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Answering Queries and Hypertree Decompositions

Answering Queries and Hypertree Decompositions. Conjunctive Queries. The problem BCQ:. Instance: < DB, Q>. Question: Has Q a nonempty result over DB?. Combined Complexity (Vardi ’82). Problems Equivalent to BCQ. Conjunctive Query Containment Query of Tuple Problem

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Answering Queries and Hypertree Decompositions

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  1. Answering Queriesand Hypertree Decompositions

  2. Conjunctive Queries The problem BCQ: Instance: < DB, Q> Question: Has Q a nonempty result over DB? Combined Complexity(Vardi ’82)

  3. Problems Equivalent to BCQ • Conjunctive Query Containment • Query of Tuple Problem • Constraint Satisfaction in AI • Clause Subsumption in Theorem Proving

  4. Example of CSP: Crossword Puzzle

  5. Complexity of BCQ • NP-complete in the general case (Chandra and Merlin ’77)NP-hard even for fixed database • Polynomial if Q has an acyclic hypergraph(Yannakakis ’81)LOGCFL-complete (in NC2) (G.L.S. ’98) Interest in larger tractable classes of CQS

  6. Is this query hard? n size of the database m number of atoms in the query m = 11 ! O(n m) • Classical methods worst-case complexity: • Despite its apparence, this query is nearly acyclic It can be evaluated in O(m·n 2·logn)

  7. Nearly Acyclic Queries • Bounded Treewidth (tw) • a measure of the cyclicity of graphs • for queries: tw(Q) = tw(G(Q)) • For fixed k: • checking tw(Q) k • Computing a tree decomposition linear time (Bodlaender’96) • Answering BCQ of treewidth k: O(nk log n) (Chekuri & Rajaraman’97) LOGCFL-complete (G.L.S.’98)

  8. Beyond treewidth Bounded Degree of Cyclicity (Gyssens & Paredaens ’84) Bounded Query width (Chekuri & Rajaraman ’97) Group together query atoms (hyperedges) instead of variables

  9. Hypertree Decomposition p(X,Y,Z), q(U,V,Z) p(X,Y,Z), q(U,V,Z) We group atoms a(X,U,W), b(Y,V,W) p(X,Y,_), c(T,W) p(X,Y,_), c(T,W) We use p(X,Y,Z) partially d(X,T) c(Y,T)

  10. j(J,X,Y,X’,Y’) a(S,X,X’,C,F), b(S,Y,Y’,C’,F’) j(_,X,Y,_,_), c(C,C’,Z) j(_,_,_,X’,Y’), f(F,F’,Z’) d(X,Z) e(Y,Z) g(X’,Z’), f(F,_,Z’) h(Y’,Z’) p(B,X’,F) q(B’,X’,F)

  11. Connectedness Condition j(J,X,Y,X’,Y’) a(S,X,X’,C,F), b(S,Y,Y’,C’,F’) j(_,X,Y,_,_), c(C,C’,Z) j(_,_,_,X’,Y’), f(F,F’,Z’) d(X,Z) e(Y,Z) g(X’,Z’), f(F,_,Z’) h(Y’,Z’) p(B,X’,F) q(B’,X’,F)

  12. Evaluating queries having bounded hypertree width k fixed Given: a database db a query Q over db such that hw(Q)  k a width k hypertree decomposition of Q • Deciding whether Q(db) is not empty is in O(n k+1log n) and complete for LOGCFL • Computing Q(db) is feasible in output-polynomial time

  13. Comparison results Hypertree Decomposition Hinge Decomposition + Tree Clustering Cycle Hypercutset HingeDecomposition Tree Clusteringw*  treewidth Biconnected Components Cycle Cutset

  14. Characterizations ofHypertree width • Logical characterization: Loosely guarded logic • Game characterization: The robber and marshals game

  15. Work in progress • Answering queries and hypertree decompositions: • A query-planner based on hypertree decompositions • Choosing the best query plan (i.e., the best decomposition) exploiting data on tables, attibute selectivity, indices, etc. • Further possible applications: • Answering queries using views

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