Containment of Relational Queries with Annotation Propagation

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Containment of Relational Queries with Annotation
Propagation

• Wang-Chiew Tan
• University of California, Santa Cruz

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Annotation Management System

• A system that is able to propagate meta-data that
  is associated with a piece of data along with the
  data as the data is being moved around
• Main feature
• To trace the provenance and flow of data

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Tracing the Provenance and Flow of Data
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Other Applications

• Keep information that cannot be otherwise stored
  in the current database design
• Highlight wrong data
• Errorneous data may be copied but the comment
  that it is wrong goes along with it
• Security
• Annotate security level of data items
• Quality metric
• Annotate quality level of data items

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Main Question

• Are the annotated outcomes the same for
  equivalent queries?
• Why this question?
• A query optimizer rewrites a query. Will the
  rewritten query have the same annotation
  propagation behavior?

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A Simple Example

• Given two relation schemas R(A,B), S(B,C)
• SELECT
• FROM R NATURAL JOIN S
• versus
• SELECT r.A, r.B, s.C
• FROM R r, S s
• WHERE r.B s.B

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In a More Concise Notation
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• Ans(x,y,z) - R(x,y), S(y,z) x ! 1, y !
  2, z ! 3
• Ans(x,y,z) - R(x,y), S(y,z), y y
• x ! 1,
  y! 2, y! 2, z ! 3
• A location is a triple (R, t, A)
• Annotations of values that reside in different
  locations but are bound to the same variable are
  unioned together
• Ans(y) - R(x,y)
• Ans(y) - S(y,z)
• Ans(2 )
• Annotations that belong to the same output
  location are unioned together

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More Examples

• Q1
• Ans(x,v) - R(x,y,u), R(x,z,v), R(t,w,z)
• Q2
• Ans(x,v) - R(p,q,v), R(x,z,v), R(t,w,z)
• First answer Ans(1, 5 )
• Second answer Ans(1, 5 )

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A sufficient condition for annotation containment

• Theorem If Q1 and Q2 are equivalent and Q1 is
  minimal, then Q1 is annotation-contained in Q2
• Intuition of proof
• If Q1 is minimal, then no proper subquery of Q1
  is equivalent to Q1
• The minimal query of Q2 is isomorphic to Q1 up to
  variable renaming. Assume that they are
  identical.
• Any valuation ? for Q1 can be simulated by a
  valuation ? h that carries annotations in the
  same way as ? of Q1 (h is the homomorphism from
  Q2 to its minimal subquery)

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Is the sufficient condition too strong?

• Is it true that if Q1 is equivalent to Q2, then
  Q1 is annotation-contained in Q2?
• Answer No.
• Is it true that if Q1 is contained in Q2 and Q1
  is minimal, then Q1 is annotation contained in
  Q2?
• Answer No.
• Q1 Ans(x) - R(x, y), S(x, y)
• Q2 Ans(x) - R(x, y)
• Ans(1 ) Ans (1 )
• Both Q1 and Q2 are minimal queries but neither Q1
  nor Q2 are annotation-contained in each other

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Necessary and Sufficient condition?
ith column
jth column
pth subgoal
Q1 H( x ) - S( x )
h(y) x, h maps the qth subgoal of Q2 to the pth
subgoal of Q1
Q2 H( y ) - S( y )
ith column
jth column
qth subgoal

• If Q1 carries an annotation of the jth column of
  some S-tuple to the output, there is a way for Q2
  to simulate this behavior via homomorphism h

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A necessary and sufficient condition for
annotation-containment via homomorphisms

• Theorem Q1 is annotation-contained in Q2 iff for
  every distinguished variable x that occurs at the
  ith column in the head and jth column of the pth
  subgoal in the body of Q1, there exists a
  homomorphism h from Q2 to Q1 such that
• h maps the body of Q2 into the body of Q1 and the
  head of Q2 to the head of Q1
• Let the qth subgoal Q2 be the preimage of the pth
  subgoal of Q1 under h. The variable that occurs
  at the jth column of the qth subgoal of Q2 is
  identical to the variable that occurs at the ith
  column in the head of Q2

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Can a single homomorphism do the job?

• Q1 Ans(x) - R(x,y), R(x,z)
• Q2 Ans(x) - R(x,y)
• Every homomorphism from Q2 to Q1 maps the body of
  Q2 to only one subgoal of Q1

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Complexity of Annotation-Containment

• Proposition It is NP-complete to decide if Q1 is
  annotation-contained in Q2

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Propagating annotations back

• If we wish to attach an annotation on a piece of
  data in the output, on which source data should
  we attach an annotation?
• The user should be given the choice
• Alert the user of a side-effect-free annotation
  when there is one

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Annotation Placement Problem

• Given the source database, the query, the output
  data that we wish to annotation, it is DP-hard to
  decide if there is a side-effect-free annotation
• Upper-bound is not DP
• Conjecture in a class slightly above DP

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Related Work

• Idea is not new though annotations were never
  explicitly stated as provenance-based Wang
  Madnick VLDB 90, Lee, Bressan Madnick WIDM
  98, Bernstein Bergstraesser IEEE Data Eng.
  99
• Annotations of Web Documents
• Annotations on genomic sequences

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Open Issues

• Are there polynomial time algorithms for deciding
  annotation-containment for the class of queries
  with bounded treewidths
• Is query minimization church-rosser?
• Exact complexity of the annotation placement
  problem?
• Annotation and propagation for XML data
• Relationship between annotation-containment and
  containment of conjunctive queries under bag
  semantics

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Open Issues (contd)

• Other annotation propagation semantics other than
  basing on provenance?
• Querying the annotations?

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Other results that do not carry over

• Query Minimization
• We can no longer minimize a query and preserve
  annotation-equivalence by discarding one subgoal
  at a time
• Answering Queries using Views
• Some classical results no longer hold
• LMSS95 if a query Q has p subgoals and a query
  Q is a complete minimal rewriting of Q using a
  set of views V, then Q has at most p subgoals

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Example

• Q A(x) - R(x,z,v), R(x,u,z), R(x,z,t),
  R(x,s,z)
• Q A(x) - R(x,u,z), R(x,z,t), R(x,s,z)
• Qmin A(x) - R(x,z,t), R(x,s,z)

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