Dynamic Ordering for Asynchronous Backtracking on DisCSPs

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Dynamic Ordering for Asynchronous Backtracking on
DisCSPs

• Roie Zivan and Amnon Meisels
• Dept. of Computer Science
• Ben-Gurion University

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Asynchronous Backtracking

• Simplifying Assumptions
• Each agent holds exactly one variable
• The constraints network is a full graph (p1 1)
• Every agent can send messages to any other agent

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ABT Strong Assumption
All Agents are ordered by a fixed order of
Priorities Yokoo 92-2000 BesBrMes 2001
2005 Hammadi 98 2002 Gomez et. al 2002
2005
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Motivation for dynamic reordering

• Results of recent studies of sequential assigning
  algorithms for DisCSPs

Nguyen Faltings 2004
Brito Meseguer 2004
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Reordering Asynchronous Backtracking

• Asynchronous Weak Commitment - AWC
• Yokoo 95 - 2000
• (exponential space)
• Hybridizing ABT and AWC into a polynomial space,
  complete protocol.
• Silaghi and Faltings 2001
• (complex algorithm with unrewarding results)

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Reordering sequential assignments algorithms
1
2
3
4
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Reordering sequential assignments algorithms
1
2
3
4
4
3
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Reordering sequential assignments algorithms
1
2
4
3
4
2
4
3
3
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Reordering sequential assignments algorithms

• Highlights
• Moving forward - choose a desirable order
• Backtracking is done in the same order as the
  forward moves

Same idea can be applied in Asynchronous
Backtracking
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Ordering Rules for Agents in ABT

• Change order only when changing assignment
• Enforce reordering only on Agents with lower
  priority

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ABT with Dynamic Ordering
A1,2,A2,1, A4,1
A1,A3,A2,A4
A3 ? 1
A3 1
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ABT with Dynamic Ordering
A1,2,A2,1, A4,1
A1,A3,A2,A4
A1,A3,A4,A2
A1,A3,A4,A2
A3 2
A3 2
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Which order is most up-to-date?

• Time-Stamping method of Nguyen Faltings 2004
• Each order is time-stamped
• A time-stamp is an array of integers of size n,
  all initialized to 0
• Each agent that assigns its variable, updates the
  time-stamp as follows
• Counters of higher priority agents are untouched
• The counter of the assigning agent is incremented
  by 1
• The counters of lower priority agents are set to
  zero
• Time-stamps are compared lexicographically

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ABT with Dynamic Ordering
A1,1,A3,1,A2,3,A4,2
A3 ? 1
A3 1
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ABT with Dynamic Ordering
A1,1,A3,1,A2,3,A4,2
A1,1,A3,1,A2,3,A4,2
A1,1,A3,1,A4,2,A2,3
A1,1,A3,2,A4,0,A2,0
A1,1,A3,1,A2,3,A4,2
A3 2
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Correctness proof (outline)

• Assumption Standard ABT is correct
• Induction on the number of agents (n)
• Assumption for every k lt n, the algorithm is
  correct
• For a single agent trivial
• For a DisCSP with n agents, assign the first
  agent and set an order for the rest
• The induced DisCSP is of size n 1 and the
  completeness holds (by induction)
• Now assign the second value to the first
  variable

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Choosing the right heuristic

• Heuristics that work well for sequential
  assignment algorithms fail for ABT_DO
  (min-domain)
• After each change of order, relevant Nogoods
  are discarded
• Nogood-triggered heuristic Ginsberg-93
• Agent changes its current order only when it
  receives a Nogood which causes an assignment
  change
• The Nogood sender is moved to a place immediately
  following that of the assigning agent
• The only agent that moves up has no conflicts
  with any lower priority ? no Nogoods are removed

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ABT_DO - Different Heuristics (20 Agents p1
0.4)
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ABT_DO - Different Heuristics (20 Agents p1
0.4)
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ABT_DO - Different Heuristics (20 Agents p1
0.7)
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ABT_DO - Different Heuristics (20 Agents p1
0.7)
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Conclusions

• Dynamic Variable/Agent reordering is a powerful
  tool in CSP solving algorithms
• Most previous studies of asynchronous
  backtracking used a fixed order of agents
• Dynamic reordering for ABT with polynomial space
  can be achieved and implemented
• The heuristic must take into consideration the
  loss of valid Nogoods
• A Nogood-triggered heuristic, avoids Nogoods loss
  and improves the performance by a large factor

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A Distributed Constraint Network
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Asynchronous Backtracking

• Motivation
• Concurrent computation.
• HighLights
• Agents always hold an assignment which is
  consistent with their view of the system.
• On backtrack operations, agents change their
  views by eliminating inconsistent values.