On the Minimization of XPath Queries

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On the Minimization of XPath Queries

• Paper by
• S. Flesca, F. Furfaro, E. Masciari
• CmpE 521 Presentation
• Emre Yurtsever 2002701372
• Muammer Yüzügüldü 2003700183

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Outline

• Introduction
• Trees Tree Patterns
• Problem Statement
• A Framework for minimizing XPath queries
• Complexity Results
• Tractability Results
• Conclusions Future Works

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Introduction

• XML Queries are usually expressed by means of
  XPath expressions.
• XPath expressions.
• A way of navigating an XML tree to return the set
  of nodes through the paths specified by the
  expression.

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Introduction contd

• An XPath expression can be represented
  graphically as a tree pattern.

An XML Tree...
5
Introduction contd

• For example
• find the titles of all the books for which at
  least one author is known.
• XPath Expression
• bib/book//author/title

Descendant Edge
Output Node
A tree pattern
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Introduction contd

• Efficiency of XPath expression depends on size.
• Optimization Minimization
• We should minimize the expression.

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Introduction contd

• Example Query
• Retrieve the editors that published thrillers
  and whose authors have written a thriller.
• query containment.
• Reduced Query
• Retrieve the editors that published thriller.

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Introduction contd

• Minimization problem for XPath fragments can be
  efficiently solved as
• It can be reduced to solve a number of instances
  of containment between pairs of tree patterns.
• For these fragments, it can be reduced to find a
  homomorphism between them.

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Trees Tree Patterns

• A tree t is a tuple (rt, Nt, Et, ?t) where
• Nt ? N, set of nodes.
• ?t Nt ? ? is a node labelling function.
• rt ? Nt is the distinguished root of t.
• Et ? Nt x Nt, set of edges.

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Trees Tree Patterns contd

• Given a tree t (rt, Nt, Et, ?t)
• Tree t (rt, Nt, Et, ?t) is the subtree if
  
• Nt ? Nt
• The edge (ni, nj) belongs to Et iff ni ? Nt, nj
  ? Nt and (ni, nj) ? Et.

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Trees Tree Patterns contd

• Definition
• A tree pattern p is a pair (tp, op), where
• tp (rp, Np, Ep, ?p) is a tree.
• Ep is partitioned into the two disjoint sets Cp
  and Dp denoting, respectively, the child and
  descendent branches
• op ? Np is a distinguished output node.

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Trees Tree Patterns contd

• Grammar for XPath expressions
• exp ? exp exp/exp exp//exp expexp
  ? .
• where ? is a symbol in ?, and the symbol .
  stands for the current node.
• Given XPath expression
• ab///c//d

a
b
d

c
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Trees Tree Patterns contd

• Given a tree t and a tree pattern p, an embedding
  e of p into t is a total function e Np ? Nt,
  such that
• e(rp) rt,
• ?(x y) ? Cp, e(y) is a child of e(x) in t,
• ?(x y) ? Dp, e(y) is a descendant of e(x) in t,
  and
• ?x ? Np, if ?p(x) a (where a ? ) then ?t(e(x))
  a.

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Trees Tree Patterns contd

• Models and Canonical Models of Tree Patterns
• The models of a tree pattern p defined over the
  alphabet ? are the trees of T? which can be
  embedded by p. The set of models of p is Mod(p)
  t ? T? p(t) ? ?
• Canonical models of a tree pattern p are models
  having the same shape as p. That is, a canonical

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Trees Tree Patterns contd

• Model and Canonical Model of a tree pattern

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Trees Tree Patterns contd

• Given two tree patterns p1, p2, we say that p1 is
  contained in p2 (p1 ? p2) iff ?t p1(t) ? p2(t).
• We say that p1 and p2 are equivalent (p1 ? p2)
  iff p1 ? p2 and p2 ? p1 (i.e. ?t p1(t) p2(t)).
• The set of patterns which are equivalent to a
  given pattern p will be denoted as Eq(p).

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Trees Tree Patterns contd

• Notations on tree patterns.

A pattern p and its subpatterns spb, spd, spa
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Trees Tree Patterns contd

• Tree pattern p whose root has 2 children

Subpattern examples
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Problem Statement

• Given a tree pattern p, construct a tree
  pattern pmin which is equivalent to p and having
  minimum size (i.e. size(pmin) minsize(p))

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Problem Statement contd

• a minimum size tree pattern equivalent to p can
  be found among the subpatterns of p
• the containment between two tree patterns p, q (p
  ? q) is equivalent to the problem of finding a
  homomorphism from q to p. A homomorphism h from a
  pattern q to a pattern p is a total mapping from
  the nodes of q to the nodes of p such that
• h preserves node types (i.e. ?u ? Nq ?q(u) ? '
  ) ?q(u) ?p(h(u)))
• h preserves structural relationships
  (i.e.whenever v is a child (resp. descendant) of
  u in q, h(v) is a child (resp. descendant) of
  h(u) in p).

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Problem Statement contd
A homomorphism between two tree patterns
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Problem Statement contd
Two tree patterns not related with homomorphism
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A framework for minimizing XPath Queries

• Two fundamental contribution
• Proving that property 1 holds for XP/, //, ,
  
• An algorithm for minimizing a tree pattern query

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Proving that Property holds for XP/, //, ,

• Various lemmas are introduced
• Lemma 1 Let p and q be two patterns with root
  r, such that p contained in q. Then, for each
  subpattern Qj element of P(q) there exists a
  subpattern Pi element of P(p) s.t Pi contained in
  Qi.

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Example for Lemma 1

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Proving that Property holds for XP/, //, ,

• Lemma 2 Let p and be two patterns rooted in r
  s.t pq and let m and n, with mgtn, be the number
  of children of r in p and, respectively, q. Then,
  there exist a set S subset of SP(p) consisting of
  m-n subpatterns spi, such that p-S p

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Example for Lemma 2
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Proving that Property holds for XP/, //, ,

• Lemma 3 Let p and q be two equivalent patterns
  rooted in r having the same number of child and
  descendant nodes of r, and let q be of minimum
  size. Then, there not exists a subpattern spk
  element of SP(p) such that p spk p

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Proving that Property holds for XP/, //, ,

• Lemma 4 Let p and q be two eqivalent patterns
  whose roots have the same number of child and
  descendant nodes, and let q be of minimum size.
  For each subpattern Pi element of P(p) there
  exists a unique subpattern Qj element of P(q)
  directly connected to rq s.t pi?qj

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Proving that Property holds for XP/, //, ,

• Lemma 5 A pattern p in XP , /, //, is not
  of minimum size iff at least one of the following
  conditions hold
• there exists a pair of subpatterns pi, pj s.t pi
  contained in pj
• there exists a subpattern pi of p which is not of
  minimum size.

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Proving that Property holds for XP/, //, ,

• Theorem 1 Given a pattern p in XP/, //, ,
  if minsize(p) k then there exists a subpattern
  pmin of p such that p pmin and size(pmin)k

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An Algorithm for tree pattern minimization

• Function Minimize(pa tree pattern)pmin a
  minimum tree pattern equivalent to p
• Begin
• pmin p
• For each pi element of P(pmin) do
• if (pmin -spi contained in pmin)
• pmin pmin spi
• SPnew 0
• For each spi element of SP(pmin) do
• SPnew SPnew Minimize(spi)
• pmin assemble (pmin, SPnew)
• return pmin
• End

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Upper Bound

• Algorithm 1 works in O(br(p2)((w1)(d1)))
• p is the size of p
• d is the number descendant edges in p
• w is one the longest chain of in p
• b is the number branches of p as b
• r is the maximum degree of any node of p

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Complexity Results

• In XPath/, //, , it is not possible to
  define an algorithm performing much better than
  Algorithm 1
• Lemmma 6 Let p be a pattern in XP/, //, ,
  and k is possitive integer. The problem of
  testing if minsize(p)gtk is NP-complete problem

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Complexity Results

• Theorem 2 Let p be a pattern in XP/, //, ,
  and k a positive integer. The problem of
  testing if there exists a pattern p equivalent
  to p such that size(p)lt k is coN-complete

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Tractability Results

• Definition A limited branched tree pattern p is
  a tree pattern in XP/, //, , such that
• Every non leaf node of p may have any number of
  children
• If a node n has k children n1...nk, then at least
  k-1 of the patterns spn, (where i element
  1...k) are linear.

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Example
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Tractability Results

• Theorem Let p a limited branched tree pattern.
  A minimum pattern pmin equivalent to p can be
  found in polynomial time. (w.r.t. The size of p)
• Linear patterns have minimum size
• The containment between pairs of linear patterns
  can be decided in polynomial time.

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Tractability Results

• Algorithm 2
• Function Minimize(pa boundend branched tree
  pattern)pmin a minimum tree pattern equivalent
  to p
• Begin
• pmin p
• B b1, ...., bm
• while(B ! 0)
• b deepest(B)
• q spb
• Redq 0
• For each pi element of P(q) do
• For each qj of P(q) do
• if ((i!j) (qi is linear) (qj not element
  of Redq) (qj contained in qi))
• Redq Redq qi
• q q Redq
• pmin replace(pmin, sqb, q)
• end while
• return pmin
• End

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Conclusion

• It has been proved the global minimality
  property, a minimum tree pattern equivalent to a
  given tree pattern p can be found amoung the
  subpatterns of p, and thus obtained by prunning
  redundant branches from p.
• It has been characerized the complexity of the
  minimization problem, showing that the
  corresponding decisional problem is
  coNP-complete.

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Conclusion

• It has been studied a tractable form of tree
  pattern which can be minimized in polynomial
  time.
• It has been provided by some algorithms proposed
  in the paper.

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Future Works

• Extending minimization framework to deal with
  XPath queries that must satisfy some constraints
  such as join conditions on tree pattern nodes.
• The introduction of these constraints makes the
  minimization problem harder, and global
  minimality property does not hold.

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Questions?

• Thank you...