1. ICDT'2001, London, UK1 Minimizing View Sets without Losing Query-Answering Power Chen Li Stanford University joint work with Mayank Bawa and Jeff Ullman

2. 2 source query answer Client cache user query A web-caching scenario Server

3. 3 Client Source relation: Book(Title, Author, Pub, Price) Cached query results: Q1(T,A,Pr) :- book(T,A,Pub,Pr) Q2(T,A,Pr) :- book(T,A,prenhall,Pr) Q3(A1,A2) :- book(T,A1,prenhall,Pr1), book(T,A2,prenhall,Pr2)

4. 4 Book(Title, Author, Pub, Price) Q2  Q1 Remove Q2? Cannot answer query: Q(T,Pr) :- book(T,smith,prenhall,Pr) What query results to remove? Cached query results: Q1(T,A,Pr) :- book(T,A,Pub,Pr) Q2(T,A,Pr) :- book(T,A,prenhall,Pr) Q3(A1,A2) :- book(T,A1,prenhall,Pr1), book(T,A2,prenhall,Pr2)

5. 5 Compute Q3 using Q2: Q3(A1,A2) :- Q2(T,A1,Pr1),Q2(T,A2,Pr2) We are not losing any query-answering power! How about removing Q3? Book(Title, Author, Pub, Price) Cached query results: Q1(T,A,Pr) :- book(T,A,Pub,Pr) Q2(T,A,Pr) :- book(T,A,prenhall,Pr) Q3(A1,A2) :- book(T,A1,prenhall,Pr1), book(T,A2,prenhall,Pr2)

6. 6 Observations: –Traditional query-containment does not help [Chandra and Merlin, 1977]. –We should consider query-answering power. General questions: –How to describe “query-answering power”? –How to minimize a view set without losing its query-answering power?

7. 7 Rest of the talk Answering queries using views Query-answering power –p-containment –Relationship with traditional query containment –Minimizing a view set p-containment relative to a set of queries Conclusion and open problems

8. 8 Answering queries using views Conjunctive queries and views: h(X) :- g 1 (X 1 ),…,g n (X n ) Example: V1(T,A,Pr) :- book(T,A,Pub,Pr) V2(T,A,Pr) :- book(T,A,prenhall,Pr) V3(A1,A2) :- book(T,A1,prenhall,Pr1), book(T,A2,prenhall,Pr2)

9. 9 Query answerability A query Q is answerable by a view set V if we can rewrite Q using views in V [LMSS95]. Example: V2(T,A,Pr) :- book(T,A,prenhall,Pr) V3(A1,A2) :- book(T,A1,prenhall,Pr1), book(T,A2,prenhall,Pr2) V3 is answerable by V2: V3(A1,A2) :- V2(T,A1,Pr1),V2(T,A2,Pr2)

10. 10 Algorithms Bucket algorithm [LRO96] Inverse-rule algorithm [DG97,Qia96] MiniCon algorithm [PL00] SVB algorithm [Mit99] CoreCover Algorithm [ALU00] Testing whether a query is answerable by a set of views is NP-complete.

11. 11 Views are expensive to maintain Require storage space. Need to be kept up-to-date. We want to minimize a given view set while keeping its query-answering power.

12. 12 p-containment A view set V is p-contained in another view set W if W can answer all the queries that are answerable by V. –“p” stands for “power.” –Denoted: V  p W Two view sets are equipotent, if V  p W and W  p V. –They have the same power to answer queries.

13. 13 Example: V1(T,A,Pr) :- book(T,A,Pub,Pr) V2(T,A,Pr) :- book(T,A,prenhall,Pr) V3(A1,A2) :- book(T,A1,prenhall,Pr1), book(T,A2,prenhall,Pr2) {v1,v2,v3}  p {v1,v2} {v1,v2}  p {v1,v2,v3} Therefore: {v1,v2,v3} and {v1,v2} are equipotent.

14. 14 Lemma: V  p W iff each view in V can be answered by W. –Implies an algorithm for testing p-containment. –Assuming view sets are finite. Theorem: Testing V  p W is NP-complete.

15. 15 p-containment and query containment V1(T,A,Pr) :- book(T,A,Pub,Pr) V2(T,A,Pr) :- book(T,A,prenhall,Pr) V3(A1,A2) :- book(T,A1,prenhall,Pr1), book(T,A2,prenhall,Pr2) Query containment does not imply p-containment {v1} and {v2} p-containment does not imply query containment {v2} and {v3}

16. 16 Minimizing a view set Keep removing views from the view set while retaining the equipotence. Might have multiple equipotent minimals V1(A) :- r(A,B) V2(B) :- r(A,B) V3(A,B) :- r(A,X),r(Y,B) {V1,V2,V3} has two equipotent minimals: {V1,V2}, {V3}

17. 17 p-containment relative to queries Queries: Q={Q1,Q2,…} V = {V1,V2,…,Vm}W = {W1,W2,…,Wn} V is p-contained in W w.r.t. Q if the queries in Q that are answerable by V are also answerable by W.

18. 18 Example of relative p-containment Relations: car(Make,Dealer) loc(Dealer,City) Queries: Q1(D,C) :- car(toyota,D),loc(D,C) Q2(D,C) :- car(honda,D), loc(D,C) Views: V = {V1,V2}, V1 = Q1, V2 = Q2 W = {W1} W1(M,D,C) :- car(M,D),loc(D,C)

19. 19 Testing relative p-containment Q is finite: test by the definition. Q is infinite?

20. 20 Parameterized queries Motivation: web search forms. A PQ is a conjunctive query with placeholders. Example: q(D) :- car($M,D),loc(D,$C) –Placeholders $M,$C, replaced by constants –Instances: q(D) :- car(toyota,D),loc(D,sf) q(D) :- car(honda,D),loc(D,pa) –The domain of each placeholder is infinite. –Thus, represent infinite number of queries.

21. 21 Q: q(D) :- car($M,D),loc(D,$C) v1(M,D,C) :- car(M,D),loc(D,C) –Answer all instances of Q. v2(M,D) :- car(M,D),loc(D,sf) –Answer some instances of Q. –Answerable instances of Q are instances of: q(D) :- car($M,D),loc(D,sf) v3(M) :- car(M,D),loc(D,sf) –Answer no instances of Q.

22. 22 Assume queries are generated by one PQ; Results easily extendable to the case with finite set of PQs. Complete answerability of a PQ using views –V can answer all instances of a PQ Q. –Example: q(D) :- car($M,D),loc(D,$C) v1(M,D,C) :- car(M,D),loc(D,C)

23. 23 An algorithm for testing complete answerability Replace each placeholder with a new distinct constant, get a canonical instance I; Test if I is answerable by V. Example: PQ: q(D) :- car($M,D),loc(D,$C) View: v1(M,D,C) :- car(M,D),loc(D,C) Canonical instance: q(D) :- car(m0,D),loc(D,c0) Rewriting: q(D) :- v1(m0,D,c0)

24. 24 Partial answerability Some instances of Q are answerable by V q(D) :- car($M,D),loc(D,$C) v2(M,D) :- car(M,D),loc(D,sf) Theorem: All the answerable instances of a PQ using V are instances of a finite set of PQs, s.t. each of them is completely answerable by V. q(D) :- car($M,D),loc(D,sf)

25. 25 a parameterized query Q All instances of Q answerable instances PQ1 PQ2 PQk … V={V1,…,Vn} An algorithm for finding the finite set of PQs.

26. 26 Testing p-containment w.r.t. PQ Find the PQs whose instances are all the instances of Q that are answerable by V. For each of the PQs, test if it is completely answerable by V. Details are in the paper.

27. 27 Conclusion Introduced p-containment, which is different from query containment. Showed how to minimize a view set without losing query-answering power. Developed an algorithm for testing relative p-containment w.r.t. instances of PQs. Extended to MCR-containment.

28. 28 Open problems Find a view subset with lowest “cost.” If views are not given, find the best views to materialize.