Skip to main content
SpringerLink
Log in

A learning automata-based heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

During the last decades, a host of efficient algorithms have been developed for solving the minimum spanning tree problem in deterministic graphs, where the weight associated with the graph edges is assumed to be fixed. Though it is clear that the edge weight varies with time in realistic applications and such an assumption is wrong, finding the minimum spanning tree of a stochastic graph has not received the attention it merits. This is due to the fact that the minimum spanning tree problem becomes incredibly hard to solve when the edge weight is assumed to be a random variable. This becomes more difficult if we assume that the probability distribution function of the edge weight is unknown. In this paper, we propose a learning automata-based heuristic algorithm to solve the minimum spanning tree problem in stochastic graphs wherein the probability distribution function of the edge weight is unknown. The proposed algorithm taking advantage of learning automata determines the edges that must be sampled at each stage. As the presented algorithm proceeds, the sampling process is concentrated on the edges that constitute the spanning tree with the minimum expected weight. The proposed learning automata-based sampling method decreases the number of samples that need to be taken from the graph by reducing the rate of unnecessary samples. Experimental results show the superiority of the proposed algorithm over the well-known existing methods both in terms of the number of samples and the running time of algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chiang TC, Liu CH, Huang YM (2007) A near-optimal multicast scheme for mobile ad hoc networks using a hybrid genetic algorithm. Expert Syst Appl 33:734–742

    Article  Google Scholar 

  2. Rodolakis G, Laouiti A, Jacquet P, Naimi AM (2008) Multicast overlay spanning trees in ad hoc networks: capacity bounds protocol design and performance evaluation. Comput Commun 31:1400–1412

    Article  Google Scholar 

  3. Boruvka O (1926) Ojistém problému minimálním (About a certain minimal problem). Praca Moravske Prirodovedecke Spolecnosti 3:37–58

    Google Scholar 

  4. Kruskal JB (1956) On the shortest spanning sub tree of a Graph and the traveling salesman problem. In: Proceedings of the American mathematical society 7(1):748–750

    Article  MathSciNet  Google Scholar 

  5. Prim RC (1957) Shortest connection networks and some generalizations. Bell Syst Tech J 36:1389–1401

    Google Scholar 

  6. Ishii H, Shiode S, Nishida T, Namasuya Y (1981) Stochastic spanning tree problem. Discrete Appl Math 3:263–273

    Article  MATH  MathSciNet  Google Scholar 

  7. Ishii H, Nishida T (1983) Stochastic bottleneck spanning tree problem. Networks 13:443–449

    Article  MATH  MathSciNet  Google Scholar 

  8. Mohd IB (1994) Interval elimination method for stochastic spanning tree problem. Appl Math Comput 66:325–341

    Article  MATH  MathSciNet  Google Scholar 

  9. Ishii H, Matsutomi T (1995) Confidence regional method of stochastic spanning tree problem. Math Comput Model 22(19–12):77–82

    Article  MATH  MathSciNet  Google Scholar 

  10. Alexopoulos C, Jacobson JA (2000) State space partition algorithms for stochastic systems with applications to minimum spanning trees. Networks 35(2):118–138

    Article  MATH  MathSciNet  Google Scholar 

  11. Katagiri H, Mermri EB, Sakawa M, Kato K (2004) A study on fuzzy random minimum spanning tree problems through possibilistic programming and the expectation optimization model. In: Proceedings of the 47th IEEE international midwest symposium on circuits and systems

  12. Almeida TA, Yamakami A, Takahashi MT (2005) An evolutionary approach to solve minimum spanning tree problem with fuzzy parameters. In: Proceedings of the international conference on computational intelligence for modelling, control and automation

  13. Hutson KR, Shier DR (2006) Minimum spanning trees in networks with varying edge weights. Ann Oper Res 146:3–18

    Article  MATH  MathSciNet  Google Scholar 

  14. Fangguo H, Huan Q (2008) A model and algorithm for minimum spanning tree problems in uncertain networks. In: Proceedings of the 3rd international conference on innovative computing information and control (ICICIC’08)

  15. Dhamdhere K, Ravi R, Singh M (2005) On two-stage stochastic minimum spanning trees. Springer, Berlin, pp 321–334

    Google Scholar 

  16. Swamy C, Shmoys DB (2006) Algorithms column: approximation algorithms for 2-stage stochastic optimization problems. ACM SIGACT News 37(1):1–16

    Article  Google Scholar 

  17. Gallager RG, Humblet PA, Spira PM (1983) A distributed algorithm for minimum weight spanning trees. ACM Trans Program Lang Syst 5:66–77

    Article  MATH  Google Scholar 

  18. Spira P (1977) Communication complexity of distributed minimum spanning tree algorithms. In: Proceedings of the second Berkeley conference on distributed data management and computer networks

  19. Dalal Y (April 1977) Broadcast protocols in packet switched computer networks. Technical Report 128. Department of Electrical Engineering, Stanford University, Stanford

    Google Scholar 

  20. Gafni E (1985) Improvements in the time complexity of two message-optimal election algorithms. In: Proceedings of the 4th symposium on principles of distributed computing (PODC), pp 175–185

    Chapter  Google Scholar 

  21. Awerbuch B (1987) Optimal distributed algorithms for minimum weight spanning tree, counting, leader election, and related problems. In: Proceedings of the 19th ACM symposium on theory of computing (STOC), pp 230–240

    Google Scholar 

  22. Garay J, Kutten S, Peleg D (1998) A sublinear time distributed algorithm for minimum-weight spanning trees. SIAM J Comput 27:302–316

    Article  MATH  MathSciNet  Google Scholar 

  23. Kutten S, Peleg D (1998) Fast distributed construction of k-dominating sets and applications. J Algorithms 28:40–66

    Article  MATH  MathSciNet  Google Scholar 

  24. Elkin M (2004) A faster distributed protocol for constructing minimum spanning tree. In: Proceedings of the ACM-SIAM symposium on discrete algorithms (SODA), pp. 352–361

    Google Scholar 

  25. Peleg D, Rabinovich V (1999) A near-tight lower bound on the time complexity of distributed MST construction. In: Proceedings of the 40th IEEE symposium on foundations of computer science (FOCS), pp 253–261

    Google Scholar 

  26. Elkin M (2004) Unconditional lower bounds on the time-approximation tradeoffs for the distributed minimum spanning tree problem. In: Proceedings of the ACM symposium on theory of computing (STOC), pp 331–340

    Google Scholar 

  27. Elkin M (2004) An overview of distributed approximation. ACM SIGACT News 35(4):40–57

    Article  Google Scholar 

  28. Khan M, Pandurangan G (2006) A fast distributed approximation algorithm for minimum spanning trees. In: Proceedings of the 20th international symposium on distributed computing (DISC)

    Google Scholar 

  29. Aggarwal V, Aneja Y, Nair K (1982) Minimal spanning tree subject to a side constraint. Comput Oper Res 9:287–296

    Article  Google Scholar 

  30. Gruber M, Hemert J, Raidl GR (2006) Neighborhood searches for the bounded diameter minimum spanning tree problem embedded in a VNS, EA and ACO. In: Proceedings of genetic and evolutionary computational conference (GECCO’2006)

    Google Scholar 

  31. Bui TN, Zrncic CM (2006) An ant-based algorithm for finding degree-constrained minimum spanning tree. In: Proceedings of the 8th annual conference on Genetic and evolutionary computation, pp 11–18

    Chapter  Google Scholar 

  32. Oncan T (2007) Design of capacitated minimum spanning tree with uncertain cost and demand parameters. Inf Sci 177:4354–4367

    Article  MathSciNet  Google Scholar 

  33. Oencan T, Cordeau JF, Laporte G (2008) A tabu search heuristic for the generalized minimum spanning tree problem. Eur J Oper Res 191(2):306–319

    Article  MATH  Google Scholar 

  34. Parsa M, Zhu Q, Garcia-Luna-Aceves JJ (1998) An iterative algorithm for delay-constrained minimum-cost multicasting. IEEE/ACM Trans Netw 6(4):461–474

    Article  Google Scholar 

  35. Salama HF, Reeves DS, Viniotis Y (1997) The Delay-constrained minimum spanning tree problem. In: Proceedings of the second IEEE symposium on computers and communications, pp 699–703

    Chapter  Google Scholar 

  36. Gouveia L, Simonetti L, Uchoa E (2009) Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs. J Math Program (in press)

  37. Sharaiha YM, Gendreau M, Laporte G, Osman IH (1998) A tabu search algorithm for the capacitated shortest spanning tree problem. Networks 29(3):161–171

    Article  MathSciNet  Google Scholar 

  38. Hanr L, Wang Y (2006) A novel genetic algorithm for degree-constrained minimum spanning tree problem. Int J Comput Sci Netw Secur 6(7A):50–57

    Google Scholar 

  39. Krishnamoorthy M, Ernst A (2001) Comparison of algorithms for the degree constrained minimum spanning tree. J Heurist 7:587–611

    Article  MATH  Google Scholar 

  40. Doulliez P, Jamoulle E (1972) Transportation networks with random arc capacities. RAIRO Oper Res 3:45–60

    MathSciNet  Google Scholar 

  41. Hutson KR, Shier DR (2005) Bounding distributions for the weight of a minimum spanning tree in stochastic networks. Oper Res 53(5):879–886

    Article  MATH  MathSciNet  Google Scholar 

  42. Thathachar MAL, Harita BR (1987) Learning automata with changing number of actions. IEEE Trans Syst Man Cybern SMG17:1095–1100

    Google Scholar 

  43. Narendra KS, Thathachar KS (1989) Learning automata: an introduction. Printice-Hall, New York

    Google Scholar 

  44. Lakshmivarahan S, Thathachar MAL (1976) Bounds on the convergence probabilities of learning automata. IEEE Trans Syst Man Cybern SMC-6:756–763

    MathSciNet  Google Scholar 

  45. Gower JC, Ross GJS (1969) Minimum spanning trees and single linkage cluster analysis. J R Stat Soc 18(1): 54–64

    MathSciNet  Google Scholar 

  46. Barzily Z, Volkovich Z, Akteke-Öztürk B, Weber GW (2009) On a minimal spanning tree approach in the cluster validation problem. Informatica 20(2):187–202

    MATH  MathSciNet  Google Scholar 

  47. Marchand-Maillet S, Sharaiha YM (1996) A minimum spanning tree approach to line image analysis. In: Proceedings of 13th international conference on pattern recognition (ICPR’96), p 225

    Chapter  Google Scholar 

  48. Li J, Yang S, Wang X, Xue X, Li B (2009) Tree-structured data regeneration with network coding in distributed storage systems. In: Proceedings of international conference on image processing, Charleston, USA, pp 481–484

    Google Scholar 

  49. Kang ANC, T Lee RC, Chang CL, Chang SK (1977) Storage reduction through minimal spanning trees and spanning forests. IEEE Trans Comput C-26:425–434

    Article  Google Scholar 

  50. Osteen RE, Lin PP (1974) Picture skeletons based on eccentricities of points of minimum spanning trees. SIAM J Comput 3:23–40

    Article  MATH  MathSciNet  Google Scholar 

  51. Graham RL, Hell P (1985) On the history of the minimum spanning tree problem. IEEE Ann Hist Comput 7(1):43–57

    Article  MATH  MathSciNet  Google Scholar 

  52. Jain A, Mamer JW (1988) Approximations for the random minimal spanning tree with applications to network provisioning. Oper Res 36:575–584

    Article  MATH  MathSciNet  Google Scholar 

  53. Torkestani Akbari J, Meybodi MR (2010) Learning automata-based algorithms for finding minimum weakly connected dominating set in stochastic graphs. Int J Uncertain Fuzziness Knowl-Based Syst (to appear)

  54. Torkestani Akbari J, Meybodi MR (2010) Mobility-based multicast routing algorithm in wireless mobile ad hoc networks: a learning automata approach. J Comput Commun 33:721–735

    Article  Google Scholar 

  55. Torkestani Akbari J, Meybodi MR (2010) A new vertex coloring algorithm based on variable action-set learning automata. J Comput Inf 29(3):1001–1020

    Google Scholar 

  56. Torkestani Akbari J, Meybodi MR (Feb. 2010) Weighted steiner connected dominating set and its application to multicast routing in wireless MANETs, Wireless personal communications, Springer, Berlin

    Google Scholar 

  57. Torkestani Akbari J, Meybodi MR (2010) An efficient cluster-based cdma/tdma scheme for wireless mobile ad-hoc networks: a learning automata approach. J Netw Comput Appl 33:477–490

    Article  Google Scholar 

  58. Torkestani Akbari J, Meybodi MR (2010) Clustering the wireless ad-hoc networks: a distributed learning automata approach. J Parallel Distrib Comput 70:394–405

    Article  Google Scholar 

  59. Torkestani Akbari J, Meybodi MR (2010) An intelligent back bone formation algorithm in wireless ad hoc networks based on distributed learning automata. J Comput Netw 54:826–843

    Article  Google Scholar 

  60. Billard EA, Lakshmivarahan S (1999) Learning in multi-level games with incomplete information. Part I. IEEE Trans Syst Man Cybern-Part B: Cybern 19:329–339

    Article  Google Scholar 

  61. Meybodi MR (1983) Learning automata and its application to priority assignment in a queuing system with unknown characteristics, Ph.D. thesis, Department of Electrical Engineering and Computer Science, University of Oklahoma, Norman, Oklahoma, USA

  62. Hashim AA, Amir S, Mars P (1986) Application of learning automata to data compression In: Narendra KS (ed.) Adaptive and learning systems. Plenum, New York, pp 229–234

    Google Scholar 

  63. Oommen BJ, Hansen ER (Aug. 1987) List organizing strategies using stochastic move-to-front and stochastic move-to-rear operations. SIAM J Comput 16:705–716

    Article  MATH  MathSciNet  Google Scholar 

  64. Unsal C, Kachroo P, Bay JS (1999) Multiple stochastic learning automata for vehicle path control in an automated highway system. IEEE Trans Syst Man Cybern-Part A 29:120–128

    Article  Google Scholar 

  65. Barto AG, Anandan P (1985) Pattern-recognizing stochastic learning automata. IEEE Trans Syst Man Cybern SMC-15:360–375

    MathSciNet  Google Scholar 

  66. Akbari Torkestani J, Meybodi MR (2010) A learning automata-based cognitive radio for clustered wireless ad-hoc networks. J. Netw. Syst. Manag. (to appear)

Download references

Author information

Authors and Affiliations

  1. Department of Computer Engineering, Islamic Azad University, Arak Branch, Arak, Iran

    Javad Akbari Torkestani

  2. Department of Computer Engineering and IT, Amirkabir University of Technology, Tehran, Iran

    Mohammad Reza Meybodi

  3. Institute for Studies in Theoretical Physics and Mathematics (IPM), School of Computer Science, Tehran, Iran

    Mohammad Reza Meybodi

Authors
  1. Javad Akbari Torkestani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohammad Reza Meybodi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Javad Akbari Torkestani.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Akbari Torkestani, J., Meybodi, M.R. A learning automata-based heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs. J Supercomput 59, 1035–1054 (2012). https://doi.org/10.1007/s11227-010-0484-1

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-010-0484-1

Keywords

Search

Navigation