Is Prüfer Code Encoding Always a Bad Idea?

  • H. HildmannEmail author
  • D. Y. Atia
  • D. Ruta
  • A. F. Isakovic
Part of the Studies in Computational Intelligence book series (SCI, volume 795)


Real world problems are often of a complexity that renders deterministic approaches intractable. In the area of applied optimization, heuristics can offer a viable alternative. While potentially forfeiting on finding the most optimal solution, these techniques return good solutions in a short time. To do so, a suitable modelling of the problem as well as an efficient mapping of the problem’s solutions into a so-called solution space is required. Since it is very common to represent solutions as graphs, algorithms that efficiently map graphs into a heuristic-friendly solutions-space are of general interest to community. For a special type of graph, namely trees (i.e., undirected, connected and acyclic graphs such as Cayley in Phil Mag 13:172–176 (1857) [13]) Prüfer Code (Prüfer in Archiv der Mathematik und Physik 27:742–744 (1918) [43]) (PC) offers a bijective encoding process that comes at a low complexity (algorithms of \(\varTheta (n)\)-complexity are known Micikevičius et al. in Linear-time algorithms for encoding trees as sequences of node labels (2007) [37]) and facilitates mapping to \(n-2\) dimensional Euclidean space. However, this encoding does not preserve properties such as e.g., locality and has therefore been shown to be sub-optimal for entire classes of problems (Gottlieb et al. in Prüfer numbers: a poor representation of spanning trees for evolutionary search (2001) [24]). We argue that Prüfer Code does preserve some characterizing properties (e.g., degree of branching and branching vertices) and that these are sufficiently relevant for certain types of problems to motivate encoding them in PC. We present our investigations and provide an example problem where PC encoding worked very well.


Encryption Code Branching Vertex Solution Space Branch Vertices Branching Node 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors are grateful for the support from the UAE ICT-Fund on the project “Biologically Inspired Network Services”. We acknowledge K. Poon (EBTIC, KUST) for bringing the I-DAS problem to our attention. HH acknowledges the hospitality of the EBTIC Institute and F. Saffre (EBTIC, KUST) during his fellowship 2017.


  1. 1.
    Almeida, M., Hildmann, H., Solmazc, G.: Distributed UAV-swarm-based real-time geomatic data collection under dynamically changing resolution requirements. In: UAV-g 2017 - International Conference on Unmanned Aerial Vehicles in Geomatics, ISPRS Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Bonn, Germany (September 2017)Google Scholar
  2. 2.
    Anderson, C., Boomsma, J.J., Bartholdi, J.J.: Task partitioning in insect societies bucket brigades. Insectes Soc. 49, 171–180 (2002)CrossRefGoogle Scholar
  3. 3.
    Atia, D.Y.: Indoor distributed antenna systems deployment optimization with particle swarm optimization. M.Sc. thesis, Khalifa University of Science, Technology (2015)Google Scholar
  4. 4.
    Atia, D.Y., Ruta, D., Poon, K., Ouali, A., Isakovic, A.F.: Cost effective, scalable design of indoor distributed antenna systems based on particle swarm optimization and prufer strings. In: IEEE 2016 Congress on Evolutionary Computation, Vancouver, Canada. IEEE, New York (July 2016)Google Scholar
  5. 5.
    Bartholdi, J.J., Eisenstein, D.D.: A production line that balances itself. Oper. Res. 44(1), 21–34 (1996)CrossRefGoogle Scholar
  6. 6.
    Berdahl, A., Torney, C.J., Ioannou, C.C., Faria, J.J., Couzin, I.D.: Emergent sensing of complex environments by mobile animal groups. Science 339(6119), 574–576 (2013)CrossRefGoogle Scholar
  7. 7.
    Blackburn, P., deRijke, M., Venema, Y.: Modal Logic. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  8. 8.
    Bonabeau, E., Dorigo, M., Theraulaz, G.: Swarm Intelligence: From Natural to Artificial Systems. SFI Studies on the Sciences of Complexity. Oxford University Press, New York (1999)zbMATHGoogle Scholar
  9. 9.
    Bonabeau, E., Dorigo, M., Theraulaz, G.: Inspiration for optimization from social insect behaviour. Nature 406, 39–42 (2000)CrossRefGoogle Scholar
  10. 10.
    Borchardt, C.W.: über eine Interpolationsformel für eine Art symmetrischer Funktionen und über deren Anwendung. In: Math. Abh. Akad. Wiss. zu Berlin, pp. 1–20. Berlin (1860)Google Scholar
  11. 11.
    Brownlee, J.: Clever Algorithms: Nature-inspired Programming Recipes. (2011)
  12. 12.
    Camazine, S., Deneubourg, J.-L., Franks, N.R., Sneyd, J., Theraulaz, G., Bonabeau, E.: Self-organization in Biological Systems. Princeton University Press, Princeton (2001)zbMATHGoogle Scholar
  13. 13.
    Cayley, A.: On the theory of the analytical forms called trees. Phil. Mag. 13, 172–176 (1857)CrossRefGoogle Scholar
  14. 14.
    Cayley, A.: volume 13 of Cambridge Library Collection - Mathematics, p. 2628. Cambridge University Press, Cambridge (July 2009)Google Scholar
  15. 15.
    Cha, S.-H.: On complete and size balanced k-ary tree integer sequences. Int. J. Appl. Math. Inf. 6(2), 67–75 (2012)Google Scholar
  16. 16.
    Cha, S.-H.: On integer sequences derived from balanced k-ary trees. In: Proceedings of American Conference on Applied Mathematics, pp. 377–381. Cambridge, MA (January 2012)Google Scholar
  17. 17.
    Diestel, R.: Graph Theory. Electronic Library of Mathematics. Springer, Berlin (2006)Google Scholar
  18. 18.
    Dorigo, M., Stützle, T.: Ant Colony Optimization. Bradford Company, Scituate (2004)zbMATHGoogle Scholar
  19. 19.
    Ducatelle, F., Di Caro, G.A., Gambardella, L.M.: Principles and applications of swarm intelligence for adaptive routing in telecommunications networks. Swarm Intell. 4(3), 173–198 (2010)CrossRefGoogle Scholar
  20. 20.
    Fraser, A., Burnell, D.G.: Computer Models in Genetics. McGraw-Hill, New York (1970)Google Scholar
  21. 21.
    Fraser, A.S.: Simulation of genetic systems by automatic digital computers 1. Introduction. Aust. J. Biol. Sci. 10, 484–491 (1957)CrossRefGoogle Scholar
  22. 22.
    Gabrys, B., Ruta, D.: Genetic algorithms in classifier fusion. Appl. Soft Comput. 6(4), 337–347 (2006)CrossRefGoogle Scholar
  23. 23.
    Ghosh, S.K., Ghosh, J., Pal, R.K.: A new algorithm to represent a given k-ary tree into its equivalent binary tree structure. J. Phys. Sci. 12, 253–264 (2008)Google Scholar
  24. 24.
    Gottlieb, J., Julstrom, B.A., Raidl, G.R., Rothlauf, F.: Prüfer numbers: a poor representation of spanning trees for evolutionary search. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2001), San Francisco, CA, USA, pp. 343–350. Morgan Kaufmann Publishers, Burlington (2001)Google Scholar
  25. 25.
    Halloy, J., Sempo, G., Caprari, G., Rivault, C., Asadpour, M., Tâche, F., Saïd, I., Durier, V., Canonge, S., Amé, J.M., Detrain, C., Correll, N., Martinoli, A., Mondada, F., Siegwart, R., Deneubourg, J.L.: Social integration of robots into groups of cockroaches to control self-organized choices. Science 318(5853), 1155–1158 (2007)CrossRefGoogle Scholar
  26. 26.
    Hildmann, H., Atia, D.Y., Ruta, D., Poon, K., Isakovic, A.F.: Nature-inspired optimization in the Era of IoT: Particle Swarm Optimization (PSO) applied to Indoor Distributed Antenna Systems (I-DAS), chapter TBD, page TBD. Springer, Berlin (2018) (forthcoming)Google Scholar
  27. 27.
    Hildmann, H. Ruta, D., Atia, D.Y., Isakovic, A.F.: Using branching-property preserving Prüfer code to encode solutions for particle swarm optimisation. In: 2017 Federated Conference on Computer Science and Information Systems (FedCSIS), pp. 429–432 (September 2017)Google Scholar
  28. 28.
    Hildmann, H., Martin, M.: Resource allocation and scheduling based on emergent behaviours in multi-agent scenarios. In: Vitoriano,, Parlier, G.H. (Eds.) Proceedings of the International Conference on Operations Research and Enterprise Systems, pp. 140–147, Lisbon, Portugal (January 2015). INSTICC, SCITEPRESSGoogle Scholar
  29. 29.
    Hildmann, H., Sebastien Nicolas, A.: self-organizing client, server allocation algorithm for applications with non-linear cost functions. In: IEEE PES Innovative Smart Grid Technologies Latin America (2015 ISGT-LA), p. 2015. Montevideo (October 2015)Google Scholar
  30. 30.
    Hildmann, H., Nicolas, S., Saffre, F.: A bio-inspired resource-saving approach to dynamic client-server association. IEEE Intell. Syst. 27(6), 17–25 (2012)CrossRefGoogle Scholar
  31. 31.
    Julstrom, B.A.: Exercises in data structures, quick decoding and encoding of prüfer strings (2005)Google Scholar
  32. 32.
    Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceedings of IEEE International Conference on Neural Networks, vol. 4, pp. 1942–1948 (November 1995)Google Scholar
  33. 33.
    Kuila, P., Jana, P.K.: Energy efficient clustering and routing algorithms for wireless sensor networks: particle swarm optimization approach. Eng. Appl. Artif. Intell. 33, 127–140 (2014)CrossRefGoogle Scholar
  34. 34.
    Lim, S., Rus, D.: Stochastic distributed multi-agent planning and applications to traffic. In: ICRA, pp. 2873–2879. IEEE, New York (2012)Google Scholar
  35. 35.
    Ma, R.-J., Yu, N.-Y., Hu, J.-Y.: Application of particle swarm optimization algorithm in the heating system planning problem. Sci. World J. (2013)Google Scholar
  36. 36.
    Macaš, M., Gabrys, B., Ruta, D., Thotská, L.: Particle swarm optimization of multiple classifier systems. In: 9th International Work-Conference on Artificial Neural Networks, pp. 333–340 (2007)Google Scholar
  37. 37.
    Micikevičius, P., Caminiti, S., Deo, N.: Linear-time algorithms for encoding trees as sequences of node labels (2007)Google Scholar
  38. 38.
    Mitri, S., Floreano, D., Keller, L.: The evolution of information suppression in communicating robots with conflicting interests. Proc. Nat. Acad. Sci. 106(37), 15786–15790 (2009)CrossRefGoogle Scholar
  39. 39.
    Mugler, A., Bailey, A.G., Takahashi, K., ten Wolde, P.R.: Membrane clustering and the role of rebinding in biochemical signaling. Biophys. J. 102(5), 1069–1078 (2012)CrossRefGoogle Scholar
  40. 40.
    Navlakha, S., Bar-Joseph, Z.: Algorithms in nature: the convergence of systems biology and computational thinking. Mol. Syst. Biol. 7, 546 (2011)CrossRefGoogle Scholar
  41. 41.
    Paulden, T., Smith, D.K.: Developing new locality results for the prüfer code using a remarkable linear-time decoding algorithm. Elect. J. Comb. 14(1) (August 2007)Google Scholar
  42. 42.
    Pearl, J.: Heuristics: Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley, The Addison-Wesley Series in Artificial Intelligence (1984)Google Scholar
  43. 43.
    Prüfer, H.: Neuer Beweis eines Satzes über Permutationen. Archiv der Mathematik und Physik 27, 742–744 (1918)zbMATHGoogle Scholar
  44. 44.
    Raman, S., Raina, G., Hildmann, H., Saffre, F.: Ant-colony based heuristics to minimize power and delay in the internet. In: IEEE International Conference on Green Computing and Communications 2013 (IEEE GreenCom 2013 WS - Greencom-Next 2013), Beijing, P.R. ChinaGoogle Scholar
  45. 45.
    Ramanan, P.V., Liu, C.L.: Permutation representation of k-ary trees. Theor. Comput. Sci. 38, 83–98 (1985)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Renfrew, D., Yu, X.H.: Traffic signal control with swarm intelligence. In: 2009 Fifth International Conference on Natural Computation, vol. 3, pp. 79–83 (August 2009)Google Scholar
  47. 47.
    Saffre, F., Furey, R., Krafft, B., Deneubourg, J.-L.: Collective decision-making in social spiders: dragline-mediated amplification process acts as a recruitment mechanism. J. Theor. Biol. 198, 507–517 (1999)CrossRefGoogle Scholar
  48. 48.
    Saffre, F., Hildmann, H., Deneubourg, J.-L.: Can individual heterogeneity influence self-organised patterns in the termite nest construction model? Swarm Intell. (October 2017)Google Scholar
  49. 49.
    Schoonderwoerd, R., Bruten, J.L., Holland, O.E., Rothkrantz, L.J.M.: Ant-based load balancing in telecommunications networks. Adapt. Behav. 5(2), 169–207 (1996)CrossRefGoogle Scholar
  50. 50.
    Taylor, J.G., Cutsuridis, V., Hartley, M., Althoefer, K., Nanayakkara, T.: Observational learning: basis, experimental results and models, and implications for robotics. Cogn. Comput. 5(3), 340–354 (2013)CrossRefGoogle Scholar
  51. 51.
    Trelea, I.C., Ioan Cristian Trelea: The particle swarm optimization algorithm: convergence analysis and parameter selection. Inf. Process. Lett. 85(6), 317–325 (2003)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Turing, A.M.: Computing machinery and intelligence (1950)Google Scholar
  53. 53.
    Werfel, J., Petersen, K., Nagpal, R.: Designing collective behavior in a termite-inspired robot construction team. Science 343(6172), 754–758 (2014)CrossRefGoogle Scholar
  54. 54.
    Zhou, L., Li, B., Wang, F.: Particle swarm optimization model of distributed network planning. JNW 8(10), 2263–2268 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • H. Hildmann
    • 1
    Email author
  • D. Y. Atia
    • 2
  • D. Ruta
    • 3
  • A. F. Isakovic
    • 4
  1. 1.Dep. de Ingeniería de Sistemas y AutomáticaUniversidad Carlos III de Madrid (UC3M)LéganesSpain
  2. 2.Khalifa University of Science and TechnologyAbu DhabiUAE
  3. 3. EBTICKhalifa University of Science and TechnologyAbu DhabiUAE
  4. 4.Physics DepartmentKhalifa University of Science and TechnologyAbu DhabiUAE

Personalised recommendations