Tree Clustering for Constraint Networks

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Tree Clustering for Constraint Networks
By Rina Dechter Judea Pearl Artificial
Intelligence, Oct 1988

• Chris Reeson
• Advanced Constraint Processing
• Fall 2009

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Overview

• Contributions of the Paper
• Context of the Paper
• Algorithms
• Tree Clustering
• Adaptive Consistency
• Relative Merits
• Conclusion
• Related algorithms

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Contributions of the Paper

• Introduces Tree Clustering (T-C) for
  backtrack-free search
• Introduces the Adaptive Consistency (A-C)
  algorithm
• Compares the two algorithms

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Context

• Two ways to restrict CSPs to make finding the
  minimal CSP efficient
• Topology of constraint graph
• Types of constraints
• T-C A-C
• Target the topology tree
• Are applicable to binary non-binary CSPs
• Two ways to transform a constraint graph into a
  tree
• Remove redundant arcs in dual graphs (join tree)
• Form larger clusters of c-variables (simulates a
  join tree)

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Definitions

• Hyper, dual, and primal graphs

Hypergraph
Dual graph
Primal graph

• The discussion focuses on primal graphs for
  simplicity

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From Join Graph to Join Tree

• Join Graph
• Start w/ the dual graph, remove redundant edges
  while maintaining the connectedness property
• Connectedness property For each two nodes
  sharing a variable, there is at least one path of
  labeled arcs containing the shared variable
• Join Tree
• When the join graph is a tree, the dual CSP can
  be solved BT free w/ directional arc consistency
• What if there isnt a join tree?
• ? The idea for Tree Clustering

AEF
ABC
A
C
AE
E
AC
ACE
CDE
CE
AEF
ABC
AE
AC
ACE
CDE
CE
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Motivating Problem
Constraint Graph
A solution
C lt A
C ? D
A ? B
C lt B
D lt A
D lt B
F gt B
E gt B
G gt F
G gt E
Domains 1, 2, 3, 4, 5
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Overview

• Contributions of the Paper
• Context of the Paper
• Algorithms
• Tree Clustering (T-C)
• Adaptive Consistency (A-C)
• Relative Merits
• Conclusion
• Related Algorithms

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Tree Clustering (T-C) Idea

• A CSP organized as a join tree can be solved
  efficiently
• Tree Clustering Algorithm
• Solves a CSP by breaking it into subproblems
• Triangulates the primal graph
• Solves subproblems combines the solutions

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Tree Clustering (T-C) Algorithm

• Triangulate the primal graph
• Identify all the maximal cliques in the primal
  chordal graph
• Form a join tree
• Solve the subproblems
• Each cluster becomes single variable
• Solve the tree problem
• Perform DAC from leaves to root
• Instantiate BT-free from root to leaves

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Tree Clustering (T-C) Costs

• Given a CSP and its primal graph generate a
  chordal primal graph O(n2)
• Identify all the maximal cliques in the primal
  chordal graph O(E)
• Form the dual graph O(n)
• Solve the sub problems O(kr) where kdomain
  size
• Solve the tree problem O(n t log t)

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Tree Clustering (T-C) Total Cost

• Dominated by O(n t log t)
• t is the largest number of solutions in a
  cluster,
• t kr
• Time O(n kr r log k) O(nr kr )
• Space O(n kr )

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Overview

• Contributions of the Paper
• Context of the Paper
• Algorithms
• Tree Clustering (T-C)
• Adaptive Consistency (A-C)
• Relative Merits
• Conclusion
• Related Algorithms

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Adaptive Consistency (A-C)

• An ordered constraint graph is backtrack-free if
  the level of directional strong consistency along
  this order is greater than the width of the
  ordered graph
• Beware
• Enforcing i-consistency for i gt 2 often requires
  the addition of constraints which increase the
  width

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Adaptive Consistency (A-C) Idea

• Given an ordering d,
• d-i-consistency is defined recursively
• letting i change dynamically from node to node
• (A-C later redefined as bucket elimination)

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Adaptive Consistency Algorithm

1. For in downto 1 do Steps 2-4
2. Compute PARENTS(Xi)
3. Connect all PARENTS(Xi)
4. Perform Consistency(Xi, PARENTS(Xi)) ? joining
   the constraints between Xi its parents
5. Build a solution BT-free in the ordering (X1, ,
   Xn)

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Adaptive Consistency Cost

• Time O(n exp(W(d) 1)), see Dechter page 109
• Space O(n kW(d))

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Overview

• Contributions of the Paper
• Context of the Paper
• Algorithms
• Tree Clustering (T-C)
• Adaptive Consistency (A-C)
• Relative Merits
• Conclusion
• Related Algorithms

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Relative Merits

• Arcs resulting from triangulation match arcs
  added by adaptive consistency, for the same
  ordering
• Every cluster in T-C is represented in A-C by a
  series of smaller constraints
• Similar bounds
• W(d) 1 the size of the largest clique
• A-C eliminates the redundancy of generated
  solutions
• T-C enumerates all solutions that A-C represents
  via constraints.

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Conclusions

• Tree clustering groups c-nodes into a tree
  capable of supporting query answering
  backtrack-free
• Useful in systems that need to answer many
  questions about a dataset and where the
  environmental conditions undergo local changes
• Recently, researchers have started looking at T-C
  for solving the CSP, see BTD by Jégou Terrioux
  (and others in soft CSPs)

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Note On Triangulation

• Find the triangulated graph w/ smallest maximum
  clique NP-hard
• Heuristics
• Operation when eliminating a node, connect all
  its neighbors, to form a clique (fill edges)
• H1 choose the node w/ smallest degree
• H2 choose the node that, after elimination,
  yields the smallest number of fill edges
• H3 Given any ordering (e.g., maximal cardinality
  ordering), moralize the graph
• Elimination order is the reverse of instantiation
  order
• Elimination order of a triangulated graph is
  called a perfect elimination scheme
• In this ordering, every node is simplicial
  forms a clique w/ its neighbors
• If you follow elimination order, no fill edges
  need to be added

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Maximal Cardinality Ordering

• An approximation of min. width ordering
• Choose a node arbitrarily a simplicial node
• Among the remaining nodes, choose the one that is
  connected to the maximum number of already chosen
  nodes, break ties arbitrarily
• Repeat
• Reverse the final order

Tsang 6.2.4 Dechter Fig 4.5
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Two Additional Algorithms

• Maximal Cliques of the triangulated graph
• Join Tree of the triangulated graph