Answering Queries Using Views

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Answering Queries Using Views
Alon Y. Halevy
Based on Levy et al. PODS 95
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Conjunctive queries
head
body

• Safe - each variable in the head appears in the
  body.

subgoal
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Views

• A view is a derived relation defined in terms of
  stored base relations.
• A materialized view is a view stored in the DB
  (like a cache).

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Query Containment

• iff for any DB, any answer to Q2 is an answer
  to Q1.
• Containment is NP-complete.
• Equivalence is defined as two-way containment.

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The problem

• Definition Given a query Q and views V1,V2,,Vm,
  find an equivalent query Q (a rewriting) that
  uses one or more of the views.

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Variants of the problem

• Q is a complete rewriting of Q if it uses just
  the views (and built-in predicates).
• Q is a locally minimal rewriting if we cannot
  remove a subgoal from Q and retain equivalence.
• Q is a globally minimal rewriting if there is no
  other rewriting with less subgoals.

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Containment mapping

• Definition A containment mapping (CM) from Q1 to
  Q2 is a mapping from the variables of Q1 to those
  of Q2 such that every subgoal in Q1 is mapped to
  a subgoal in Q2.
• Example Q1 q(X)-p(X,Y),p(Y,Z),p(Z,W).
• Q2 q(X)-p(X,Y),p(Y,X).
• CM X/X, Y/Y, Z/X, W/Y

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Rewriting vs. containment

• Proposition 1 Let Q and V be conjunctive
  queries. There is a rewriting of Q using V iff
• i.e. if V is empty for a DB then so is Q.

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Corollaries

• Proposition 1 provides a complete
  characterization of the rewriting problem.
• We can use known results from containment to the
  rewriting problem
• Complexity results
• Practical techniques for finding rewritings

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Results

• Complexity
• Proposition 2 If Q but not V may contain
  built-in predicates, then finding a rewriting
  of Q using V is NP-complete.
• Practical solution using Containment Mappings
  (Chandra Merlin 1977)

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Using containment mappings

• Theorem For conjunctive queries (without
  built-in predicates), there is a containment
  mapping between Q1 and Q2 iff
• Finding a CM is NP-complete.
• We can find a rewriting by finding a CM from V
  to Q that only maps the bodies.
• Then, just add Vs head to Qs body.

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Minimal rewriting

• Lemma 3 (technical)
• If Q is a locally minimal rewriting of Q using
  V, then Q contains no new relation subgoals.
• If Q has a rewriting then it has a rewriting with
  no new variables.
• Lemma 4 A locally minimal rewriting Q of Q has
  no more subgoals than Q.

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Complexity of complete rewriting

• Proposition 5 Finding a complete rewriting is
  NP-complete.
• the problem of exact-cover by 3-sets

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The picture so far

• To find a rewriting
• 1. Seek a CM from the views to the query. Add
  these view heads to the query.
• 2. Minimize by removing redundant subgoals.
• Both problems are independently NP-complete.

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Conclusions

• Using views to answer queries is an important
  problem. Especially for information integration
  on the web.
• Query containment and containment mappings
  provide the key for solving the problem.
• The variants of the problem are NP-complete. This
  is not too bad, since queries are usually short.