Conjunctive Queries
Conjunctive Queries
Conjunctive Queries
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
ans a(<br />
S,<br />
X , X ', C,<br />
F)<br />
b(<br />
S,<br />
Y,<br />
Y ', C',<br />
F')<br />
c(<br />
C,<br />
C',<br />
Z)<br />
d(<br />
X , Z)<br />
<br />
e(<br />
Y,<br />
Z)<br />
f ( F,<br />
F',<br />
Z')<br />
g(<br />
X ', Z')<br />
h(<br />
Y ', Z')<br />
<br />
j(<br />
J,<br />
X , Y,<br />
X ', Y ') p(<br />
B,<br />
X ', F)<br />
q(<br />
B',<br />
X ', F)<br />
B B’<br />
X X’ C F<br />
J Z Z’<br />
S<br />
Y Y’ C’ F’<br />
Nearly Acyclic <strong>Queries</strong> & CSPs<br />
• Bounded Treewidth (tw)<br />
– a measure of the cyclicity of graphs<br />
– for queries: tw(Q) = tw(G(Q))<br />
• For fixed k:<br />
– checking tw(Q) k<br />
– Computing a tree decomposition<br />
linear time<br />
(Bodlaender’96)<br />
Deciding CSP of treewidth k:<br />
It can be evaluated in O(m·n 2· logn)<br />
O(n k log n) [Dechter; Chekuri & Rajaraman’97; Kolaitis & Vardi, 98]<br />
LOGCFL-complete (G.L.S.’98)<br />
Primal graphs of<br />
CSPs/<strong>Queries</strong><br />
ans Enrolled(S,C,R) Teaches(P,C,A) Parent(P,S)<br />
S<br />
R<br />
C<br />
A<br />
Hypergraph H(Q)<br />
P<br />
R<br />
S<br />
C<br />
P<br />
A<br />
Primal graph G(Q)<br />
FurtherGraphRepresentations<br />
S<br />
X1<br />
2<br />
S 1<br />
Decther, 92<br />
X 2<br />
S 2<br />
Dual Graph<br />
X 2 S 3<br />
X 1<br />
X 3 X 2<br />
S<br />
S 4 1<br />
X 2 X 4 X 1<br />
S<br />
S 1<br />
3<br />
X 2<br />
S<br />
X 2<br />
5 X S4<br />
3<br />
Seidel, 81 S 3<br />
X 4<br />
S 4<br />
Incidence Graph<br />
X 5<br />
(Hidden variable encoding)
Change language