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Physarum Can Compute Shortest Paths: Convergence Proofs and Complexity Bounds

  • Luca Becchetti
  • Vincenzo Bonifaci
  • Michael Dirnberger
  • Andreas Karrenbauer
  • Kurt Mehlhorn
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)

Abstract

Physarum polycephalum is a slime mold that is apparently able to solve shortest path problems. A mathematical model for the slime’s behavior in the form of a coupled system of differential equations was proposed by Tero, Kobayashi and Nakagaki [TKN07]. We prove that a discretization of the model (Euler integration) computes a (1 + ε)-approximation of the shortest path in O( m L (logn + logL)/ε 3) iterations, with arithmetic on numbers of O(log(nL/ε)) bits; here, n and m are the number of nodes and edges of the graph, respectively, and L is the largest length of an edge. We also obtain two results for a directed Physarum model proposed by Ito et al. [IJNT11]: convergence in the general, nonuniform case and convergence and complexity bounds for the discretization of the uniform case.

Keywords

Short Path Equilibrium Point Short Path Problem Slime Mold Physarum Polycephalum 
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.

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References

  1. [BD97]
    Baldauf, S.L., Doolittle, W.F.: Origin and evolution of the slime molds (Mycetozoa). Proc. Natl. Acad. Sci. USA 94, 12007–12012 (1997)CrossRefGoogle Scholar
  2. [BMV12]
    Bonifaci, V., Mehlhorn, K., Varma, G.: Physarum can compute shortest paths. Journal of Theoretical Biology 309, 121–133 (2012); A preliminary version of this paper appeared at SODA 2012, pp. 233–240MathSciNetCrossRefGoogle Scholar
  3. [Bol98]
    Bollobás, B.: Modern Graph Theory. Springer, New York (1998)zbMATHCrossRefGoogle Scholar
  4. [Bon13]
    Bonifaci, V.: Physarum can compute shortest paths: A short proof. Information Processing Letters 113(1-2), 4–7 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [IJNT11]
    Ito, K., Johansson, A., Nakagaki, T., Tero, A.: Convergence properties for the Physarum solver. arXiv:1101.5249v1 (January 2011)Google Scholar
  6. [Kir10]
    Kirby, B.J.: Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  7. [LaS76]
    LaSalle, J.B.: The Stability of Dynamical Systems. SIAM (1976)Google Scholar
  8. [MO07]
    Miyaji, T., Ohnishi, I.: Mathematical analysis to an adaptive network of the Plasmodium system. Hokkaido Mathematical Journal 36(2), 445–465 (2007)MathSciNetGoogle Scholar
  9. [MO08]
    Miyaji, T., Ohnishi, I.: Physarum can solve the shortest path problem on Riemannian surface mathematically rigourously. International Journal of Pure and Applied Mathematics 47(3), 353–369 (2008)MathSciNetzbMATHGoogle Scholar
  10. [NIU+07]
    Nakagaki, T., Iima, M., Ueda, T., Nishiura, Y., Saigusa, T., Tero, A., Kobayashi, R., Showalter, K.: Minimum-risk path finding by an adaptive amoebal network. Physical Review Letters 99(068104), 1–4 (2007)Google Scholar
  11. [NYT00]
    Nakagaki, T., Yamada, H., Tóth, Á.: Maze-solving by an amoeboid organism. Nature 407, 470 (2000)CrossRefGoogle Scholar
  12. [SM03]
    Süli, E., Mayers, D.: Introduction to Numerical Analysis. Cambridge University Press (2003)Google Scholar
  13. [Ste04]
    Steele, J.: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press (2004)Google Scholar
  14. [TKN07]
    Tero, A., Kobayashi, R., Nakagaki, T.: A mathematical model for adaptive transport network in path finding by true slime mold. Journal of Theoretical Biology 244, 553–564 (2007)MathSciNetCrossRefGoogle Scholar
  15. [You]

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Luca Becchetti
    • 1
  • Vincenzo Bonifaci
    • 2
  • Michael Dirnberger
    • 3
  • Andreas Karrenbauer
    • 3
  • Kurt Mehlhorn
    • 3
  1. 1.Dipartimento di Informatica e SistemisticaSapienza Università di RomaItaly
  2. 2.Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti”Consiglio Nazionale delle RicercheRomeItaly
  3. 3.Max Planck Institute for InformaticsSaarbrückenGermany

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