The Physarum Computer

Transcription

1 The Physarum Computer Kurt Mehlhorn Max Planck Institute for Informatics and Saarland University joint work with Vincenzo Bonifaci and Girish Varma paper available on my homepage August 30, 2011

2 The Physarum Computer Physarum, a slime mold, single cell, several nuclei builds evolving networks Nakagaki, Yamada, Tóth, Nature 2000 show video Kurt Mehlhorn 2/25

3 2008 Ig Nobel Prize For achievements that first make people LAUGH then make them THINK COGNITIVE SCIENCE PRIZE: Toshiyuki Nakagaki, Ryo Kobayashi, Atsushi Tero, Ágotá Tóth for discovering that slime molds can solve puzzles. REFERENCE: "Intelligence: Maze-Solving by an Amoeboid Organism," Toshiyuki Nakagaki, Hiroyasu Yamada, and Ágota Tóth, Nature, vol. 407, September 2000, p Kurt Mehlhorn 3/25

4 Mathematical Model (Tero et al.) G = (V, E) undirected graph each edge e has a positive length L e (fixed) and a positive diameter D e (t) (dynamic) send one unit of flow from s 0 to s 1 in an electrical network where resistance of e equals Q e (t) is flow across e at time t Dynamics: R e (t) = L e /D e (t). D e (t) = dd e(t) = Q e (t) D e (t). dt 1 and 3 links Tero et al., J. of Theoretical Biology, , 2007 Kurt Mehlhorn 4/25

5 Mathematical Model II: The Node Potentials electrical flows are driven by node potentials Q e = D e (p u p v )/L e is flow on edge { u, v } from u to v flow conservation gives n equations, one for each vertex u D e (p u p v )/L e = δ u e={ u,v } E δ u = ±1 if u { s 0, s 1 } and δ u = 0, otherwise together with p s1 = 0, the above defines the p v s uniquely can be computed by solving a linear system Kurt Mehlhorn 5/25

6 Mathematical Model II: The Node Potentials electrical flows are driven by node potentials Q e = D e (p u p v )/L e is flow on edge { u, v } from u to v flow conservation gives n equations, one for each vertex u D e (p u p v )/L e = δ u e={ u,v } E δ u = ±1 if u { s 0, s 1 } and δ u = 0, otherwise together with p s1 = 0, the above defines the p v s uniquely can be computed by solving a linear system Kurt Mehlhorn 5/25

7 Computer Experiments (Discrete Time) initialize potentials while true do update diameters: D e (t + 1) = D e (t) + ɛ( Q e (t) D e (t)) recompute potentials end while In simulations, the system converges (Miyaji/Ohnishi 07/08) e on shortest s 0 -s 1 path: D e converges to 1 e not on shortest path: D e converges to 0 Miyaji/Ohnishi ran simulations only on small graphs We ran experiments on thousands of graphs of size up to 50,000 vertices and 200,000 edges. Confirmed their findings. Kurt Mehlhorn 6/25

8 The Questions Does system convergence for all (!!!) initial conditions? How fast is the convergence? Details of the convergence process? Beyond shortest paths? Inspiration for distributed algorithms? Kurt Mehlhorn 7/25

9 Convergence against Shortest Path Theorem (Convergence) Dynamics converge against shortest path, i.e., D e 1 for edges on shortest source-sink path and D e 0 otherwise. this assumes that shortest path is unique; otherwise converge against a flow of value 1 using only shortest source-sink paths Miyaji/Onishi previously proved convergence for planar graphs with source and sink on the same face Kurt Mehlhorn 8/25

10 Parallel Links (Miyaji/Ohnishi 07) s0... e1 s1 ek parallel links with lengths L 1 < L 2 <... < L k D 1 1, D 2,..., D k 0 p s0 p s1 L 1 but D 2,..., D k 1 do not necessarily converge monotonically Kurt Mehlhorn 9/25

11 What did Evolution Optimize? Evolution optimized dynamics so as to achieve a global objective. Which? (Lyapunov Function) First idea: the energy of the flow e Q e e decreases over time not true, even for parallel links Theorem For the case of parallel links: i Q il i, i D il i / i D i, and (p s p t ) i D il i decrease over time computer experiment: the obvious generalizations (replace i by e ) to general graphs do not work Kurt Mehlhorn 10/25

12 What did Evolution Optimize? Evolution optimized dynamics so as to achieve a global objective. Which? (Lyapunov Function) First idea: the energy of the flow e Q e e decreases over time not true, even for parallel links Theorem For the case of parallel links: i Q il i, i D il i / i D i, and (p s p t ) i D il i decrease over time computer experiment: the obvious generalizations (replace i by e ) to general graphs do not work Kurt Mehlhorn 10/25

13 What did Evolution Optimize? Evolution optimized dynamics so as to achieve a global objective. Which? (Lyapunov Function) First idea: the energy of the flow e Q e e decreases over time not true, even for parallel links Theorem For the case of parallel links: i Q il i, i D il i / i D i, and (p s p t ) i D il i decrease over time computer experiment: the obvious generalizations (replace i by e ) to general graphs do not work Kurt Mehlhorn 10/25

14 What did Evolution Optimize? Evolution optimized dynamics so as to achieve a global objective. Which? (Lyapunov Function) First idea: the energy of the flow e Q e e decreases over time not true, even for parallel links Theorem For the case of parallel links: i Q il i, i D il i / i D i, and (p s p t ) i D il i decrease over time computer experiment: the obvious generalizations (replace i by e ) to general graphs do not work Kurt Mehlhorn 10/25

15 A not so Obvious Generalization s0... e1 s1 ek i D i L i i D i e D e L e value of min s 0 -s 1 cut with cap e = D e Kurt Mehlhorn 11/25

16 What did Evolution Optimize? Computer experiment: e V := D el e value of min s 0 -s 1 cut with cap e = D e decreases Theorem (Lyapunov Function) V + ( e δ({ s 0 }) D e 1) 2 decreases. Derivative of V (essentially) satisfies V c (D e Q e ) 2. e Proof is brute-force except for applications of min-cut-max-flow and... Kurt Mehlhorn 12/25

17 What did Evolution Optimize? Computer experiment: e V := D el e value of min s 0 -s 1 cut with cap e = D e decreases Theorem (Lyapunov Function) V + ( e δ({ s 0 }) D e 1) 2 decreases. Derivative of V (essentially) satisfies V c (D e Q e ) 2. e Proof is brute-force except for applications of min-cut-max-flow and... Kurt Mehlhorn 12/25

18 What did Evolution Optimize? Computer experiment: e V := D el e value of min s 0 -s 1 cut with cap e = D e decreases Theorem (Lyapunov Function) V + ( e δ({ s 0 }) D e 1) 2 decreases. Derivative of V (essentially) satisfies V c (D e Q e ) 2. e Proof is brute-force except for applications of min-cut-max-flow and... Kurt Mehlhorn 12/25

19 Convergence against Shortest Path Corollary (Convergence) Dynamics converge against shortest path, i.e., D e 1 for edges on shortest s-t path and D e 0 otherwise. this assumes that shortest path is unique; otherwise,... Miyaji/Onishi previously proved convergence for planar graphs. Kurt Mehlhorn 13/25

20 Statespace = R^E V t V decreases and stays positive V 0 V c e (D e Q e ) 2 D e Q e goes to zero for all e Q e = (D e /L e ) e and hence e L e for Q e and t large s0 s 1 converges to length of some source-sink path s0 s 1 converges to length of shortest path... Kurt Mehlhorn 14/25

21 Stable Topology (Miyaji/Ohnishi) How fast is the convergence? Definition: An edge e = (u, v) stabilizes if for all ε > 0 either p u (T ) p v (T ) ε for all large T or p v (T ) p u (T ) ε for all large T. p v (T ) p u (T ) ε for all large T. slightly more general than Miyaji/Ohnishi Definition: A network stabilizes if all edges stabilize Kurt Mehlhorn 15/25

22 A Path with Fixed Potential Difference a v b assume p a and p b are fixed L(P) length of path from a to b. define f = (p a p b )/L(P) and assume f < 1 then for all edges of p: D decays like exp((f 1)t) p v converges to p b + (p a p b )dist(v, b)/l(p) Kurt Mehlhorn 16/25

23 Stable Topology III Theorem If network stabilizes, network converges as defined next. decompose undirected G into paths: P 0 = shortest s-t path for v P 0 : p v dist(v, t) for e P 0 : D e 1 assume P 0,..., P i 1 are defined. Then P i has endpoints a and b on P 0... P i 1 internal nodes and edges are fresh maximizes f i := (p a p b )/L(P i ) this is less than one for v P i : p v p b + (p a p b )dist(v, b)/l(p i ) for e P i : D e 0, exponentially with rate f i 1. direct edges in P i in direction from from a to b Kurt Mehlhorn 17/25

24 Open Problems Do networks stabilize? If so, after what time? More generally, how long does it take for the dynamics to converge? Kurt Mehlhorn 18/25

25 Wheatstone Graph s simplest graph where flow directions are not clear b a direction of flow on e???? L d e t c R potentials evolve non-monotonically; run NonMonotone state space is cyclic; run TwoChanges Theorem Wheatstone network stabilizes Kurt Mehlhorn 19/25

26 Wheatstone Graph s simplest graph where flow directions are not clear b a direction of flow on e???? L d e t c R potentials evolve non-monotonically; run NonMonotone state space is cyclic; run TwoChanges Theorem Wheatstone network stabilizes Kurt Mehlhorn 19/25

27 Wheatstone: Middle Edge Stabilizes R i = L i /D i = resistance of edge i. x a = R a /(R a + R c ), similarly for b. if x a < x b, direction of e is RL if x a > x b, direction of e is LR x a = L a /(L a + L c ), similarly for b. split [0, 1] into S = [0, x a ), M = [x a, x b ] and L = [x b, 1] consider evolvement of (x a, x b ) in S S, both grow: in M M, x a decreases and x b grows in L L, both shrink assume x a x b Kurt Mehlhorn 20/25

28 Wheatstone: Middle Edge Stabilizes split [0, 1] into S = [0, xa) M = [xa, xb ] L = [xb, 1] S x_b S M L RL RL consider evolvement of (x a, x b ) x_a M LR RL in S S, both grow: in M M, x a decreases and x b grows L LR LR in L L, both shrink Kurt Mehlhorn 21/25

29 Wheatstone: Middle Edge Stabilizes split [0, 1] into S = [0, xa) M = [xa, xb ] L = [xb, 1] S x_b S M L RL RL consider evolvement of (x a, x b ) x_a M LR RL in S S, both grow: in M M, x a decreases and x b grows L LR LR in L L, both shrink Kurt Mehlhorn 21/25

30 Wheatstone: Middle Edge Stabilizes split [0, 1] into S = [0, x a) M = [x a, x b ] x_b S M L L = [x b, 1] consider evolvement of (x a, x b ) in S S, both grow: x_a S M LR RL RL RL LR RL in M M, x a decreases and x b grows L LR LR in L L, both shrink Kurt Mehlhorn 21/25

31 Wheatstone: Middle Edge Stabilizes split [0, 1] into S = [0, x a) M = [x a, x b ] L = [x b, 1] consider evolvement of (x a, x b ) in S S, both grow: in M M, x a decreases and x b grows in L L, both shrink x_a S M L x_b S M L LR RL RL LR LR LR RL RL what if system stays in S S or L L? Kurt Mehlhorn 21/25

32 The Transportation Problem undirected graph G = (V, E) b : V R such that v b v = 0 v supplies flow b v if b v > 0 v extracts flow b v if b v < 0 find a cheapest flow where cost of sending x units across an edge of length L is Lx Dynamics of Physarum solves transportation problem. D e s converge against a mincost solution of transportation problem. Kurt Mehlhorn 22/25

33 Open Problems and Related Work Open Problems show: flow directions stabilize show: convergence is exponential Physarum apparently can do more, e.g., network design. Prove it. inspiration for the design of distributed algorithms Related Work Ito/Johansson/Nakagaki/Tero: Convergence Properties for the Physarum Solver, January 28th, 2011, they change Q e into Q e and prove convergence for all graphs Kurt Mehlhorn 23/25

34 Network Design: Science 2010 Kurt Mehlhorn 24/25

35 Natural Computation Humans and Animals are not Turing Machines Part of their computational capabilities is based on their bodies Other Models of Computation are Relevant Suggestions for distributed algorithms CS methods can help analyzing such systems, do not leave it to physicists and biologists Kurt Mehlhorn 25/25