1. Asynchronous Partitioning Framework
The Open University
of Israel
Vitaliy Freidovich
Department of Mathematics and Computer Science
Vice President R&D of
Amnon Meisels
Ben-Gurion University
Of the Negev
Department of Computer Science

2. Missing features of DCSP algorithms
Order of search unrelated to structure of constraints graph
Structure of constraints graph can enhance the use of the fail first principle
Utilization of graph structure in distributed search needs a method for agent cooperation and coordination

3. A better idea might be Analyze cooperatively the constraints graph
Find regions which are the most heavily constrained, and partition the graph by those regions
Solve the local problem inside each region
Achieve globally consistent assignment by propagating relevant assignments between the different regions

4. Advantages Enhancing the Fail First Principle
Concentrating on the toughest parts of the problem first
Improving performance by allowing parallel search inside different regions

5. Asynchronous Partitioning Framework
A general framework for agent cooperation and coordination
Composed of 4 distinct components:
GroupPartition algorithm - pluggable
Partition into disjoint groups
Select a group leader
LocalSearch algorithm - pluggable
GlobalSearch algorithm - pluggable
Agents are represented by their group leaders
Coordination engine
Adapt to assignments in higher priority groups
Ignore constraints with lower priority groups

6. Asynchronous Partitioning Framework
Requirements:
GroupPartition algorithm:
Form disjoint groups
Each group must contain at least one ‘fully’ connected agent
Elect one such agent to be a group leader
Assign injective priorities to groups
Order the agents inside each group
LocalSearch algorithm:
Synchronous - exactly one token
Allocate each agent an agent view
Update the agent view upon reception of a token
GlobalSearch algorithm:
The above requirements enable correctness

7. Asynchronous Group Partition
A first implementation of APF
Phases:
Initialization
Priority calculation
Group partitioning
Group ordering
Search for a solution
SBT for LocalSearch
CBJ for GlobalSearch
Phases 2-4 implement GroupPartition

8. An Example 3-coloring problem
Find an assignment out of: {Red, Green, Blue}, such that no two connected agents are painted with the same color

9. Phase 1: Initialization
Main goal: Initialize agent priority
Pai = ki + ∑j=1 ki (CommonNeighbors (ai, aj))
ki – number of neighbors of ai
Initially: Pai = ki

10. Phase 2: Priority calculation
a4 a6 a7 a8,a9 1 a3
P=4 a4
P=2
a4 a6 a7 a8,a9 a5
P=5 a9
P=4
a4 a6 a7 a8,a9 2
a4 a6 a7 a8,a9
a4 a6 a7 a8,a9 1 a2
P=1 a6
P=3 2 a7
P=3
Each agent sends a list of its neighbors, to all its neighbors
Upon receiving this message, each agent replies with the number of joint neighbors
The originating agent, adds the replies to its priority, and notifies all the agents about its priority

11. Phase 2: Priority calculation
Each agent sends a list of its neighbors, to all its neighbors
Upon receiving this message, each agent replies with the number of joint neighbors
The originating agent, adds the replies to its priority, and notifies all the agents about its priority

12. Phase 3: Group partitioning
Each agent performs a local search in its neighborhood
When an agent receives such a request:
If group leader: approves
If not a group leader: denies
If not determined: wait until is, and reply
The search continues until:
Some agent had approved the request
The current agent has the highest priority

13. Phase 3: Group partitioning
JOIN
JOIN a3
P=4 a4
P=2 a5
P=11 a9
P=8
JOIN
W:{a1}
W:{a10}
JOIN
W:{a9}
JOIN a2
P=1 a6
P=5 a7
P=7
W:{a3}

14. Phase 3: Group partitioning
W:{a1} a4
P=2 a5
P=11
W:{a9,a6} a9
P=8
W:{a10} a2
P=1 a6
P=5 a7
P=7
W:{a3}

15. Phase 3: Group partitioning
W:{a1} a4
P=2 GL a5
P=11 a9
P=8
NOT_JOINED
JOINED
W:{a10}
JOINED a2
P=1
NOT_JOINED a6
P=5 a7
P=7
W:{a3}

16. Phase 3: Group partitioning
JOINED GL a3
P=4 a4
P=2 GL a5
P=11 a9
P=8
JOIN
JOINED a2
P=1 a6
P=5 a7
P=7

17. Phase 3: Group partitioning
JOIN
a10
P=2 a1
P=2
NOT_JOINED a8
P=4
JOIN
JOINED GL a3
P=4 a4
P=2 GL a5
P=11 a9
P=8
JOIN
JOINED
JOINED
JOIN a2
P=1 a6
P=5 a7
P=7
a1 and a10 have the same priority
a1 has a higher lexicographical priority

18. Phase 3: Group partitioningGL
a10
P=2 a8
P=4 GL a3
P=4 a4
P=2 GL a5
P=11 a9
P=8 a2
P=1 a6
P=5 a7
P=7

19. Phase 4: Group ordering Group Priority  Group Leader Priority
Inner Group Priority (IGP):
Calculated for each agent inside a group
IGPai = PGai + ∑jPcgij
IGPai – The Inner Group Priority of agent ai
PGai – The Priority of the Group of agent ai
Pcgij – The Priority of Connected Group j –The priority of the jth Group, to which agent ai is connected
In this way, agents connected to a greater number of bigger groups will have a higher IGP

20. Phase 4: Group ordering
GL a10
P=2 a1
P=2
IGP=4+2 a8
P=4
IGP=17
IGP=6
IGP=11 GL a3
P=4
GL a5
P=11 a9
P=8 a4
P=2
IGP=
IGP=26
IGP=15
IGP=11
IGP=13 a2
P=1 a6
P=5 a7
P=7
IGP=4
IGP=4
IGP=15
IGP=11
Each agent notifies the other agents in its group about its IGP
The agents in each group are ordered by their calculated IGP
Equal priorities are resolved lexicographically

21. Phase 4: Group ordering GL a10 a1 O=2 a8 O=1 O=6 GL a3 O=1 GL a5 a9 a4
IGP=6
O=2 a8
IGP=17
O=1
IGP=11
O=6 GL a3
IGP=26
O=1
GL a5 a9 a4
IGP=15
O=1
IGP=11
O=4
IGP=13
O=3 a2
IGP=4
O=3 a6
IGP=15
O=2 a7
IGP=11
O=5

22. Phase 5: Search for solution
The search proceeds in two levels in parallel:
SBT as APF’s LocalSearch
CBJ as APF’s GlobalSearch
Inside each group:
The token is being expanded through the group leader
Keeps consistency with:
Inter-group constraints
Higher priority intra-group constraints
Lower priority intra-group constrained agents are notified upon value changes

23. Phase 5: Search for solution
A global token is moved between the group leaders
When a group leader receives the global token:
Group is consistent: advance
Group is inconsistent: backtrack
Groups’ state hasn’t been determined: wait
Termination:
Last group is consistent
First group is inconsistent
Detailed example: Appendix (No time )

24. AGP-CBJ A second implementation of APF
Identical to AGP; the LocalSearch algorithm is upgraded to be CBJ
Was formally proven to be correct

25. Experimental Evaluation
Three sets of experiments on randomly generated problems:
(n=10, k=10, 0.1≤p1≤0.9, 0.1≤p2≤0.9)
(n=15, k=10, 0.1≤p1≤0.9, 0.1≤p2≤0.9)
(n=20, k=10, 0.1≤p1≤0.9, 0.1≤p2≤0.9)
For each combination 10 instances
2430 problems in total

26. How important is LocalSearch?

27. Partitioning – What good is it?
AGP-CBJ
Partitioning appears to be beneficial
1 Group → Most likely an almost Clique

28. APF – Where is it best?
We expect it to be best for low values of p1

29. Why static partitioning?

30. Where is coordination best ?

31. Where is coordination best ?
P1=0.4

32. Conclusions
Preliminary results indicate that partitioning into groups can be beneficial
Most efficient for sparse problems and for highly dense problems
Upgrading LocalSearch & GlobalSearch may improve performance
AFC-CBJ
Can APF’s requirements be relaxed ?
ABT

33. Thank You!

34. Appendix – AGP’s Search For Solution

35. Future work Upgrading LocalSearch & GlobalSearch
AFC-CBJ
Relaxing APF’s requirements
ABT
Upgrading heuristics
Partitioning priorities
Group ordering heuristic
Upgrading the partitioning strategy
Limiting the number of agents in a group?

36. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1 a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a10 is the first agent to start execution
a10 as a group leader:
Not the first one - generates no global token
Generates a new Token
Sends Token to the first agent in its group

37. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1 a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a10 as an agent:
Finds consistent assignment
Returns token to its group leader
a10 as a group leader:
Enters the consistent state

38. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
Group Leader a5 starts execution
First one - Generates a new GlobalToken
Generates a new Token
Sends Token to the first agent in its group

39. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
V_C
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a4:
Finds consistent assignment
Notifies a3
Returns Token to a5

40. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a3 – group leader:
First generates a new Token
Sends to first agent
a3 – agent:
Finds consistent assignment
Returns Token to group leader

41. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
V_C
Token
Token
V_C_H a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a3 - Group leader:
Forwards the value change message to the addressee
a3 - agent:
Updates agent view
Notifies group leader that value change was handled

42. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
R_T
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a3 - Group leader:
Asks a3 to regenerate new token
a3 - agent:
Regenerates Token
Returns Token to group leader

43. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a3 – group leader:
Sends Token to a3
a3 – agent:
Finds consistent assignment

44. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a3 – group leader:
Sends Token to a3
a3 – agent:
Finds consistent assignment
Returns Token to group leader

45. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

46. Phase 5: Search for solution
V_C
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

47. Phase 5: Search for solution
V_C
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

48. Phase 5: Search for solution
V_C
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
V_C
Token
Token a2
P=1
O=3 a6
P=5 O=2
V_C a7
P=7 O=5

49. Phase 5: Search for solution
V_C
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4
Token a9
P=8
O=3
V_C a4
P=2
O=1
Token
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

50. Phase 5: Search for solution
V_C
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3
V_C a4
P=2
O=1
Token
V_C
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

51. Phase 5: Search for solution
V_C
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
V_C_H
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3
V_C a4
P=2
O=1
Token
Token
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

52. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
Token
V_C_H
GlobalToken
Token
GL a5
P=11 O=4 a9
P=8
O=3
R_T GL a3
P=4
O=1 a4
P=2
O=1
Token
Token
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

53. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

54. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2
Token a8
P=4 O=6
GlobalToken GL a3
P=4
O=1
V_C_H
Token
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token
R_T
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

55. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

56. A more complicated example
Demonstrating the backtracking mechanisms of AGP:
a10 is prohibited to be assigned any value different from Red
Execution continues as before, till a10 handles value changes from a1 and a9

57. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2
B_Token
Token a8
P=4 O=6
C={a9}
GlobalToken GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
State after token had been regenerated in a10‘s group
a10 tries to find consistent assignment:
Checks first against a9, as it is in the highest priority group
a9 eliminates all values
Execution continues as before till a10 receives the global token

58. Phase 5: Search for solution
B_GlobalToken
GlobalToken
GL a10
P=2
O=1 a1
P=2
O=2
R_C_S a8
P=4 O=6
C={a9} GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token
Token
B_TOKEN
R_T a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
a10 as a group leader:
Requests conflict sets from all agents in its group
a10 as an agent:
Returns its conflict set
Backtracks GlobalToken with C={a9}

59. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
V_C
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

60. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
Token
B_Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

61. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
V_C
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

62. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3
V_C a4
P=2
O=1
Token
Token
B_Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

63. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3
V_C a4
P=2
O=1
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5
V_C

64. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3
V_C a4
P=2
O=1
Token
V_C
Token
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

65. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
V_C
Token
V_C
Token a2
P=1
O=3
B_Token a6
P=5 O=2 a7
P=7 O=5

66. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
V_C
Token
Token
V_C
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

67. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
V_C
V_C
Token
V_C a2
P=1
O=3
Token a6
P=5 O=2 a7
P=7 O=5

68. Phase 5: Search for solution
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
GlobalToken
V_C
V_C GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token
V_C
V_C
Token
V_C a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

69. Phase 5: Search for solution
B_GlobalToken
GL a10
P=2
O=1 a1
P=2
O=2 a8
P=4 O=6
Token
R_C_S
C={a1}
GlobalToken
B_Token
V_C
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3
V_C a4
P=2
O=1
V_C
B_Token
R_T
V_C
Token
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5

70. Phase 5: Search for solution
V_C
GL a10
P=2
O=1 a1
P=2
O=2
Token a8
P=4 O=6
GlobalToken
Token
Token GL a3
P=4
O=1
GL a5
P=11 O=4 a9
P=8
O=3 a4
P=2
O=1
Token a2
P=1
O=3 a6
P=5 O=2 a7
P=7 O=5