the fly; Incomplete and uncertain results must be admitted; > THE PROBLEM

8. Peer-to-Peer OLAP FACT PEERS DIMENSION PEERS AGGREGATE P2P OLAP QUERIES COMPLETE AND CERTAIN QUERY ANSWERS MODEL (DEFINES): AUTONOMOUS PEER DATA MANAGEMENT THREE PHASE PEER TO PEER COORDINATION COOPERATIVE QUERY ANSWERING ARCHITECTURE (INVOLVES):

9. OUTLINE: CHARACTERIZATION PROBLEM AND PROPOSAL FACT INTEGRATION DIMENSION INTEGRATION AGGREGATE QUERIES CONCLUSIONS Aggregate Queries in Peer-to-Peer OLAP

10. Fact Integration GENERIC FACT FACT PEERS TYPES OF FACT: IS-A RELATIONSHIP FACT CONCILIATION PHASE: SOURCE PEER PUBLISHES GENERIC FACT DEFINITION AND DIMENSIONAL STRUCTURE LISTENING PEER GENERIC FACT AGREEMENT AND DIMENSION PEERS DEFINITION

11. OUTLINE: CHARACTERIZATION PROBLEM AND PROPOSAL FACT INTEGRATION DIMENSION INTEGRATION AGGREGATE QUERIES CONCLUSIONS Aggregate Queries in Peer-to-Peer OLAP

12. Dimension Integration LEVEL HIERARCHY INTEGRATION MEMBER HIERARCHY INTEGRATION. CONSISTS IN: CORRESPONDENCE DEFINITION AMONG DIMENSION LEVELS REVISION/MAPPING DEFINITION AMONG DIMENSION INSTANCES COMPRISES: INVOLVES: A PAIR OF DIMENSION PEERS

13. Level Hierarchy Integration LEVEL CORRESPONDENCE APPLIES ON SCHEMAS ESTABLISHES HOW A PAIR OF LEVELS ON DIFFERENT PEER DIMENSIONS ARE RELATED IS PRODUCED/UPDATED DURING A SCHEMA CONCILIATION PHASE IS MATERIALIZED AS METADATA IN CORRESPONDENCE TABLES

14. ORDER PRESERVING LEVEL CORRESPONDENCE Benefit Type Funding Class All Tax Discharge Category Loan Type All Charity Modality Benefit Type Level Hierarchy Integration

15. A LEVEL CORRESPONDENCE THAT DO NOT PRESERVE ORDER IS NOT ADMISSIBLE Benefit Type Funding Class All Tax Discharge Category Loan Types All Charity Modality Benefit Type Level Hierarchy Integration WRONG

16. Member Hierarchy Integration INTEGRATION BY MAPPING APPLIES ON INSTANCES ESTABLISHES HOW A PAIR OF MEMBERS OF CORRESPONDING LEVELS ARE RELATED IS PRODUCED/UPDATED DURING A MAPPING ACQUISITION PHASE MUST BE PRECEDED BY AT LEAST ONE SCHEMA CONCILIATION PHASE IS MATERIALIZED AS METADATA IN MAPPING TABLES

17. For each member m of a level l, such that map (l:m) is defined, if there exists some member m’ of level l’, satisfying roll-up (l:m) = l’:m’ and level l’ is in dom(Correspondence) then roll-up (map (l:m) ) = map (l’:m’). Member Hierarchy Integration MAPPINGS: HOMOMORPHISM PROPERTY l:m l':m’ map roll-up

18. Member m’ in level l’ is conflicting, it cannot be mapped. An approach based on mapping exclusively is not always effective. Member Hierarchy Integration HOMOMORPHISM MAY NOT BE ALWAYS GRANTED l:m 1 l':m’ map roll-up l:m 2 roll-up

19. MAPPINGS DO NOT SUFFICE: REVISIONS MAY BE NECESSARY Member Hierarchy Integration l:m 1 l':m’ l:m 2 Conflicting Member REVISIONS AFFECT THE VIEW A PEER HAS OF THE HIERARCHY OF ITS ACQUAINTANCE ONLY LOCAL ACQUAINTANCE

20. A REVISION BY SPLITTING MAY BE USED TO REPAIR CONFLICTS GIVING WAY TO MAPPABLE MEMBERS Member Hierarchy Integration l:m 1 l':m 2 ’ l:m 2 l:m 1 ’ LOCAL ACQUAINTANCE EXAMPLE OF A REVISION: CONFLICTING MEMBER SPLIT Non-Conflicting Members

21. A REVISION BY RECLASSIFYING MAY BE AN ALTERNATIVE TO RESTORE HOMOMORPHISM Member Hierarchy Integration l:m 1 l:m 2 l:m’ LOCAL ACQUAINTANCE l:m 3 l':m” EXAMPLE OF A REVISION: CONFLICTING MEMBER RECLASSIFICATION Non-Conflicting Members

22. PRODUCES AND BROADCASTS REVISION AND MAPPING DEFINITIONS TO POTENTIAL ACQUAINTANCES REVISE AND MAP APPROACH: LOCAL PEER: Member Hierarchy Integration ACQUAINTANCE: REVISES ITS OWN HIERARCHIES PRODUCING A REVISED INSTANCE (REVISED ROLL-UPS) WITH RESPECT TO THE LOCAL PEER STORE INFORMATION ON MAPPINGS IN METADATA MAPPING TABLES

23. Whenever some member m 2 ’ of a level l’ is not mapped, a bottom-up completion approach for query answering is employed: information on non-mapped members and their roll-ups is stored in metadata completion tables. Member Hierarchy Integration BOTTOM-UP COMPLETION APPROACH l:m 1 l':m 2 ’ map Incomplete roll-up l:m 2 roll-up l':m 1 ’ Non-Mapped Member

24. OUTLINE: CHARACTERIZATION PROBLEM AND PROPOSAL FACT INTEGRATION DIMENSION INTEGRATION AGGREGATE QUERIES CONCLUSIONS Aggregate Queries in Peer-to-Peer OLAP

25. P2P OLAP Queries Syntactical Structure (Datalog Style): query( Z 1,..., Z n, aggr(M), Set of Peers)  Generic Fact(X 1,..., X n, M ), rollup dimension d 1 from bottom level to desired level l 1 ( X 1, Z 1 ),..., rollup dimension d n from bottom level to desired level l n ( X n, Z n );

26. GENERATES A QUERY FOR EACH RELEVANT PEER (INCLUDING THE LOCAL PEER); GENERATED QUERIES ARE PROPAGATED TO RELEVANT PEERS; QUERIES FOR RELEVANT PEERS STEM FROM THE REWRITING OF THE SUBMITTED P2P OLAP QUERY; THE REWRITING PROCESS INTRODUCES REFERENCES TO FACT PEERS, REVISED ROLL-UPS, AND MAPPING AND COMPLETION TABLES; RESULTS OF PROPAGATED QUERIES ARE COLLECTED AND AGGREGATED LOCALLY TO PRODUCE THE FINAL QUERY ANSWER; QUERY ANSWERS MAY BE UNCERTAIN AND INCOMPLETE DUE TO BOTTOM-UP COMPLETION. Query Evaluation Process

27. Query Processing Local PeerRelevant Peer Facttables QUERY Rewriting Evaluation Partial Result RevisedRollups Metadata Mappingtables Integration Answer Completiontables

28. Aggregate Queries in Peer-to-Peer OLAP GENERIC FACTS FACT CONCILIATION PHASE HIERARCHY LEVEL CORRESPONDENCE SCHEMA CONCILIATION PHASE REVISE AND MAP APPROACH BOTTOM-UP COMPLETION MAPPING ACQUISITION PHASE P2P OLAP QUERIES QUERY REWRITING AND EVALUATION CONCLUSIONS: MAIN POINTS DISCUSSED