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Asynchronous Partitioning Framework

Asynchronous Partitioning Framework. Vitaliy Freidovich. Department of Mathematics and Computer Science Vice President R&D of www.TheCTO.co.il/en Vitaliy@TheCTO.co.il. Ben-Gurion University Of the Negev. The Open University of Israel. Amnon Meisels.

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Asynchronous Partitioning Framework

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  1. Asynchronous Partitioning Framework Vitaliy Freidovich Department of Mathematics and Computer Science Vice President R&D of www.TheCTO.co.il/en Vitaliy@TheCTO.co.il Ben-Gurion University Of the Negev The Open University of Israel Amnon Meisels Department of Computer Science www.cs.bgu.ac.il/~am am@cs.bgu.ac.il

  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: • Synchronous - exactly one token • 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 a10 a1 a8 a3 a4 a5 a9 a2 a6 a7

  9. Phase 1: Initialization • Main goal: Initialize agent priority • Pai = ki + ∑j=1ki (CommonNeighbors(ai, aj)) • ki – number of neighbors of ai • Initially: Pai = ki a10 P=2 a1 P=2 a8 P=2 a3 P=4 a4 P=2 a5 P=5 a9 P=4 a2 P=1 a6 P=3 a7 P=3

  10. 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 a10 P=2 a1 P=2 a8 P=2 a4 a6 a7 a8,a9 1 a3 P=4 a4 P=2 a5 P=5 a9 P=4 a4 a6 a7 a8,a9 a4 a6 a7 a8,a9 0 2 a4 a6 a7 a8,a9 a4 a6 a7 a8,a9 1 a2 P=1 a6 P=3 a7 P=3 2

  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 a10 P=2 a1 P=2 a8 P=4 a3 P=4 a4 P=2 a5 P=11 a9 P=8 a2 P=1 a6 P=5 a7 P=7

  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 a10 P=2 a1 P=2 a8 P=4 JOIN JOIN a4 P=2 a5 P=11 a9 P=8 a3 P=4 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 a10 P=2 a1 P=2 a8 P=4 a4 P=2 a9 P=8 a3 P=4 W:{a1} a5 P=11 W:{a9,a6} W:{a10} a2 P=1 a6 P=5 a7 P=7 W:{a3}

  15. Phase 3: Group partitioning a10 P=2 a1 P=2 a8 P=4 a4 P=2 a9 P=8 a3 P=4 W:{a1} GL a5 P=11 NOT_JOINED JOINED W:{a10} JOINED a2 P=1 a6 P=5 NOT_JOINED a7 P=7 W:{a3}

  16. Phase 3: Group partitioning a10 P=2 a1 P=2 a8 P=4 JOINED a4 P=2 a9 P=8 GL a5 P=11 GL a3 P=4 JOINED JOIN 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 a4 P=2 a9 P=8 GL a5 P=11 GL a3 P=4 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 partitioning GL a10 P=2 a1 P=2 a8 P=4 a4 P=2 a9 P=8 GL a5 P=11 GL a3 P=4 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 GLa10 P=2 a1 P=2 a8 P=4 IGP=4+2 IGP=17 IGP=6 IGP=11 GLa5 P=11 a9 P=8 GL a3 P=4 a4 P=2 IGP=4+11+11 IGP=11 IGP=15 IGP=26 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 GLa10 a1 IGP=6 O=2 IGP=17 O=1 a8 IGP=11 O=6 GLa5 a9 GL a3 IGP=26 O=1 a4 IGP=15 O=1 IGP=11 O=4 IGP=13 O=3 a2 IGP=4 O=3 a7 a6 IGP=15 O=2 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 GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 a2 P=1 O=3 a6 P=5O=2 a7 P=7O=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 GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 a2 P=1 O=3 a6 P=5O=2 a7 P=7O=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 • Group Leader a5 starts execution • First one - Generates a new GlobalToken • Generates a new Token • Sends Token to the first agent in its group GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5

  39. Phase 5: Search for solution GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token V_C a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5 • a4: • Finds consistent assignment • Notifies a3 • Returns Token to a5

  40. Phase 5: Search for solution GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token V_C Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=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 GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token V_C Token V_C_H a2 P=1 O=3 a6 P=5O=2 a7 P=7O=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 GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token R_T Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=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 GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5 • a3 – group leader: • Sends Token to a3 • a3 – agent: • Finds consistent assignment

  44. Phase 5: Search for solution GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5 • a3 – group leader: • Sends Token to a3 • a3 – agent: • Finds consistent assignment • Returns Token to group leader

  45. Phase 5: Search for solution GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5

  46. Phase 5: Search for solution V_C GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken Token GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5

  47. Phase 5: Search for solution V_C GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 Token Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5

  48. Phase 5: Search for solution V_C GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 a4 P=2 O=1 V_C Token Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5 V_C

  49. Phase 5: Search for solution V_C GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 Token V_C a4 P=2 O=1 Token V_C a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5

  50. Phase 5: Search for solution V_C GLa10 P=2 O=1 a1 P=2 O=2 a8 P=4O=6 Token GlobalToken GLa5 P=11O=4 a9 P=8 O=3 GL a3 P=4 O=1 V_C a4 P=2 O=1 Token V_C Token a2 P=1 O=3 a6 P=5O=2 a7 P=7O=5

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