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TEMPLATE: ADRIAN WILKEAnt Colony Optimization          University of Paderborn          Bio-Inspired Networking Seminar   ...
Ant Colony OptimizationTalk focus: Routing                                                            TEMPLATE: ADRIAN WIL...
History: Double bridge experiments[Goss et al., 1989]                                                                     ...
History: Double bridge experiments[Deneubourg et al., 1990]                                                               ...
From real                  behaviour to artificial systems                                                                 ...
Introductory examplePositive feedback                                                                                     ...
Ant System[Dorigo et al., 1996]                                                                                           ...
Ant System: Transition probabilityPath selection                                                                 TEMPLATE:...
Ant System: Transition probabilityChoice of the next node                                                                 ...
Ant System: Transition probabilityChoice of the next node                                                                 ...
Ant System: Transition probabilityChoice of the next node                                                                 ...
Ant System: Transition probabilityChoice of the next node                                                                 ...
Ant System: Transition probabilityChoice of the next node                                                                 ...
Ant System: Transition probabilityChoice of the next node                                                                 ...
Ant System: Transition probabilityChoice of the next node                                                                 ...
Ant System: Transition probabilityChoice of the next node                                                                 ...
Ant SystemUpdating pheromone                                                                                         TEMPL...
Ant SystemUpdating pheromone                                                                                              ...
Ant SystemUpdating pheromone                                                                                              ...
Ant SystemUpdating pheromone                                                                                              ...
Ant SystemAlgorithm: All together                                                             TEMPLATE: ADRIAN WILKE  Requ...
Ant SystemAlgorithm: All together                                                                             TEMPLATE: AD...
Ant SystemAlgorithm: All together                                                                                         ...
Ant SystemAlgorithm: All together                                                                                         ...
Ant SystemSome results                                                                                 TEMPLATE: ADRIAN WI...
From Ant                  System to AntNet                                                                                ...
AntNet: Data structures[Caro and Dorigo, 1998]                                                                            ...
AntNetAlgorithm in short                                                                                      TEMPLATE: AD...
From AntNet to Mobile                   ad hoc networks                                                                   ...
From AntNet to Mobile                   ad hoc networks                                                                   ...
AntHocNet[Di Caro et al., 2005]                                                                                           ...
AntHocNetReactive path setup                                                                                        TEMPLA...
AntHocNetPheromone updates                                                                                                ...
AntHocNetProactive probing & exploration                                                                           TEMPLAT...
AntHocNetProactive maintenance                                                                                            ...
AntHocNetProactive maintenance                                                                                            ...
From AntHocNet to HOPNET                                                                                                  ...
HOPNET: Routing tables[Wang et al., 2009]                                                                                 ...
HOPNET example                                                                                             TEMPLATE: ADRIA...
HOPNET example                                                                                             TEMPLATE: ADRIA...
HOPNET example                                                                                                    TEMPLATE...
HOPNET example                                                                                            TEMPLATE: ADRIAN...
HOPNET example                                                                                                 TEMPLATE: A...
HOPNET: Pheromone updates ISource update algorithm                                                                        ...
HOPNET: Pheromone updates IIUpdates by backward ants                                                                      ...
HOPNETResults: Overhead and Scalability                                                             TEMPLATE: ADRIAN WILKE...
Ant Colony OptimizationSummary                                                                        TEMPLATE: ADRIAN WIL...
Thank you for your attention                                                                                              ...
References I                                                                                                   TEMPLATE: A...
References II                                                                                                     TEMPLATE...
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Ant Colony Optimization: Routing

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  • Or, for the Traveling salesman/salesperson problem take a look at ACOTSPJava. http://adibaba.github.io/ACOTSPJava/
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  • You're welcome :) If your're looking for a related paper check "Peter Janacik, Dalimir Orfanus, and Adrian Wilke. A Survey of Ant Colony Optimization-Based Approaches to Routing in Computer Networks. In Fourth International Conference on Intelligent Systems, Modelling and Simulation (ISMS), pages 427-432, 2013." http://dx.doi.org/10.1109/ISMS.2013.20
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  • Thanks a lot Adrian.
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Transcript of "Ant Colony Optimization: Routing"

  1. 1. TEMPLATE: ADRIAN WILKEAnt Colony Optimization University of Paderborn Bio-Inspired Networking Seminar January 18, 2011 Adrian Wilke
  2. 2. Ant Colony OptimizationTalk focus: Routing TEMPLATE: ADRIAN WILKEAnt Colony Optimization - Adrian Wilke January 18, 2011 2
  3. 3. History: Double bridge experiments[Goss et al., 1989] TEMPLATE: ADRIAN WILKE r = ratio of long to short branchAnt Colony Optimization - Adrian Wilke January 18, 2011 3
  4. 4. History: Double bridge experiments[Deneubourg et al., 1990] TEMPLATE: ADRIAN WILKE Analogy Students on a campus: The more a point is passed, the more the vegetation is trampled.Ant Colony Optimization - Adrian Wilke January 18, 2011 4
  5. 5. From real behaviour to artificial systems TEMPLATE: ADRIAN WILKE • Stigmergy: Indirect • Possible fields? communication through Optimization problems, modifications in environment routing [Caro and Dorigo, 1998] • Routing: Direct data flow • Where to store pheromone from source to destination in distributed systems? (Shortest path) • Evaporation? (Pheromone • Congestion? decreases over time) (Too much load)Ant Colony Optimization - Adrian Wilke January 18, 2011 5
  6. 6. Introductory examplePositive feedback TEMPLATE: ADRIAN WILKE (a) (b) (c) (d) Exploring Pheromone Additional Shortest trail node pathAnt Colony Optimization - Adrian Wilke January 18, 2011 6
  7. 7. Ant System[Dorigo et al., 1996] TEMPLATE: ADRIAN WILKE “A new general-purpose heuristic algorithm [...] to solve different optimization problems” Covered topics • Path selection: Choice of the next node • Updating pheromone • Ant System algorithmAnt Colony Optimization - Adrian Wilke January 18, 2011 7
  8. 8. Ant System: Transition probabilityPath selection TEMPLATE: ADRIAN WILKE ??Ant Colony Optimization - Adrian Wilke January 18, 2011 8
  9. 9. Ant System: Transition probabilityChoice of the next node TEMPLATE: ADRIAN WILKE  α β [τij (t)] ·[νij ]   α β if j ∈ alloweda a pij (t) = [τiy (t)] ·[νiy ] (1)  y ∈alloweda 0 otherwise Ant Colony Optimization - Adrian Wilke January 18, 2011 9
  10. 10. Ant System: Transition probabilityChoice of the next node TEMPLATE: ADRIAN WILKE  α β [τij (t)] ·[νij ]   α β if j ∈ alloweda a pij (t) = [τiy (t)] ·[νiy ] (1)  y ∈alloweda 0 otherwise  τij (t) is the intensity of trail on edge (i, j) at time t.Ant Colony Optimization - Adrian Wilke January 18, 2011 9
  11. 11. Ant System: Transition probabilityChoice of the next node TEMPLATE: ADRIAN WILKE  α β [τij (t)] ·[νij ]   α β if j ∈ alloweda a pij (t) = [τiy (t)] ·[νiy ] (1)  y ∈alloweda 0 otherwise  τij (t) is the intensity of trail on edge (i, j) at time t. dij is the euclidean distance i − j between two nodes i and j. 1 νij = dij is called the visibility. Smaller distance → higher visibility.Ant Colony Optimization - Adrian Wilke January 18, 2011 9
  12. 12. Ant System: Transition probabilityChoice of the next node TEMPLATE: ADRIAN WILKE  α β [τij (t)] ·[νij ]   α β if j ∈ alloweda a pij (t) = [τiy (t)] ·[νiy ] (1)  y ∈alloweda 0 otherwise  τij (t) is the intensity of trail on edge (i, j) at time t. dij is the euclidean distance i − j between two nodes i and j. 1 νij = dij is called the visibility. Smaller distance → higher visibility. α is the weight of the trail (Simulation: α = 1). β is the weight of the visibility (Simulation: β = 5).Ant Colony Optimization - Adrian Wilke January 18, 2011 9
  13. 13. Ant System: Transition probabilityChoice of the next node TEMPLATE: ADRIAN WILKE  α β [τij (t)] ·[νij ]   α β if j ∈ alloweda a pij (t) = [τiy (t)] ·[νiy ] (1)  y ∈alloweda 0 otherwise  τij (t) is the intensity of trail on edge (i, j) at time t. dij is the euclidean distance i − j between two nodes i and j. 1 νij = dij is called the visibility. Smaller distance → higher visibility. α is the weight of the trail (Simulation: α = 1). β is the weight of the visibility (Simulation: β = 5). a Denominator of fraction ensures that pij (t) = 1. j∈NAnt Colony Optimization - Adrian Wilke January 18, 2011 9
  14. 14. Ant System: Transition probabilityChoice of the next node TEMPLATE: ADRIAN WILKE  α β [τij (t)] ·[νij ]   α β if j ∈ alloweda a pij (t) = [τiy (t)] ·[νiy ] (1)  y ∈alloweda 0 otherwise  τij (t) is the intensity of trail on edge (i, j) at time t. dij is the euclidean distance i − j between two nodes i and j. 1 νij = dij is called the visibility. Smaller distance → higher visibility. α is the weight of the trail (Simulation: α = 1). β is the weight of the visibility (Simulation: β = 5). a Denominator of fraction ensures that pij (t) = 1. j∈N tabua is the tabu list of nodes, which were alredy visited by the ath ant. alloweda N − tabua is the list of nodes, which were not visited by the ath ant.Ant Colony Optimization - Adrian Wilke January 18, 2011 9
  15. 15. Ant System: Transition probabilityChoice of the next node TEMPLATE: ADRIAN WILKE α β [τij (t)] ·[νij ] ... = α β α=1 [τiy (t)] ·[νiy ] y ∈alloweda β=5 Greedy: ??Ant Colony Optimization - Adrian Wilke January 18, 2011 10
  16. 16. Ant System: Transition probabilityChoice of the next node TEMPLATE: ADRIAN WILKE α β [τij (t)] ·[νij ] ... = α β α=1 [τiy (t)] ·[νiy ] y ∈alloweda β=5 Greedy: ?? (Image source: [Dorigo et al., 1996])Ant Colony Optimization - Adrian Wilke January 18, 2011 10
  17. 17. Ant SystemUpdating pheromone TEMPLATE: ADRIAN WILKE S S ? ? S D D DAnt Colony Optimization - Adrian Wilke January 18, 2011 11
  18. 18. Ant SystemUpdating pheromone TEMPLATE: ADRIAN WILKE Q a La if ath ant uses edge (i, j) in its tour ∆τij = (2) 0 otherwise Q is a constant, “its influence was found to be negligible.” La is the length of the tour of the ath ant.Ant Colony Optimization - Adrian Wilke January 18, 2011 12
  19. 19. Ant SystemUpdating pheromone TEMPLATE: ADRIAN WILKE Q a La if ath ant uses edge (i, j) in its tour ∆τij = (2) 0 otherwise Q is a constant, “its influence was found to be negligible.” La is the length of the tour of the ath ant. m a ∆τij = ∆τij (3) a=1Ant Colony Optimization - Adrian Wilke January 18, 2011 12
  20. 20. Ant SystemUpdating pheromone TEMPLATE: ADRIAN WILKE Q a La if ath ant uses edge (i, j) in its tour ∆τij = (2) 0 otherwise Q is a constant, “its influence was found to be negligible.” La is the length of the tour of the ath ant. m a ∆τij = ∆τij (3) a=1 τij (t + n) = ρ · τij (t) + ∆τij (4) ρ ρ < 1 is a constant , which represents the evaporation of trail.Ant Colony Optimization - Adrian Wilke January 18, 2011 12
  21. 21. Ant SystemAlgorithm: All together TEMPLATE: ADRIAN WILKE Require: NCMAX NC ← 0 and t ← 0 and s ← 1 ∆τij ← 0 τij (t) ← c shortestTour ← ∞ for all ants a = 1 to m do tabua (s) = choose start node end forAnt Colony Optimization - Adrian Wilke January 18, 2011 13
  22. 22. Ant SystemAlgorithm: All together TEMPLATE: ADRIAN WILKE Require: NCMAX NC ← 0 and t ← 0 and s ← 1 ∆τij ← 0 τij (t) ← c shortestTour ← ∞ for all ants a = 1 to m do tabua (s) = choose start node end for repeat { // Each cycle repeat s ←s +1 for all ants a = 1 to m do a move a to next node j by pij (t) insert node j in tabua (s) end for until s = nAnt Colony Optimization - Adrian Wilke January 18, 2011 13
  23. 23. Ant SystemAlgorithm: All together TEMPLATE: ADRIAN WILKE Require: NCMAX for all ants a = 1 to m do NC ← 0 and t ← 0 and s ← 1 compute length La of tabua ∆τij ← 0 shortestTour ← update(tabua , La ) τij (t) ← c end for shortestTour ← ∞ for all edges (i, j) and all ants a = 1 to m do for all ants a = 1 to m do compute ∆τij a tabua (s) = choose start node compute ∆τij end for end for repeat { // Each cycle for all edges (i, j) do compute τij (t + n) repeat ∆τij ← 0 s ←s +1 end for for all ants a = 1 to m do a move a to next node j by pij (t) insert node j in tabua (s) end for until s = nAnt Colony Optimization - Adrian Wilke January 18, 2011 13
  24. 24. Ant SystemAlgorithm: All together TEMPLATE: ADRIAN WILKE Require: NCMAX for all ants a = 1 to m do NC ← 0 and t ← 0 and s ← 1 compute length La of tabua ∆τij ← 0 shortestTour ← update(tabua , La ) τij (t) ← c end for shortestTour ← ∞ for all edges (i, j) and all ants a = 1 to m do for all ants a = 1 to m do compute ∆τij a tabua (s) = choose start node compute ∆τij end for end for repeat { // Each cycle for all edges (i, j) do compute τij (t + n) repeat ∆τij ← 0 s ←s +1 end for for all ants a = 1 to m do a move a to next node j by pij (t) NC ← NC + 1 and t ← t + n insert node j in tabua (s) move a to tabua (1) and clear tabua (2 . . . n) end for until s = n } until NC < NCMAX and no stagnation return shortestTourAnt Colony Optimization - Adrian Wilke January 18, 2011 13
  25. 25. Ant SystemSome results TEMPLATE: ADRIAN WILKE • Oliver30 problem (30 nodes TSP) • Columns: basic heuristic and improvements • Entries: Solutions • Ant System outperformed 2-opt But: Longer computational time • Best-known solution in less than 400 cycles, 100 cycles for values under 430 • As effective or better than some general-purpose heuristicsAnt Colony Optimization - Adrian Wilke January 18, 2011 14
  26. 26. From Ant System to AntNet TEMPLATE: ADRIAN WILKE Ant System AntNet • Focus: General-purpose • Focus: Internet Protocol heuristic with irregular topology • Global data structure • Routing tableAnt Colony Optimization - Adrian Wilke January 18, 2011 15
  27. 27. AntNet: Data structures[Caro and Dorigo, 1998] TEMPLATE: ADRIAN WILKE Routing table Traffic model Queues destinations destinations low priority mean neighbours variance - data packets time - forward ants high priority probabilistic traffic distribution desirability backward ants node identifiers s k x elapsed time 0 5 8 xAnt Colony Optimization - Adrian Wilke January 18, 2011 16
  28. 28. AntNetAlgorithm in short TEMPLATE: ADRIAN WILKE • At time intervals: Forward ant is launched to a destination Choice of destination: Data flow to possible targets • Next node on a tour is chosen by the routing table entries and queue lengths of neighbours • At nodes on path, forward ant memorizes the current node ID and the time from source node • At a destination the forward ant is destroyed and a backward ant is launched • Backward ants use the known paths back and update the routing table entries and the traffic model on path nodesAnt Colony Optimization - Adrian Wilke January 18, 2011 17
  29. 29. From AntNet to Mobile ad hoc networks TEMPLATE: ADRIAN WILKE AntNet • Introduced: Routing tables • Approach works for IP • What about dynamic networks with leaving nodesAnt Colony Optimization - Adrian Wilke January 18, 2011 18
  30. 30. From AntNet to Mobile ad hoc networks TEMPLATE: ADRIAN WILKE AntNet MANETs • Introduced: Routing tables • Mobile ad hoc networks • Approach works for IP • Restriction: Limited bandwidth • No central component, no fixed infractructure • Dynamic structure: Nodes are joining and leaving → Paths can become useless • What about dynamic networks with leaving nodesAnt Colony Optimization - Adrian Wilke January 18, 2011 18
  31. 31. AntHocNet[Di Caro et al., 2005] TEMPLATE: ADRIAN WILKE An approach for MANETs • Hybrid approach: • Reactive path setup phase • Proactive maintenance • Also of interest: • Pheromone updatesAnt Colony Optimization - Adrian Wilke January 18, 2011 19
  32. 32. AntHocNetReactive path setup TEMPLATE: ADRIAN WILKE • Node s wants to reach d • Same generation: only best • Unicast, if pheromone • First hop different: available. Else: broadcast Acceptance factor 0.9Ant Colony Optimization - Adrian Wilke January 18, 2011 20
  33. 33. AntHocNetPheromone updates TEMPLATE: ADRIAN WILKE Pheromone calculated by backward ants using a known Path P Mathematical Informal ˆi ˆi i Tmac = α · Tmac + (1 − α) · tmac Time to send a packet form a node i (Running average) ˆi i ˆi Ti+1 = (Qmac + 1) · Tmac Time for packets in queue (+1) ˆ n−1 ˆ i TP = i=1 Ti+1 Time for whole path ˆ T i +h·T i τd = d 2 hop Consider unloaded conditions i i Tnd = γ · Tnd + i (1 − γ) · τd Final routing table entry (Running average)Ant Colony Optimization - Adrian Wilke January 18, 2011 21
  34. 34. AntHocNetProactive probing & exploration TEMPLATE: ADRIAN WILKE • No broadcast: • Probe of path • Broadcast: • Improve paths • Path variationsAnt Colony Optimization - Adrian Wilke January 18, 2011 22
  35. 35. AntHocNetProactive maintenance TEMPLATE: ADRIAN WILKE HelloMessages Link failures • Every 1 sec: HelloMessage • Neighbors which lost best (Just senders address) path: Also broadcast • Received HelloMessage LinkFailureNotification → Add in routing table Failed transmission Link failures • Broadcast PathRepairAnt • 2 HelloMessages missing: (like reactive forward ant) • Remove from routing table • But: Max. 2 broadcasts • Broadcast LinkFailureNotification • LinkFailureNotification: List of destinations to which best path lost & new best delayAnt Colony Optimization - Adrian Wilke January 18, 2011 23
  36. 36. AntHocNetProactive maintenance TEMPLATE: ADRIAN WILKE HelloMessages Link failures • Every 1 sec: HelloMessage • Neighbors which lost best (Just senders address) path: Also broadcast • Received HelloMessage LinkFailureNotification → Add in routing table Failed transmission Link failures • Broadcast PathRepairAnt • 2 HelloMessages missing: (like reactive forward ant) • Remove from routing table • But: Max. 2 broadcasts • Broadcast LinkFailureNotification Task for you: • LinkFailureNotification: List • Count the broadcasts of destinations to which best on this page! path lost & new best delayAnt Colony Optimization - Adrian Wilke January 18, 2011 23
  37. 37. From AntHocNet to HOPNET TEMPLATE: ADRIAN WILKE AntHocNet HOPNET • Reactive path setup phase • Hybrid approach • Proactive maintenance • Avoid overhead and • Results: Better than AODVa achieve scalability except for overhead • Idea: Use Zones • Inside zones: Proactively maintained • Between zones: Reactive part, on demand a AODV: Ad-hoc on-demand distance vector routing, “reference algorithm in area”Ant Colony Optimization - Adrian Wilke January 18, 2011 24
  38. 38. HOPNET: Routing tables[Wang et al., 2009] TEMPLATE: ADRIAN WILKE Intrazone routing table Interzone routing table • Inside zones • Between zones • Proactively maintained • Reactive part, on demand • Size: Nodes in zone × • Used, if destination not in Node degree (neighbours) Intrazone routing table Pheromone Destination Visited times Sequence number Hops Between node and Path nodes in zone Expire Sequence number Incremented each time f./b.-ants are generatedAnt Colony Optimization - Adrian Wilke January 18, 2011 25
  39. 39. HOPNET example TEMPLATE: ADRIAN WILKE S wants to send data to T. (Reworked example from paper)Ant Colony Optimization - Adrian Wilke January 18, 2011 26
  40. 40. HOPNET example TEMPLATE: ADRIAN WILKE T not in zone. S sends external forward ant to pheripheral nodes.Ant Colony Optimization - Adrian Wilke January 18, 2011 27
  41. 41. HOPNET example TEMPLATE: ADRIAN WILKE D sends external forward ant to K and M, the other ants are destroyed.Ant Colony Optimization - Adrian Wilke January 18, 2011 28
  42. 42. HOPNET example TEMPLATE: ADRIAN WILKE M sends external forward ant to pheripheral nodes, except of S and A.Ant Colony Optimization - Adrian Wilke January 18, 2011 29
  43. 43. HOPNET example TEMPLATE: ADRIAN WILKE T is in intrazone routing tables. Two paths were found. A backward ant is launched.Ant Colony Optimization - Adrian Wilke January 18, 2011 30
  44. 44. HOPNET: Pheromone updates ISource update algorithm TEMPLATE: ADRIAN WILKE Internal forward ants update pheromone during movement vs Vs Vi Vj vi vl ϕ(vi , vs ) = ϕ(vi , vs ) + T (vs , vi ) + w (vi , vj ) constant Pheromone(neighbor , source)+ = time(vs , vi ) + timeOnLink ϕ(vl , vs ) = ϕ(vl , vs ) · (1 − E ), ∀ l = iAnt Colony Optimization - Adrian Wilke January 18, 2011 31
  45. 45. HOPNET: Pheromone updates IIUpdates by backward ants TEMPLATE: ADRIAN WILKE Backward ants update pheromone vD vb VS Vk Vb VD vl ϕ(vb , vD ) = ϕ(vb , vD ) + T (vS , vD ) − T (vS , vk ) ϕ(vl , vD ) = ϕ(vl , vD ) · (1 − E ), ∀ l = bAnt Colony Optimization - Adrian Wilke January 18, 2011 32
  46. 46. HOPNETResults: Overhead and Scalability TEMPLATE: ADRIAN WILKEAnt Colony Optimization - Adrian Wilke January 18, 2011 33
  47. 47. Ant Colony OptimizationSummary TEMPLATE: ADRIAN WILKEOverview • History: Double bridge experiments • Ant System: First approach • AntNet: Wired IP networks • AntHocNet: First MANET approach • HOPNET: MANET with zones To take away • Stigmergy: Indirect communication through modifications in environment • Routing table for pheromone • Proactive and reactive partsAnt Colony Optimization - Adrian Wilke January 18, 2011 34
  48. 48. Thank you for your attention TEMPLATE: ADRIAN WILKE 1 1 Source: http://xkcd.com/638/, Found: http://www.idsia.ch/˜gianni/Ant Colony Optimization - Adrian Wilke January 18, 2011 35
  49. 49. References I TEMPLATE: ADRIAN WILKE[Bonabeau et al., 1999] Bonabeau, E., Dorigo, M., and Theraulz, G. (1999). Swarm Intelligence. Oxford University Prress.[Caro and Dorigo, 1998] Caro, G. D. and Dorigo, M. (1998). AntNet: Distributed Stigmergetic Control for Communications Networks. Journal of Artificial Intelligence Research, 9:317–365.[Deneubourg et al., 1990] Deneubourg, J.-L., Aron, S., Goss, S., and Pasteels, J. M. (1990). The Self-Organizing Exploratory Pattern of the Argentine Ant. Journal of lnsect Behavior, 3(2).[Di Caro et al., 2005] Di Caro, G., Ducatelle, F., and Gambardella, L. M. (2005). AntHocNet: an adaptive nature-inspired algorithm for routing in mobile ad hoc networks. European Transactions on Telecommunications, 16(5):443–455.[Dorigo et al., 1996] Dorigo, M., Maniezzo, V., and Colorni, A. (1996). Ant system: optimization by a colony of cooperating agents. Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Transactions on, 26(1):29–41.[Dorigo and Stutzle, 2004] Dorigo, M. and Stutzle, T. (2004). ¨ ¨ Ant Colony Optimization. MIT press.Ant Colony Optimization - Adrian Wilke January 18, 2011 36
  50. 50. References II TEMPLATE: ADRIAN WILKE[Dressler and Akan, 2010] Dressler, F. and Akan, O. B. (2010). A survey on bio-inspired networking. Computer Networks, 54(6):881–900.[Goss et al., 1989] Goss, S., Aron, S., Deneubourg, J. L., and Pasteels, J. M. (1989). Self-organized Shortcuts in the Argentine Ant. Naturwissenschaften, 76:579–581.[Wang et al., 2009] Wang, J., Osagie, E., Thulasiraman, P., and Thulasiram, R. K. (2009). HOPNET: A hybrid ant colony optimization routing algorithm for mobile ad hoc network. Ad Hoc Networks, 7(4):690–705.Ant Colony Optimization - Adrian Wilke January 18, 2011 37
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