Refining the Basic Constraint Propagation Algorithm

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Refining the Basic Constraint Propagation
Algorithm

• Christian Bessière and Jean-Charles Régin

Presented by Sricharan Modali
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Outline

• AC3
• Two refinements
• AC2000
• AC2001
• Experiments
• Analytical comparison of AC2001 AC6
• Conclusion

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Introduction

• Importance of constraint propagation
• Propagation scheme of most of existing constraint
  solving engines
• Constraint oriented, or
• Variable oriented
• AC3 is a generic algorithm
• AC4, AC6 AC7 value oriented propagation

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Importance of AC3

• When you know constraint semantics, use special
  propagation algorithms (e.g., all-diff,
  functional)
• When nothing is known about constraints, use a
  generic AC algorithm (e.g., AC1, 2, 3, 4, 6 or 7)
  
• AC3 does not require maintaining a specific data
  structures during search, in contrast to AC4, AC6
  AC7
• ? Authors focus on AC3 generic and light weight

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Contribution

• Modify AC3 into AC2000 AC2001
• More efficient with heavy propagation
• Light weight data structures
• Variable-oriented
• Dominate AC3 ( CC CPU time)

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AC2000 vs. AC3

• AC2000, like AC3, is free of any data structure
  to be maintained during search
• (not really authors use ?(Xi) per variable)
• Regarding human cost of implementation AC2000
  needs 5 more lines than AC3

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AC2001 vs. AC3

• AC2001 needs an extra data structure, an integer
  for each value-constraint pair Last(Xi,vi,Xj)
• Achieves optimal worst-case time complexity
• Human cost (implementation) AC2001 needs
  management of additional data structure (?(Xi),
  Last(Xi,vi,Xj))

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Constraint network

• P (X, D, C)
• X is a set of n variables X1, , Xn
• D is a set of domains D(X1), , D(Xn)
• C is a set of e binary constraints between pairs
  of variables.
• Constraint check verifying whether or not a
  given pair of values (vi,vj) is allowed by Cij

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Arc consistent value

• vi is an arc-consistent value on Cij
• vi ? D(Xi) ? ?vj ? D(Xj) (vi,vj) ? Cij
• vj is called support for (Xi,vi) on Cij

Xj
Xi
vj
vi
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Viable value

• Viable value vi ? D(Xi) is viable ? it has
  support in all neighboring D(Xj)
• Arc consistent CSP if all the values in all the
  domains are viable.

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AC3

• A variable-oriented propagation scheme
• Difference with Mackworth 77
• Instead of handling a queue for the constraints
  to be propagated,
• it has a queue of the variables whose domain has
  been modified.
• This AC3 terminates whenever any domain is empty

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A (bad) example for AC3
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AC3 overdoes it

• Revise3(Xj,Xi) removes vj from D(Xj)
• AC puts Xj in Q
• Propagate3 calls Revise3(Xi,Xj) for every
  constraint Cij involving Xj
• Revise3(Xi,Xj) will look for a support for every
  value in D(Xi) even when vj was not a support!

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Enhancement in AC2000

• Instead of blindly looking for a support for a
  value vi?D(Xi) each time D(Xj) is modified, it is
  done only when needed

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AC2000

• In addition to Q, ?(Xj) is used
• ?(Xj) contains the values removed from D(Xj)
  since the last propagation of Xj
• When calling Revise2000(Xi,Xj,t) a check is made
  to see if vi has a support in ?(Xj)

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Example
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How AC2000 operates

• The larger ?(Xj),
• the closer it gets in size to D(Xj)
• the more expensive the process is
• the more likely for vi to have a support in ?(Xj)
• Hence lazymode is used only when ?(Xj) is
  sufficiently smaller than D(Xj)
• Use of lazymode is controlled with Ratio
• ?(Xj) / D(Xj) lt Ratio, use lazymode
• ?(Xj) / D(Xj) ? Ratio, use ?lazymode

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Analysis of AC2000

• Assumption AC3 is correct
• Prove Lazymode of AC2000 does not lead to
  arc-inconsistent values in the domain
• The only way the search for support for a value
  vi in D(Xi) is skipped is when vi is not
  supported by values in ?(Xj)
• ?(Xj) contains all values last deleted from D(Xj)
  ? vi has exactly the same set of supports as
  before on Cij
• Looking again for a support for vi is useless as
  it remains consistent with Cij

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Space complexity of AC2000

• It is bounded by the sizes of Q and ?
• Q is O(n), ? is O(nd)
• d is the size of the largest domain
• Overall complexity O(nd)

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Time Complexity of AC2000

• The main change is in Revise2000, where both
  ?(Xj) and D(Xj) are examined instead of only
  D(Xj)
• This leads to a worst case where d2 checks are
  performed in Revise2000
• Hence the overall time complexity is O(ed3) since
  Revise2000 can be called d times per constraint.

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AC2000 too overdoes it..

• In AC2000 we have to look again for a support for
  vi on Cij
• If we can remember the support found for vi in
  D(Xj) the last time Cij is revised
• Next time we need to check whether or not this
  last support belongs to ?(Xj).

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AC2001 saves more..

• A new data structure Last(Xi,vi,Xj) is used to
  store the value that supports vi
• The function Revise2001 always runs in lazymode,
  except during the initialization phase.
• Further, when supports are checked in a given
  ordering ltd (i.e., sorted) we know that there
  isnt any support for vi before Last(Xi,vi,Xj) in
  D(Xj).

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Example
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Space complexity of AC2001

• Is bounded above by the size of Q, ? and Last
• Q is in O(n) ? is in O(nd)
• But Last is in O(ed)
• Since each value vi has a Last pointer for each
  constraint involving Xi.
• This gives the overall complexity of O(ed)

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Time Complexity of AC2001

• As in AC3 AC2000, Revise2001 can be called d
  times per constraint.
• But at each call to Revise2001(Xi,Xj,t)for each
  value vi ? D(Xi)
• There will be a test on the Last(Xi,vi,Xj)
• And a search for support on D(Xj) greater than
  Last(Xi,vi,Xj)
• The overall time complexity is then bounded above
  by d(dd)2e, which is in O(ed2)
• O(ed2) is optimal
• AC2001 is the first optimal arc-consistency
  algorithm proposed in the literature that is free
  of any lists of supported values.

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Experiments

• To see if AC2000 and AC2001 are effective vs.
  AC3, compare CC CPU
• Context pre-processing search (MAC)
• The goal is not to compete with AC6/AC7
• An improvement (even small) w.r.t AC3 is
  significant

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AC as a preprocessing

• The chance of having some propagations are very
  small on real instances
• Hence only one real world instance is
  considered
• Other instances are randomly generated to fall in
  the phase transition region
• Ratio of 0.2 is taken (no justification given)

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Parameters

• ltN, D, C/p1, T/p2gt
• N number of variables
• D size of the domain
• C number of constraints
• P1 density of constraints 2C/N.(N-1)
• T number of forbidden tuples
• P2 tightness of the forbidden tuples T/D2

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Results

• Low density (p10.045)
• Instance 1 under-constrained (p20.5)
• Instance 2 over-constrained (p2 0.94)
• High tightness (p20.918, 0.875)
• Instance 3 sparse (p10.045)
• Instance 4 dense (p11.0)

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Observation
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Maintaining Arc consistency during search

• MAC-3, MAC-2000, MAC-2001
• Experiments carried over all the instances
  contained in FullRLFAP archive for which more
  than 2 secs is necessary to find a solution or to
  prove that none exists
• Ratio is again 0.2 (no justification given)

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Results
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Observations

• There is a slight gain of MAC2000 over MAC3
• Except for SCEN11
• On SCEN11 it is seen that MAC2000 outperforms
  MAC3 for ratio 0.1
• MAC2001 outperforms MAC3 with 9 times less CC and
  2 times less cpu time

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Restrictions

• Comparison is between algorithms with simple data
  structures
• Note that to solve SCEN11
• MAC-6 (MAC AC6) 14.69 sec
• MAC3 needs 39.50 sec
• MAC2000 needs 38.22 sec
• MAC2001 needs 22.69 sec

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AC2001 vs. AC6

• Time complexity and space complexity of AC2001 is
  equal to that of AC6
• What are the differences between AC6 and AC2001?
• Property1 CC same!
• Property2 Difference is in the effort of
  maintaining specific data structures
• Authors give condition who wins when

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Conclusion

• Two refinements to AC3 AC2000 AC2001
• AC2000 improves slightly over AC3, w/o
  maintenance of any new data structure
• AC2001 needs an additional data structure Last
• AC2001 achieves optimal worst-case time complexity

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Thanks