Swarm Intelligence

, Volume 6, Issue 2, pp 117–150 | Cite as

Distributed graph coloring: an approach based on the calling behavior of Japanese tree frogs

  • Hugo Hernández
  • Christian BlumEmail author


Graph coloring—also known as vertex coloring—considers the problem of assigning colors to the nodes of a graph such that adjacent nodes do not share the same color. The optimization version of the problem concerns the minimization of the number of colors used. In this paper we deal with the problem of finding valid graphs colorings in a distributed way, that is, by means of an algorithm that only uses local information for deciding the color of the nodes. The algorithm proposed in this paper is inspired by the calling behavior of Japanese tree frogs. Male frogs use their calls to attract females. Interestingly, groups of males that are located near each other desynchronize their calls. This is because female frogs are only able to correctly localize male frogs when their calls are not too close in time. The proposed algorithm makes use of this desynchronization behavior for the assignment of different colors to neighboring nodes. We experimentally show that our algorithm is very competitive with the current state of the art, using different sets of problem instances and comparing to one of the most competitive algorithms from the literature.


Distributed graph coloring Calling behavior of Japanese tree frogs 



The authors greatly appreciate the help by Enrico Malaguti who kindly provided experimental results of the centralized algorithm from Malaguti et al. (2008) for all problem instances.

Furthermore, we would like to thank Martin Middendorf who made us aware of the literature dealing with the calling behavior of Japanese tree frogs.

Finally, it is mandatory to mention the work of the Editor, Marco Dorigo, and the anonymous reviewers, who did a great job in helping to fine-tune the paper.

This work was supported by grant TIN2007-66523 (FORMALISM) of the Spanish government, and by the EU project FRONTS (FP7-ICT-2007-1). In addition, C. Blum acknowledges support from the Ramón y Cajal program of the Spanish Government, and H. Hernández acknowledges support from the Comissionat per a Universitats i Recerca del Departament d’Innovació, Universitats i Empresa de la Generalitat de Catalunya and from the European Social Fund.


  1. Aihara, I. (2009). Modeling synchronized calling behavior of Japanese tree frogs. Physical Review E, 80(1), 11–18. CrossRefGoogle Scholar
  2. Aihara, I., Kitahata, H., Yoshikawa, K., & Aihara, K. (2008). Mathematical modeling of frogs’ calling behavior and its possible application to artificial life and robotics. Artificial Life and Robotics, 12(1), 29–32. CrossRefGoogle Scholar
  3. Avanthay, C., Hertz, A., & Zufferey, N. (2003). A variable neighborhood search for graph coloring. European Journal of Operational Research, 151(2), 379–388. MathSciNetzbMATHCrossRefGoogle Scholar
  4. Barenboim, L., & Elkin, M. (2010). Deterministic distributed vertex coloring in polylogarithmic time. In SIGACT-SIGOPS 2010—Proceedings of the 29th ACM symposium on principles of distributed computing (pp. 410–419). New York: ACM Press. CrossRefGoogle Scholar
  5. Battiti, R., Bertossi, A. A., & Brunato, M. (2000). Distributed saturation degree methods for code assignment in multihop radio networks. In WSDAAL 2000—Proceedings of the 5th workshop on distributed systems: algorithms, architectures and languages (pp. 18–20). Google Scholar
  6. Blöchliger, I., & Zufferey, N. (2008). A graph coloring heuristic using partial solutions and a reactive tabu scheme. Computers & Operations Research, 35(3), 960–975. MathSciNetzbMATHCrossRefGoogle Scholar
  7. Blum, C., & Merkle, D. (Eds.) (2008). Swarm intelligence: introduction and applications. Natural computing. Berlin: Springer. zbMATHGoogle Scholar
  8. Bonabeau, E., Dorigo, M., & Theraulaz, G. (1999). Swarm intelligence: from natural to artificial systems. New York: Oxford University Press. zbMATHGoogle Scholar
  9. Braunstein, A., Mulet, R., Pagnani, A., Weigt, M., & Zecchina, R. (2003). Polynomial iterative algorithms for coloring and analyzing random graphs. Physics Review E, 68, 15 pp. Google Scholar
  10. Bui, M., Butelle, F., & Lavault, C. (2004). A distributed algorithm for constructing a minimum diameter spanning tree. Journal of Parallel and Distributed Computing, 64(5), 571–577. zbMATHCrossRefGoogle Scholar
  11. Cardei, M., MacCallum, E. D., & Cheng, X. (2002). Wireless sensor networks with energy efficient organization. Journal of Interconnection Networks, 3(4), 213–229. CrossRefGoogle Scholar
  12. Degesys, J., & Nagpal, R. (2008). Towards desynchronization of multi-hop topologies. In S. Brueckner, P. Robertson, & U. Bellur (Eds.), SASO 2008—Proceedings of the 2nd IEEE international conference on self-adaptive and self-organizing systems (pp. 129–138). Piscataway: IEEE Press. CrossRefGoogle Scholar
  13. Dorigo, M., & Birattari, M. (2007). Swarm intelligence. Scholarpedia, 2(9), 1462. CrossRefGoogle Scholar
  14. Dorne, R., & Hao, J. K. (1998). A new genetic local search algorithm for graph coloring. In A. Eiben, T. Bäck, M. Schoenauer, & H.-P. Schwefel (Eds.), Lecture notes in computer science: Vol. 1498. PPSN 1998—Proceedings of the 5th international conference on parallel problem solving from nature (pp. 745–755). Berlin: Springer. CrossRefGoogle Scholar
  15. Finocchi, I., Panconesi, A., & Silvestri, R. (2005). An experimental analysis of simple, distributed vertex coloring algorithms. Algorithmica, 41(1), 1–23. MathSciNetzbMATHCrossRefGoogle Scholar
  16. Center for Discrete Mathematics and Theoretical Computer Science (2006). Dimacs implementation challenges.
  17. Fraigniaud, P., Gavoille, C., Ilcinkas, D., & Pelc, A. (2009). Distributed computing with advice: information sensitivity of graph coloring. Distributed Computing, 21(6), 395–403. CrossRefGoogle Scholar
  18. Galinier, P., & Hao, J.-K. (1999). Hybrid evolutionary algorithms for graph coloring. Journal of Combinatorial Optimization, 3, 379–397. MathSciNetzbMATHCrossRefGoogle Scholar
  19. Gavoille, C., Klasing, R., Kosowski, A., Kuszner, L., & Navarra, A. (2009). On the complexity of distributed graph coloring with local minimality constraints. Networks, 54(1), 12–19. MathSciNetzbMATHCrossRefGoogle Scholar
  20. Guo, C., Zhong, L. C., & Rabaey, J. M. (2001). Low power distributed mac for ad hoc sensor radio networks. In GLOBECOM 2001—IEEE global telecommunications conference, Vol. 5 (pp. 2944–2948). Piscataway: IEEE Press. Google Scholar
  21. Hansen, J., Kubale, M., Kuszner, Ł., & Nadolski, A. (2004). Distributed largest-first algorithm for graph coloring. In M. Danelutto, M. Vanneschi, & D. Laforenza (Eds.), Lecture notes in computer science: Vol. 3149. Euro-Par 2004—Proceedings of the 10th international European conference on parallel and distributed computing (pp. 804–811). Berlin: Springer. Google Scholar
  22. Herman, T., & Tixeuil, S. (2004). A distributed TDMA slot assignment algorithm for wireless sensor networks. In S. Nikoletseas & J. D. P. Rolim (Eds.), Lecture notes in computer science: Vol. 3121. ALGOSENSORS 2004—Proceedings of 1st international workshop on algorithmic aspects of wireless sensor networks (pp. 45–58). Berlin: Springer. CrossRefGoogle Scholar
  23. Hernández, H., & Blum, C. (2011). Implementing a model of Japanese tree frogs’ calling behavior in sensor networks: a study of possible improvements. In N. Krasnogor & P. L. Lanzi (Eds.), BIS-WSN 2011—Proceedings of the 1st international GECCO workshop on bio-inspired solutions for wireless sensor networks, Vol. 2 (pp. 615–622). New York: ACM Press. Google Scholar
  24. Hertz, A., Plumettaz, M., & Zufferey, N. (2008). Variable space search for graph coloring. Discrete Applied Mathematics, 156(13), 2551–2560. MathSciNetzbMATHCrossRefGoogle Scholar
  25. Karp, R. M. (1972). Reducibility among combinatorial problems. Complexity of Computer Computations, 40(4), 85–103. MathSciNetCrossRefGoogle Scholar
  26. Keshavarzian, A., Lee, H., & Venkatraman, L. (2006). Wakeup scheduling in wireless sensor networks. In MobiHoc 06—Proceedings of the 7th ACM international symposium on mobile ad-hoc networking and computing (pp. 322–333). New York: ACM Press. CrossRefGoogle Scholar
  27. Kosowski, A., & Kuszner, Ł. (2006). On greedy graph coloring in the distributed model. In W. E. Nagel, W. V. Walter, & W. Lehner (Eds.), Lecture notes in computer science: Vol. 4128. Euro-Par 2006—Proceedings of the 12th international European conference on parallel and distributed computing (pp. 592–601). Berlin: Springer. Google Scholar
  28. Kroc, L., Sabharwal, A., & Selman, B. (2009). Counting solution clusters in graph coloring problems using belief propagation. In D. Koller, D. Schuurmans, Y. Bengio, & L. Bottou (Eds.), Proceedings of NIPS 2008—22nd annual conference on neural information processing systems (pp. 873–880). Cambridge: MIT Press. Google Scholar
  29. Kubale, M., & Kuszner, Ł. (2002). A better practical algorithm for distributed graph coloring. In PARELEC 02—Proceedings of the international conference on parallel computing in electrical engineering (pp. 72–75). Washington: IEEE Computer Society. CrossRefGoogle Scholar
  30. Kuhn, F., & Wattenhofer, R. (2006). On the complexity of distributed graph coloring. In PODC 2006—Proceedings of the 25th annual ACM symposium on principles of distributed computing (pp. 7–15). New York: ACM Press. Google Scholar
  31. Lee, S. A. (2008). Firefly inspired distributed graph coloring algorithms. In H. R. Arabnia & Y. Mun (Eds.), PDPTA 2008—Proceedings of the international conference on parallel and distributed processing techniques and applications (pp. 211–217). CSREA Press. Google Scholar
  32. Lee, S. A. (2010). k-Phase oscillator synchronization for graph coloring. Mathematics in Computer Science, 3(1), 61–72. MathSciNetzbMATHCrossRefGoogle Scholar
  33. Lee, S. A., & Lister, R. (2008). Experiments in the dynamics of phase coupled oscillators when applied to graph coloring. In ACSC 2008—Proceedings of the 31st Australasian conference on computer science (pp. 83–89). Darlinghurst: Australian Computer Society, Inc. Google Scholar
  34. Lu, G., Sadagopan, N., Krishnamachari, B., & Goel, A. (2005). Delay efficient sleep scheduling in wireless sensor networks. In K. Makki & E. Knightly (Eds.), INFOCOM 2005—Proceedings of the IEEE 24th international conference on computer communications (pp. 2470–2481). Piscataway: IEEE Press. Google Scholar
  35. Lü, Z., & Hao, J. K. (2010). A memetic algorithm for graph coloring. European Journal of Operational Research, 203(1), 241–250. MathSciNetzbMATHCrossRefGoogle Scholar
  36. Lynch, N. A. (2009). Distributed algorithms. San Mateo: Morgan Kaufmann. Google Scholar
  37. Malaguti, E., Monaci, M., & Toth, P. (2008). A metaheuristic approach for the vertex coloring problem. Journal on Computing, 20(2), 302–316. MathSciNetGoogle Scholar
  38. Malaguti, E., Monaci, M., & Toth, P. (2011). An exact approach for the vertex coloring problem. Discrete Optimization, 8(2), 174–190. MathSciNetCrossRefGoogle Scholar
  39. Malaguti, E., & Toth, P. (2010). A survey on vertex coloring problems. International Transactions in Operational Research, 17(1), 1–34. MathSciNetzbMATHCrossRefGoogle Scholar
  40. Maneva, E., Mossel, E., & Wainwright, M. J. (2007). A new look at survey propagation and its generalizations. Journal of the ACM, 54(4), 17. MathSciNetCrossRefGoogle Scholar
  41. Moscibroda, T., & Wattenhofer, R. (2008). Coloring unstructured radio networks. Distributed Computing, 21(4), 271–284. CrossRefGoogle Scholar
  42. Mutazono, A., Sugano, M., & Murata, M. (2009). Frog call-inspired self-organizing anti-phase synchronization for wireless sensor networks. In INDS 2009—Proceedings of the 2nd international workshop on nonlinear dynamics and synchronization (pp. 81–88). Piscataway: IEEE Press. Google Scholar
  43. Panagopoulou, P., & Spirakis, P. (2008). A game theoretic approach for efficient graph coloring. In S.-H. Hong, H. Nagamochi, & T. Fukunaga (Eds.), Lecture notes in computer science: Vol. 5369. ISAAC 2008—Proceedings of the 19th international symposium on algorithms and computation (pp. 183–195). Berlin: Springer. Google Scholar
  44. Pearl, J. (1986). Fusion, propagation, and structuring in belief networks. Artificial Intelligence, 29(3), 241–288. MathSciNetzbMATHCrossRefGoogle Scholar
  45. Santi, P. (2005a). Topology control in wireless ad hoc and sensor networks. ACM Computer Surveys, 37, 164–194. CrossRefGoogle Scholar
  46. Santi, P. (2005b). Topology control in wireless ad hoc and sensor networks. Chichester: Wiley. CrossRefGoogle Scholar
  47. Wells, K. D. (1977). The social behaviour of anuran amphibians. Animal Behaviour, 25, 666–693. CrossRefGoogle Scholar
  48. Zhang, W., Wang, G., Xing, Z., & Wittenburg, L. (2005). Distributed stochastic search and distributed breakout: properties, comparison and applications to constraint optimization problems in sensor networks. Artificial Intelligence, 161(1–2), 55–87. MathSciNetzbMATHCrossRefGoogle Scholar
  49. Zivan, R. (2008). Anytime local search for distributed constraint optimization. In AAMAS 08—The 7th international joint conference on autonomous agents and multiagent systems, Vol. 3 (pp. 1449–1452). Richland: International Foundation for Autonomous Agents and Multiagent Systems. Google Scholar

Copyright information

© Springer Science + Business Media, LLC 2012

Authors and Affiliations

  1. 1.ALBCOM Research GroupUniversitat Politècnica de CatalunyaBarcelonaSpain

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