1. Dynamic Ordering for Asynchronous Backtracking on DisCSPs
Roie Zivan and Amnon Meisels
Dept. of Computer Science
Ben-Gurion University

2. A Distributed Constraint Network
11/30/2018
Dynm Ordr ABT - CP-2005

3. Agents’ local network Agent1 V1 [0,1,2,3] V6 [0,1] V4 [0,1,2]
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Dynm Ordr ABT - CP-2005

4. A “realistic” example S1 S1 S2 S2 S3 S3 Two wards in a Hospital.
Nurses – ward 1: Dina, Ela, Flora ward 2: Aviva, Bila, Gila
Aviva, Bila, Dina are ‘Senior’ nurses.
In every shift there should be at least one senior nurse.
Local Constraint
Inter Constraint
One senior nurse S1 S1 S2 S2 S3 S3
First ward
Second ward
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Dynm Ordr ABT - CP-2005

5. Solving DisCSPs A2 A1
V1 = 2
V2 = ?
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Dynm Ordr ABT - CP-2005

6. Sequential (synchronous) BT
An-1 An
[V1=2],[V2=1],[V3=?]…
Current Partial Assignment
(CPA)
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Dynm Ordr ABT - CP-2005

7. 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.
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Dynm Ordr ABT - CP-2005

8. Asynchronous Backtracking
Simplifying Assumptions:
Each agent holds exactly one variable
Every agent can send messages to any other agent
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Dynm Ordr ABT - CP-2005

9. Asynchronous Backtrack (ABT)
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Dynm Ordr ABT - CP-2005

10. Backtrack in ABT AgentView of Agent A4: A1=2, A3=1 A4=1 A1=2, A3=1
Nogood
Current Assignment A3
A4=1
A1=2, A3=1
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Dynm Ordr ABT - CP-2005

11. Backtracking generates links
[A1=2], [A3=1]
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Dynm Ordr ABT - CP-2005

12. ABT – Strong Assumption
All Agents are ordered by a fixed order of Priorities
[Yokoo ]
[BesBrMes 2001 – 2005]
[Hammadi 98 – 2002]
[Gomez et. al 2002 – 2005]
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Dynm Ordr ABT - CP-2005

13. Motivation for dynamic reordering
Results of recent studies of sequential assigning algorithms for DisCSPs:
Nguyen & Faltings 2004
Brito & Meseguer 2004
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Dynm Ordr ABT - CP-2005

14. A2 A1 Why is it a problem? [1] [1] V1 ≠ V2 V1 ≠ V2 V1 = 1 11/30/2018
Dynm Ordr ABT - CP-2005

15. A2 A1 Why is it a problem? [1] [1] V1 ≠ V2 V1 ≠ V2 V1 = 1 11/30/2018
Dynm Ordr ABT - CP-2005

16. A1 A2 Why is it a problem? [1] [1] V1 ≠ V2 V1 ≠ V2 V1 = 1 V2 = 1
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Dynm Ordr ABT - CP-2005

17. A2 A1 Why is it a problem? [1] [1] V1 ≠ V2 V1 ≠ V2 V2 = 1 V1 = 1
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Dynm Ordr ABT - CP-2005

18. Dynamic ordering can be a problem
[1]
[1]
V1 ≠ V2
V1 ≠ V2
V2 = 1
V1 = 1
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Dynm Ordr ABT - CP-2005

19. Reordering Asynchronous Backtracking
Asynchronous Weak Commitment - AWC
[Yokoo ]
(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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Dynm Ordr ABT - CP-2005

20. Reordering – in standard backtracking1 2 3 4
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Dynm Ordr ABT - CP-2005

21. Reordering – sequential assignments algorithms1 2 3 4 4 3
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Dynm Ordr ABT - CP-2005

22. Reordering – sequential assignments algorithms1 2 4 3 4 2 4 3 3
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Dynm Ordr ABT - CP-2005

23. Same idea can be applied in Asynchronous Backtracking
Reordering – standard backtracking
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
11/30/2018
Dynm Ordr ABT - CP-2005

24. Ordering Rules for Agents in ABT
Change order only when changing assignment
Enforce reordering only on Agents with lower priority
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Dynm Ordr ABT - CP-2005

25. ABT with Dynamic Ordering
[A1,2],[A2,1], [A4,1]
A1,A3,A2,A4
A3 ≠ 1
A3 = 1
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Dynm Ordr ABT - CP-2005

26. ABT with Dynamic Ordering
[A1,2],[A2,1], [A4,1]
A1,A3,A2,A4
A1,A3,A4,A2
A3 = 2
A1,A3,A4,A2
A3 = 2
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Dynm Ordr ABT - CP-2005

27. 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
11/30/2018
Dynm Ordr ABT - CP-2005

28. ABT with Dynamic Ordering
[A1,1],[A3,1],[A2,3],[A4,2]
A3 ≠ 1
A3 = 1
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Dynm Ordr ABT - CP-2005

29. ABT with Dynamic Ordering
[A1,1],[A3,2],[A4,0],[A2,0]
[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],[A3,1],[A2,3],[A4,2]
A3 = 2
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Dynm Ordr ABT - CP-2005

30. 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 newly formed lower priority agents
 few Nogoods are removed
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Dynm Ordr ABT - CP-2005

31. ABT_DO - Different Heuristics (20 Agents p1 = 0.4)
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Dynm Ordr ABT - CP-2005

32. ABT_DO - Different Heuristics (20 Agents p1 = 0.4)
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Dynm Ordr ABT - CP-2005

33. ABT_DO - Different Heuristics (20 Agents p1 = 0.7)
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Dynm Ordr ABT - CP-2005

34. ABT_DO - Different Heuristics (20 Agents p1 = 0.7)
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Dynm Ordr ABT - CP-2005

35. Nogoods removed by Different Heuristics ( p1 = 0.4)
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Dynm Ordr ABT - CP-2005

36. Nogoods removed by Different Heuristics (p1 = 0.7)
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Dynm Ordr ABT - CP-2005

37. 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
Number of messages decreases also
11/30/2018
Dynm Ordr ABT - CP-2005