Self-stabilizing Grundy Coloring Using Game Theory and Heuristics Ordering

Document Type : Original Article

Authors

Abstract

Transient faults in distributed systems can be occurred in many situations like process failure and security
attacks. A self-stabilizing algorithm, regardless of the initial state, converges in finite time to legitimate
states and tolerates transient faults. In this paper, we propose a self-stabilizing Grundy coloring using some
concepts/results in the game theory. The proposed algorithm deals with autonomous networks, where nodes
do not have identifiers. By using Nash equilibrium, we prove our proposed algorithm converges in O(m)
moves, where m is the number of network edges. Simulation results indicate that heuristics ordering leads
to decrease the number of colors up to 18% and increase the Convergence up to 5%.

Keywords


[1]
N. A. Lynch, “Distributed Algorithms, Morgan Kaufmann,” 1996.##
[2]
L. Wang and R. Ranjan, “Processing distributed internet of things data in clouds,” IEEE Cloud Computing, vol. 2, pp.     76–80, 2015.##
[3]
D. Peleg, “Distributed Computing: A Locality-Sensitive Approach,” SIAM, 2000.##
[4]
N. Guellati and H. Kheddouci, “A survey on self-stabilizing algorithms for independence, domination, coloring, and matching in graphs,” Journal of Parallel and Distributed Computing, vol. 70, pp. 406–415, 2010.##
[5]
L. H. Yen, J. Y. Huang, and V. Turau, “Designing self-stabilizing systems using game theory,” ACM Trans. Auton. Adapt. Syst., vol. 11, pp. 1–27, 2016.##
[6]
L. H. Yen and Z. L. Chen, “Game-theoretic approach to self-stabilizing distributed formation of minimal multi-dominating sets,” IEEE Trans. Parallel Distrib. Syst., vol. 25, pp.         3201–3210, 2014.##
[7]
E. W. Dijkstra, “Self-stabilizing systems in spite of distributed control,” Communications of the ACM, vol. 17, pp. 643–644, 1974.##
[8]
S. Devismes and C. Johnen, “Silent self-stabilizing BFS tree algorithms revisited,” Journal of Parallel and Distributed Computing, vol. 97, pp. 11–23, 2016.##
[9]
J. Cohen, J. Lefèvre, K. Maâmra, L. Pilard and D. Sohier, “A self-stabilizing algorithm for maximal matching in anonymous networks,” Parallel Processing Letters, vol. 26, pp. 49–59, 2016.##
[10]
Y. Fu and X. Xiaoping, “Self-stabilized distributed network distance prediction,” IEEE/ACM Transactions on Networking, vol. 25, pp. 451–464, 2017.##
[11]
L. Blin and S. Tixeuil, “Compact deterministic self-stabilizing leader election on a ring,” Distributed Computing, vol. 1998, pp. 1-28, 2017.##
[12]
A. K. Datta, S. Devismes, and L. L. Larmore, “Self-stabilizing silent disjunction in an anonymous network,” Theoretical Computer Science, vol. 665, pp. 51–72, 2017.##
[13]
J. Behnamian, “Graph colouring-based algorithm to parallel jobs scheduling on parallel factories,” International Journal of Computer Integrated Manufacturing, vol. 29, pp. 622–635, 2015.##
[14]
B. Yüceoğlu, G. Şahin and S. P. van Hoesel, “A column generation based algorithm for the robust graph coloring problem,” Discrete Applied Mathematics, vol. 217, pp.340–352, 2017.##
[15]
C. Zhao, X. Xu, Z. Gao, and L. Huang, “A coloring-based cluster resource allocation for ultra dense network,” in IEEE International Conference on Signal Processing, Communications and Computing, Hong Kong, pp. 1-5, 2016.##
[16]
S. Basloom, A. Nazar, G. Aldabbagh, M. Abdullah, and N. Dimitriou, “Resource allocation using graph coloring for dense cellular networks,” in International Conference on Computing, Networking and Communications, Kauai, pp. 1-5, 2016.##
[17]
L. Barenboim, M. Elkin, S. Pettie, and J. Schneider, “The locality of distributed symmetry breaking,” Journal of the ACM (JACM), vol. 63, 2016. doi: 10.1145/2903137##
[18]
A. S. Sairam, S. Roy, and R. Sahay, “Coloring networks for attacker identification and response,” Security and Communication Networks, vol. 8, pp. 751–768, 2015.##
[19]
B. L. Hartnell and C. M. Mynhardt, "Independent protection in graphs," Discrete Mathematics, vol. 335, pp. 100–109, 2014.##
[20]
M. Gradinariu and S. Tixeuil, “Self-stabilizing vertex coloring of arbitrary graphs,” in International conference on Principles of Distributed Systems, Paris, pp. 55-70, 2000.##
[21]
A. Mansouri and M. S. Bouhlel, “Efficient self-stabilizing grundy coloring algorithms,” in 2016 Future Technologies Conference FTC 2016, San Francisco, pp. 199-205, 2016.##
[22]
P. N. Panagopoulou and P. G. Spirakis, “A game theoretic approach for efficient graph coloring,” in International Symposium on Algorithms and Computation, Berlin, pp. 183-195, 2008.##
[23]
I. Chatzigiannakis, C. Koninis, P. N. Panagopoulou, and P. G. Spirakis, “Distributed game-theoretic vertex coloring,” in International Conference On Principles Of Distributed Systems, Tozeur, pp. 103-118, 2010.##
[24]
A. R´enyi. and P. Erdos, “On random graphs,” Publ. Math. Debr., vol. 6, pp. 290–297, 1959.##
[25]
D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature, vol. 393, pp. 440–442, 1998.##
[26]
A.-L. Barabási and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, pp. 509–512, 1999.##
[27]
D. J. A. Welsh and M. B. Powell, “An upper bound for the chromatic number of a graph and its application to timetabling problems,” The Computer Journal, vol. 10, pp. 85–86, 1967.##
[28]
S. T. Hedetniemi, D. P. Jacobs, and P. K. Srimani, “Linear time self-stabilizing colorings,” Information Processing Letters, vol. 87, pp. 251–255, 2003.##
[29]
W. Goddard, S. T. Hedetniemi, D. P. Jacobs, and P. K. Srimani, “Self-stabilizing algorithm for orderings and colorings,” Int. J. Found. Comput. Sci., vol. 16, pp. 19–36, 2005.##
[30]
S. Bernard, S. Devismes, M. G. Potop-Butucaru, and S. Tixeuil, “Optimal deterministic self-stabilizing vertex coloring in unidirectional anonymous networks,” in IEEE International Symposium on Parallel & Distributed Processing, New York, pp. 1-8, 2009.##
[31]
S. Bernard, S. Devismes, K. Paroux, M. Potop-Butucaru, and S. Tixeuil, “Probabilistic self-stabilizing vertex coloring in unidirectional anonymous networks,” in International Conference on Distributed Computing and Networking, Kolkata, pp.167-177, 2010.##
[32]
A. Kosowski and Ł. Kuszner, “Self-stabilizing algorithms for graph coloring with improved performance guarantees,” in Artificial Intelligence and Soft Computing, London, pp.    1150-1159, 2006.##
[33]
W. Hasenplaugh, T. Kaler, T. B. Schardl, and C. E. Leiserson, “Ordering heuristics for parallel graph coloring,” in Proceedings of the 26th ACM Symposium on Parallelism in Algorithms and Architectures , Prague, pp. 166-177, 2014.##
[34]
C. A. Christen and S. M. Selkow, “Some perfect coloring properties of graphs,” Journal of Combinatorial Theory, Series B, vol. 27, pp. 49–59, 1979.##
[35]
V. Bilò, A. Fanelli, M. Flammini, and L. Moscardelli, “Graphical congestion games,” Algorithmica, vol. 61, pp.    274–297, 2011.##
[36]
D. Monderer and L. S. Shapley, “Potential games,” Games and Economic Behavior, vol. 14, pp. 124–143, 1996.##
[37]
J. C. Miller and A. Hagberg, “Efficient generation of networks with given expected degrees,” in International Workshop on Algorithms and Models for the Web-Graph, Atlanta, pp.     115-126, 2011.##
  • Receive Date: 21 April 2017
  • Revise Date: 20 February 2019
  • Accept Date: 19 September 2018
  • Publish Date: 23 July 2018