1 / 54

Ant Based Optimization for Multiway Graph Partition

Ant Based Optimization for Multiway Graph Partition. Ezra Nugroho February 28 th , 2005. Outline. Problem Definition Ant Based Optimization (ABO) and Ant Colony Optimization (ACO) ABO for Multiway Graph Partitioning (ABMGP) Conclusion. Outline. Problem Definition

saad
Download Presentation

Ant Based Optimization for Multiway Graph Partition

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Ant Based Optimization for Multiway Graph Partition Ezra Nugroho February 28th, 2005

  2. Outline • Problem Definition • Ant Based Optimization (ABO) and Ant Colony Optimization (ACO) • ABO for Multiway Graph Partitioning (ABMGP) • Conclusion Ezra Nugroho

  3. Outline • Problem Definition • Ant Based Optimization (ABO) and Ant Colony Optimization (ACO) • ABO for Multiway Graph Partitioning (ABMGP) • Conclusion Ezra Nugroho

  4. Problem: Multiway Graph Partition • Input: • G = (V, E): an undirected graph • k: an integer from 2 to |V | • Output: • S1, …, Sk:k disjointed subsets of V such that: • Si∩ Sj = ø for i ≠ j • USi = V • Si≠ø for i = 1, …, k • |Sl| - |Ss| ≤ 1, Sl the largest subset, Ss the smallest one • Goal: • The crossing edges between subsets are minimized Ezra Nugroho

  5. Problem: Further Definitions • Other name: Graph k-way Partition Problem • (α, k)-Partition Problem, also called Ratio Cut Problem • Allows unbalanced partition • |Sl| ≤ α |V |, Sl the largest subgraph • 2-way Partition Problem is often called the Bisection Problem • The size of the set of crossing edges is called the cutsize Ezra Nugroho

  6. 2-way Partition Ezra Nugroho

  7. 2-way Partition Cutsize = 6 Ezra Nugroho

  8. 2-way Partition Cutsize = 6 Ezra Nugroho

  9. 2-way Partition Ezra Nugroho

  10. 2-way Partition Cutsize = 2 Ezra Nugroho

  11. Problem: Complexity • Multiway Graph Partition Problem is NP-hard • Approximating good cut size for general graph is NP-hard • The bisection problem is NP-hard, even for bipartite graphs • Exact polynomial algorithm exists for trees and planar graphs with O(log n) optimal cutsize Ezra Nugroho

  12. Problem: Complexity …(cont) • Let |V | = n, the size of search space for the k-way partition problem is • Grows very fast as n increases Ezra Nugroho

  13. Problem: Applications • Circuit Design • Scheduling for parallel processing • Initial Process for Divide and Conquer algorithms • General Layout: Network design, Routing, etc. Ezra Nugroho

  14. Problem: Existing Algorithms • Kernighan–Lin algorithm for bisection • Recursive Kernighan–Lin • Pairwise Kernighan–Lin • Cyclic k-way Partitioning Algorithm • Other Heuristics and Hybrid Algorithms • Genetic Algorithms • Simulated Annealing • Tabu Search • Hybrid Ant Based Algorithm Ezra Nugroho

  15. Outline • Problem Definition • Ant Based Optimization (ABO) and Ant Colony Optimization (ACO) • ABO for Multiway Graph Partitioning (ABMGP) • Conclusion Ezra Nugroho

  16. ABO/ACO: Similarities • Agent-based Optimization • Inspired by ant colonies • Inspired by other animals → ‘animats’ • Animats use pheromones to send signals to others • Good in identifying possible good solution • Cannot fine-tune solutions Ezra Nugroho

  17. Ant Based Optimization (ABO) • Multi Agents, one solution • Animats move in parallel • Problem domain: clustering, landscape identification Ezra Nugroho

  18. Ant Colony Optimization (ACO) • One agent, one solution • Usually Animats move serially • Problem domain: path problems Ezra Nugroho

  19. ABO/ACO : Applications • ABO: • k-Cardinality Tree problem • Clique problem • Bisection problem • Etc. • ACO: • TSP • Quadratic Assignment • Routing • Etc. Ezra Nugroho

  20. Outline • Problem Definition • Ant Based Optimization (ABO) and Ant Colony Optimization (ACO) • ABO for Multiway Graph Partitioning (ABMGP) • Conclusion Ezra Nugroho

  21. ABMGP: Algorithm Overview • Phase 1 • 20 rounds of finding good candidate solutions and locally optimize them • Phase 2 • Fully optimize the best candidate solution Ezra Nugroho

  22. ABMGP: Algorithm Detail • Phase 1 • 20 rounds of finding good candidate solutions and locally optimize them • Phase 2 • Fully optimize the best candidate solution Ezra Nugroho

  23. ABMGP: Algorithm Detail • Phase 1: • Create initial k-way partition • for round 1 to 20 • for iteration 1 to max • Place animats in vertices • if (progressing) • Run animats • else • Perform random jumps (jolt) • end if-else • Rebalance with certain probability • Record good partition • end • Push the best solution from this round closer to an optimum • end • Phase 2: • Fully optimize the absolute best solution Ezra Nugroho

  24. ABMGP: Algorithm Detail • Create initial k-way partition • for round 1 to 20 • for iteration 1 to max • Place animats in vertices • if (progressing) • Run animats • else • Perform random jumps (jolt) • end if-else • Rebalance with certain probability • Record good partition • end • Push the best solution from this round closer to an optimum • end • Fully optimize the absolute best solution Ezra Nugroho

  25. ABMGP: Animat Activation • Ideas: • We use k colonies of territorial animats, each territory is a subgraph • Animats try to enlarge their territory by attacking other animats in different vertices • Animats use pheromones to communicate with each other to encourage collaborative decisions: defending vertices, attacking vertices • An Animat captures a vertex v if it defeats the last animat in v Ezra Nugroho

  26. ABMGP: Animat Activation No Way! We want that vertex! Ezra Nugroho

  27. ABMGP: Animat Activation Ezra Nugroho

  28. ABMGP: Animat Activation Attacker wins, defender is removed Ezra Nugroho

  29. ABMGP: Animat Activation Ezra Nugroho

  30. ABMGP: Animat Activation Attacker looses, it waits for next iteration Ezra Nugroho

  31. ABMGP: Animat Activation Ezra Nugroho

  32. ABMGP: Animat Activation Attacker wins, defender is removed Ezra Nugroho

  33. ABMGP: Animat Activation Attacker wins, defender is removed Ezra Nugroho

  34. ABMGP: Animat Activation vertex is captured Ezra Nugroho

  35. ABMGP: Animat Activation The colonies are updated Ezra Nugroho

  36. ABMGP: Algorithm Detail • Create initial k-way partition • for round 1 to 20 • for iteration 1 to max • Place animats in vertices • if (progressing) • Run animats • else • Perform random jumps (jolt) • end if-else • Rebalance with certain probability • Record good partition • end • Push the best solution from this round closer to an optimum • end • Fully optimize the absolute best solution Ezra Nugroho

  37. ABMGP: Jolts • Pairs of vertices from different colonies are randomly chosen • Swap the owner • Adjusts the pheromone levels Ezra Nugroho

  38. ABMGP: Algorithm Detail • Create initial k-way partition • for round 1 to 20 • for iteration 1 to max • Place animats in vertices • if (progressing) • Run animats • else • Perform random jumps (jolt) • end if-else • Rebalance with certain probability • Record good partition • end • Push the best solution from this round closer to an optimum • end • Fully optimize the absolute best solution Ezra Nugroho

  39. ABMGP: Algorithm Detail • Kernighan–Lin algorithm for Bisection • Starting with a bisection (A, B) • Cut size is reduced by swapping pairs of vertices • Identify pairs of vertices (a, b) that reduce the cut size the most • Vertices in these pairs are swapped • Repeat Ezra Nugroho

  40. 2-way Partition Cutsize = 6 Ezra Nugroho

  41. 2-way Partition A’ A (A, A’) gain = 0 Cutsize = 6 Ezra Nugroho

  42. 2-way Partition B A’ A (A, A’) gain = 0 (B, B’) gain = 4 B’ Cutsize = 6 Ezra Nugroho

  43. 2-way Partition B A A’ B’ Cutsize = 2 Ezra Nugroho

  44. ABMG: Algorithm Detail • Weaker derivation of Pairwise Kernighan–Lin Algorithm • Cut size between subgraph pairs are computed • Pairs are sorted according to their cut sizes • Run KL for bisection algorithm for each pair • Some vertex-swaps are considered Ezra Nugroho

  45. ABMGP: Algorithm Detail • Phase 1 • 20 rounds of finding good candidate solutions and locally optimize them • Phase 2 • Fully optimize the best candidate solution Ezra Nugroho

  46. ABMG: Algorithm Detail • Full Pairwise Kernighan–Lin Algorithm • Cut size between subgraph pairs are computed • Pairs are sorted according to their cut sizes • Run KL for bisection algorithm for each pair • All possible vertex-swaps are considered Ezra Nugroho

  47. ABMGP: Test Bed • 40 Graphs from 8 different classes • Size from 100 – 5252 • Average degree from 2 to 36 • Random and geometric Ezra Nugroho

  48. ABMGP: Implementation • ABMGP is implemented in C++ • 3.06 Ghz Xeon Processor • 2 G of RAM • Linux • 100 runs for each graph Ezra Nugroho

  49. ABMGP: Result for bisection Ezra Nugroho

  50. ABMGP: Result 4-way Partition Ezra Nugroho

More Related