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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
Q Q
1
2
 db Q (db)  Q (db)
1
2
Query of Tuple Problem
t  Q(db)
?
Constraint Satisfaction in AI
CSP  BCQ
Clause Subsumption in Theorem Proving
 s.t. C  D
?
Example of CSP: Crossword
Puzzle
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
Is this query hard?
ans  a( S , X , X ' , C , F )  b( S , Y , Y ' , C ' , F ' )  c(C , C ' , Z )  d ( X , Z ) 
e(Y , Z )  f ( F , F ' , Z ' )  g ( X ' , Z ' )  h(Y ' , Z ' ) 
j ( J , X , Y , X ' , Y ' )  p ( B, X ' , F )  q ( B ' , X ' , F )
n
m
size of the database
number of atoms in the query
• Classical methods worst-case complexity:
m = 11 !
O(n m)
• Despite its apparence, this query is nearly acyclic
It can be evaluated in O(m·n 2· logn)
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)
Beyond treewidth
Bounded Degree of Cyclicity
(Gyssens & Paredaens ’84)
Bounded Query width
(Chekuri & Rajaraman ’97)
Group together query atoms
(hyperedges) instead of variables
Hypertree Decomposition
We group atoms
p(X,Y,Z), q(U,V,Z)
a(X,U,W), b(Y,V,W)
We use p(X,Y,Z) partially
p(X,Y,_), c(T,W)
d(X,T)
c(Y,T)
ans  a( S , X , X ' , C , F )  b( S , Y , Y ' , C ' , F ' )  c(C , C ' , Z )  d ( X , Z ) 
e(Y , Z )  f ( F , F ' , Z ' )  g ( X ' , Z ' )  h(Y ' , Z ' ) 
j ( J , X , Y , X ' , Y ' )  p ( B, X ' , F )  q ( B ' , X ' , F )
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)
d(X,Z)
e(Y,Z)
j(_,_,_,X’,Y’), f(F,F’,Z’)
g(X’,Z’), f(F,_,Z’)
p(B,X’,F)
q(B’,X’,F)
h(Y’,Z’)
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)
d(X,Z)
e(Y,Z)
j(_,_,_,X’,Y’), f(F,F’,Z’)
g(X’,Z’), f(F,_,Z’)
p(B,X’,F)
q(B’,X’,F)
h(Y’,Z’)
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+1 log n) and complete for LOGCFL
Computing Q(db) is feasible in
output-polynomial time
Comparison results
Hypertree Decomposition
Hinge Decomposition
+
Tree Clustering
Hinge
Decomposition
Cycle Hypercutset
Tree Clustering
w*  treewidth
Biconnected Components
Cycle Cutset
Characterizations of
Hypertree width
Logical characterization:
Loosely guarded logic
Game characterization:
The robber and marshals game
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