Physarum Can Compute Shortest Paths: Convergence Proofs and Complexity Bounds
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.
KeywordsShort Path Equilibrium Point Short Path Problem Slime Mold Physarum Polycephalum
Unable to display preview. Download preview PDF.
- [IJNT11]Ito, K., Johansson, A., Nakagaki, T., Tero, A.: Convergence properties for the Physarum solver. arXiv:1101.5249v1 (January 2011)Google Scholar
- [LaS76]LaSalle, J.B.: The Stability of Dynamical Systems. SIAM (1976)Google Scholar
- [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
- [SM03]Süli, E., Mayers, D.: Introduction to Numerical Analysis. Cambridge University Press (2003)Google Scholar
- [Ste04]Steele, J.: The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press (2004)Google Scholar