Tree Decomposition methods Chapter 9

1
Tree Decomposition methodsChapter 9

• ICS-275
• Spring 2007

2
Graph concepts reviewsHyper graphs and dual
graphs

• A hypergraph is H (V,S) , V v_1,..,v_n and a
  set of subsets Hyperegdes SS_1, ..., S_l .
• Dual graphs of a hypergaph The nodes are the
  hyperedges and a pair of nodes is connected if
  they share vertices in V. The arc is labeled by
  the shared vertices.
• A primal graph of a hypergraph H (V,S) has V
  as its nodes, and any two nodes are connected by
  an arc if they appear in the same hyperedge.
• if all the constraints of a network R are binary,
  then its hypergraph is identical to its primal
  graph.

3
Acyclic Networks

• The running intersection property
  (connectedness) An arc can be removed from the
  dual graph if the variables labeling the arcs are
  shared along an alternative path between the two
  endpoints.
• Join graph An arc subgraph of the dual graph
  that satisfies the connectedness property.
• Join-tree a join-graph with no cycles
• Hypertree A hypergraph whose dual graph has a
  join-tree.
• Acyclic network is one whose hypergraph is a
  hypertree.

4
Solving acyclic networks

• Algorithm acyclic-solving applies a tree
  algorithm to the join-tree. It applies (a little
  more than) directional relational arc-consistency
  from leaves to root.
• Complexity acyclic-solving is O(r l log l)
  steps, where r is the number of constraints and l
  bounds the number of tuples in each constraint
  relation
• (It assumes that join of two relations when ones
  scope is a subset of the other can be done in
  linear time)

5
Example

• Constraints are R_ABC R_AEF R_CDE
  (0,0,1) (0,1,0)(1,0,0) and R_ACE (1,1,0)
  (0,1,1) (1,0,1) .
• d (R_ACE,R_CDE,R_AEF,R_ABC).
• When processing R_ABC, its parent relation is
  R_ACE we generate \pi_ACE (R_ACE x
  R_ABC), yielding R_ACE (0,1,1)(1,0,1).
• processing R_AEF we generate relation R_ACE
  \pi_ACE ( R_ACE x R_AEF ) (0,1,1).
• processing R_CDE we generate R_ACE
  \pi_ACE ( R_ACE x R_CDE ) (0,1,1).
• A solution is generated by picking the only
  allowed tuple for R_ACE, A0,C1,E1, extending
  it with a value for D that satisfies R_CDE,
  which is only D0, and then similarly extending
  the assignment to F0 and B0, to satisfy R_AEF
  and R_ABC.

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Recognizing acyclic networks

• Dual-based recognition
• perform maximal spanning tree over the dual graph
  and check connectedness of the resulting tree.
• Dual-acyclicity complexity is O(e3)
• Primal-based recognition
• Theorem (Maier 83) A hypergraph has a join-tree
  iff its primal graph is chordal and conformal.
• A chordal primal graph is conformal relative to
  a constraint hypergraph iff there is a one-to-one
  mapping between maximal cliques and scopes of
  constraints.

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Primal-based recognition

• Check cordality using max-cardinality ordering.
• Test conformality
• Create a join-tree connect every clicque to an
  earlier clique sharing maximal number of variables

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Tree-based clustering

• Convert a constraint problem to an acyclic-one
  group subset of constraints to clusters until we
  get an acyclic problem.
• Hypertree embedding of a hypergraph H (X,H) is
  a hypertree S (X, S) s.t., for every h in H
  there is h_1 in S s.t. h is included in h_1.
• This yield algorithm join-tree clustering
• Hypertree partitioning of a hypergraph H (X,H)
  is a hypertree S (X, S) s.t., for every h in H
  there is h_1 in S s.t. h is included in h_1 and
  X is the union of scopes in h_1.

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Join-tree clustering

• Input A constraint problem R (X,D,C) and its
  primal graph G (X,E).
• Output An equivalent acyclic constraint problem
  and its join-tree T (X,D, C )
• 1. Select an d (x_1,...,x_n)
• 2. Triangulation(create the induced graph along
  d and call it G )
• for jn to 1 by -1 do
• E ? E U (i,k) (i,j) in E,(k,j) in E
• 3. Create a join-tree of the induced graph G
• a. Identify all maximal cliques (each
  variable and its parents is a clique).
• Let C_1,...,C_t be all such cliques,
• b. Create a tree-structure T over the
  cliques
• Connect each C_i to a C_j (j lt I) with
  whom it shares largest subset of variables.
• 4. Place each input constraint in one clique
  containing its scope, and let
• P_i be the constraint subproblem associated
  with C_i.
• 5. Solve P_i and let R'_i be its set of
  solutions.
• 6. Return C' R'_1,..., R'_t
• the new set of constraints and their join-tree,
  T.

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Example of tree-clustering
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Complexity of JTC

• THEOREM 5 (complexity of JTC)
• join-tree clustering is O(r k (w(d)1)) time
  and O(nk(w(d)1)) space, where k is the
  maximum domain size and w(d) is the induced
  width of the ordered graph.
• The complexity of acyclic-solving is O(nw(d) log
  k k(w(d)1))

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Unifying tree-decompositions

• Let PltX,D,Cgt be an automated reasoning problem.
  A tree decomposition is ltT,?,?gt, such that
• T(V,E) is a tree
• ? associates a set of variables ?(v)?X with each
  node
• ? associates a set of functions ?(v)?C with each
  node
• such that
• 1. ?Ri?C, there is exactly one v such that
  Ri??(v) and scope(Ri)??(v).
• 2. ?x?X, the set v?Vx??(v) induces a connected
  subtree.

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HyperTree Decomposition

• Let PltX,D,Cgt be an automated reasoning problem.
  A tree decomposition is ltT,?,?gt, such that
• T(V,E) is a tree
• ? associates a set of variables ?(v)?X with each
  node
• ? associates a set of functions ?(v)?C with each
  node
• such that
• 1. ?Ri?C, there is exactly one v such that
  Ri??(v) and scope(Ri)??(v).
• 1a. ?v, ?(v) ? scope(?(v)).
• 2. ?x?X, the set v?Vx??(v) induces a connected
  subtree.

w (tree-width) is maxv?V ?(v)
hw (hypertree width) is maxv?V ?(v) sep
(max separator size) is max(u,v) sep(u,v)
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Example of two join-trees again
Tree decomposition
hyperTree- decomposition
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Cluster Tree Elimination

• Cluster Tree Elimination (CTE) works by passing
  messages along a tree-decomposition
• Basic idea
• Each node sends one message to each of its
  neighbors
• Node u sends a message to its neighbor v only
  when u received messages from all its other
  neighbors

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Constraint Propagation
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Implementation tradeoffs

• Implementing Equation 1 The particular
  implementation of equation (1) in CTE can vary.
• One option is to generate the combined relation
  (1Ri2clusterv(u) Ri) before sending messages to
• neighbor v. The other option, which we assume
  here, is that the message sent to each neighbor
  is
• created without recording the relation
  (1Ri2clusterv(u) Ri). Rather, each tuple in the
  join is projected
• on the separator immediately after being created.
  This will yields a better memory utilization.
• Furthermore, when u sends a message to v its
  cluster may contain the message it received from
• v. Thus in a synchronized message passing we can
  allow a single enumeration of the tuples in
• cluster(u) when the messages are sent back
  towards the leaves, each of which be projected in
• parallel on the separators of the outgoing
  messages.
• The output of CTE is the original
  tree-decomposition where

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Cluster-Tree Elimination (CTE)
m(u,w) ?sep(u,w) ?j m(vj,u) ? ?(u)
?(u) ?(u)
...
m(v1,u)
m(vi,u)
m(v2,u)
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Cluster Tree Elimination - properties

• Correctness and completeness Algorithm CTE is
  correct, i.e. it computes the exact joint
  probability of every single variable and the
  evidence.
• Time complexity O ( deg ? (nN) ? d w1 )
• Space complexity O ( N ? d sep)
• where deg the maximum degree of a node
• n number of variables ( number of CPTs)
• N number of nodes in the tree decomposition
• d the maximum domain size of a variable
• w the induced width
• sep the separator size
• Time and space by hyperwidth only if hypertree
  decomposition
• time and O(N tw) space,

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Cluster Tree Elimination - properties

• Correctness and completeness Algorithm CTE is
  correct, i.e. it computes the exact joint
  probability of a single variable and the
  evidence.
• Time complexity O ( deg ? (rN) ? kw1 )
• Space complexity O ( N ? d sep)
• where deg the maximum degree of a node
• nr number of of CPTs
• N number of nodes in the tree decomposition
• k the maximum domain size of a variable
• w the induced width
• sep the separator size
• JTC is O ( r ? k w1 ) time and space

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Example of CTE message propagation
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Join-Tree Decomposition(Dechter and Pearl 1989)

• Each function in a cluster
• Satisfy running intersection property
• Tree-width number of variables in a cluster-1
• Equals induced-width

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Cluster Tree Elimination
A B C R(a), R(b,a), R(c,a,b)
1
join
project
BC
B C D F R(d,b), R(f,c,d),h(1,2)(b,c)
2
sep(2,3)B,F elim(2,3)C,D
BF
B E F R(e,b,f), h(2,3)(b,f)
3
EF
E F G R(g,e,f)
4
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CTE Cluster Tree Elimination
Time O ( exp(w1 )) Space O (
exp(sep)) Time O(exp(hw)) (Gottlob et. Al.,
2000)
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Decomposable subproblems

• DEFINITION 8 (decomposable subproblem) Given a
  constraint problem R (XDC) and a
• subset of variables Y µ X, a subproblem over Y ,
  RY (YDY CY ), is decomposable relative
• to R iff sol(RY ) ¼Y sol(R) where sol(R) is the
  set of all solutions of network R

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Distributed relational arc-consistency example
A B
1 2
1 3
2 1
2 3
3 1
3 2
A
1
2
3
A
The message that R2 sends to R1 is R1 updates
its relation and domains and sends messages to
neighbors
A C
1 2
3 2
B
C
D
F
B C F
1 2 3
3 2 1
G
A B D
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
D F G
1 2 3
2 1 3
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Distributed Arc-Consistency

• DR-AC can be applied to the dual problem of any
  constraint network.

b) Constraint network
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DR-AC on a dual join-graph
A
1
2
3
A B
1 2
1 3
2 1
2 3
3 1
3 2
A C
1 2
3 2
B C F
1 2 3
3 2 1
A B D
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
D F G
1 2 3
2 1 3
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Iteration 1
A
1
2
3
A
1
2
3
A
1
3
A
1
2
3
B
A B
1 2
1 3
2 1
2 3
3 1
3 2
A
1
2
3
B
1
2
3
1
3
A C
1 2
3 2
A
1
2
3
C
2
B C F
1 2 3
3 2 1
C
2
B
1
2
3
F
1
3
A
1
2
3
D
1
2
B
1
2
3
A B D
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
D F G
1 2 3
2 1 3
F
1
3
D
1
2
3
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Iteration 1
A
1
3
A B
1 3
2 1
2 3
3 1
A C
1 2
3 2
B C F
1 2 3
3 2 1
A B D
1 3 2
2 3 1
3 1 2
3 2 1
D F G
2 1 3
31
Iteration 2
A
1
3
A
1
2
3
A
1
3
A
1
2
3
A B
1 3
2 1
2 3
3 1
A
1
3
B
1
2
3
A C
1 2
3 2
A
1
3
C
2
B C F
1 2 3
3 2 1
C
2
F
1
B
1
3
A B D
1 3 2
2 3 1
3 1 2
3 2 1
A
1
3
D
2
B
1
3
D F G
2 1 3
F
1
3
D
1
2
32
Iteration 2
A
1
3
A B
1 3
3 1
A C
1 2
3 2
B C F
3 2 1
A B D
1 3 2
3 1 2
D F G
2 1 3
33
Iteration 3
A
1
3
A
1
3
A
1
3
A
1
3
B
3
A
1
3
A B
1 3
3 1
B
1
3
A
1
3
C
2
A C
1 2
3 2
C
2
F
1
B C F
3 2 1
B
1
3
A
1
3
D
2
B
1
3
A B D
1 3 2
3 1 2
F
1
D
2
D F G
2 1 3
34
Iteration 3
A
1
3
A B
1 3
A C
1 2
3 2
B C F
3 2 1
A B D
1 3 2
3 1 2
D F G
2 1 3
35
Iteration 4
A
1
A
1
3
A
1
3
A
1
3
B
3
A
1
3
A B
1 3
B
1
3
A
1
3
C
2
A C
1 2
3 2
C
2
F
1
B C F
3 2 1
B
3
A
1
3
D
2
B
1
3
A B D
1 3 2
3 1 2
F
1
D
2
D F G
2 1 3
36
Iteration 4
A
1
A B
1 3
A C
1 2
3 2
B C F
3 2 1
A B D
1 3 2
D F G
2 1 3
37
Iteration 5
A
1
A
1
A
1
A
1
B
3
A
1
B
3
A B
1 3
A
1
C
2
A C
1 2
3 2
C
2
F
1
B C F
3 2 1
B
3
B
3
A
1
D
2
A B D
1 3 2
F
1
D
2
D F G
2 1 3
38
Iteration 5
A
1
A B
1 3
A C
1 2
B C F
3 2 1
A B D
1 3 2
D F G
2 1 3
39
Join-tree clustering is a restricted
tree-decompositionexample
40
Adaptive-consistency as tree-decomposition

• Adaptive consistency is a message-passing along a
  bucket-tree
• Bucket trees each bucket is a node and it is
  connected to a bucket to which its message is
  sent.
• The variables are the clicue of the triangulated
  graph
• The funcions are those placed in the initial
  partition

41
Bucket EliminationAdaptive Consistency (Dechter
and Pearl, 1987)
RCBE
RDBE ,
RE
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From bucket-elimination to bucket-tree propagation
43
The bottom up messages
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Adaptive-Tree-Consistency as tree-decomposition

• Adaptive consistency is a message-passing along a
  bucket-tree
• Bucket trees each bucket is a node and it is
  connected to a bucket to which its message is
  sent.
• Theorem A bucket-tree is a tree-decomposition
• Therefore, CTE adds a bottom-up message passing
  to bucket-elimination.
• The complexity of ATC is O(r deg k(w1))

45
Join-Graphs and Iterative Join-graph propagation
(IJGP)