Jaroslaw Kutylowski 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Maintaining Communication Between an Explorer and a Base Station Miroslaw Dynia Jaroslaw Kutylowski Pawel Lorek Friedhelm Meyer auf der Heide

Jaroslaw Kutylowski 2 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Problem statement Robots moving through terrain (exploring, working …) Base station serves as supply base station robot

Jaroslaw Kutylowski 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Problem statement Robots moving through terrain (exploring, working …) Base station serves as supply Robots should self-organize to fulfill their tasks a communication network is a necessary primitive How to maintain such a communication network?

Jaroslaw Kutylowski 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Problem statement Large distances between robots –Mobile relay stations support communication links base station robot

Jaroslaw Kutylowski 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Problem statement Large distances between robots –Mobile relay stations support communication links New approach for communication on long distances Necessary in complicated terrain (mountains…) Related to backbones in networks (GSM infrastructure) But: mobile, adaptive and ad-hoc

Jaroslaw Kutylowski 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Problem statement Large distances between robots –Mobile relay stations support communication links Robots move –Communication network must react to dynamics Relays are costly –Use as few as possible Need for a strategy for mobile relay stations Self-organizing robots, organic system local strategy, no communication, simple (no memory)

Jaroslaw Kutylowski 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Agenda 1.Model 2.Go-To-The-Middle strategy 3.Analysis for static case proof outline 4.Analysis for dynamic case review over experiments theoretical results 5.Further results & open questions

Jaroslaw Kutylowski 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Model Plane One explorer One base station Relay stations arranged in a chain base station explorer

Jaroslaw Kutylowski 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Model Plane One explorer One base station Relay stations arranged in a chain Two neighbored relay stations in distance at most d Relay stations should arrange on line between explorer and base station Static setting – Explorer and base station stand still Dynamic setting – Explorer moves

Jaroslaw Kutylowski 10 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Model Why one explorer makes sense? to get an understanding of the problem for multiple explorers an efficient solution to the one- explorer problem is necessary base station robot base station explorer

Jaroslaw Kutylowski 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Go-To-The-Middle Strategy every relay station moves to the middle position between its neighbors discrete time steps all stations move in parallel relay i relay i+1 relay i+2

Jaroslaw Kutylowski 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Go-To-The-Middle Strategy every relay station moves to the middle position between its neighbors discrete time steps all stations move in parallel Properties simple memoryless biologically inspired – bird flocks related to formation control

Jaroslaw Kutylowski 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (static) Key question given a valid configuration of relay stations between the explorer and base station what is the number of Go-To-The-Middle rounds necessary to get the relays next to the optimal line? base station explorer

Jaroslaw Kutylowski 14 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (static) for each relay consider its distance from the line between explore and base station describe the distances as a vector v = (d 1,…,d n ) base station explorer didi

Jaroslaw Kutylowski 15 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (static) vector v after applying one step of Go-To-The-Middle v’ = v A n x n matrix A ½ ½½ ½½ ½½ ½½ ½ vector v after applying t steps of Go-To-The-Middle v’ = v A t

Jaroslaw Kutylowski 16 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (static) At the beginning v i ≤ n We look for a t such that v i A t ≤ 1 and so A t ≤ 1/n Then the distance of each station to the optimal line is at most 1 consider a random walk on a line with reflecting barriers ½½½½½½½½ ½ ½

Jaroslaw Kutylowski 17 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (static) each element of line is a state probability distribution to be in a particular state at beginning w = (w 1,…,w n ) the same probability distribution after t steps of random walk w’ = w B t there are results stating that B t < 1/n for t=c n 2 log n (elementary Markov Chain theory) ½½½½½½½½ ½ ½

Jaroslaw Kutylowski 18 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (static) ½½ ½½ ½½ ½½ ½½ ½½ matrix B random walk on a line and GTM have common background in t=c n 2 log n we have A t <1/n

Jaroslaw Kutylowski 19 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (static) what is the number of Go-To-The-Middle rounds necessary to get the relays next to the optimal line? quite a lot ≈ n 2 log n maybe such bad configurations do not come up in practice? analysis of Go-To-The-Middle in the dynamic case

Jaroslaw Kutylowski 20 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (dynamic) Model base station stands still explorer moves explorer starts moving next to base station whenever needed explorer deploys new relays one GTM-step for one step of explorer Analysis goal monitor the number of relay stations used compare to the number needed for a perfect line ratio R

Jaroslaw Kutylowski 21 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (dynamic) Experimental evaluation explorer moves on a circle around base station –hard case –for every distance, the number of relay stations reaches a stability point –ratio R grows linearly with the distance of explorer to base station

Jaroslaw Kutylowski 22 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (dynamic) Experimental evaluation explorer performs a (bayesian) random walk on plane –ratio R remains constant

Jaroslaw Kutylowski 23 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Go-To-The-Middle Analysis (dynamic) Model explorer deploys new relay stations only when moving away from base station explorer waits when distance to last relay station is too large Analysis what is the speed of the explorer? (how much must he wait?) Result speed of explorer ≈1/d with d the distance to base station

Jaroslaw Kutylowski 24 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Further results & open questions Further (unpublished) results reducing the locality and simplicity (to some extent) one can obtain much better performances extension to terrain with obstacles Open questions can one improve the performance without sacrificing locality and simplicity? general lower bound for local strategies? multiple explorers

Jaroslaw Kutylowski 25 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Heinz Nixdorf Institute & Computer Science Institute University of Paderborn Fürstenallee Paderborn, Germany Tel.: +49 (0) 52 51/ Fax: +49 (0) 52 51/ Thank you for your attention!