maintaining communication between an explorer and a base station

Jaroslaw Kutylowski 1

HEINZ NIXDORF INSTITUTEUniversity 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



Algorithms and ComplexityProblem statement

• Robots moving through terrain (exploring, working …)• Base station serves as supply

base station

robot




• 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?




• Large distances between robots– Mobile relay stations support communication links

base station

robot





• 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





• 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)



Algorithms and ComplexityAgenda

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



Algorithms and ComplexityModel

• Plane• One explorer • One base station• Relay stations arranged in a chain

base station

explorer




• 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




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



Algorithms and ComplexityGo-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



Algorithms and ComplexityGo-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



Algorithms and ComplexityGo-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 stationexplorer




• for each relay consider its distance from the line between explore and base station

• describe the distances as a vector

v = (d1,…,dn)

base stationexplorer

di




vector v after applying one step of Go-To-The-Middle

v’ = v A

n x n matrix A½

½ ½

½ ½

½ ½

½ ½

½

vector v after applying t steps of Go-To-The-Middlev’ = v At




• At the beginning vi ≤ n

• We look for a t such that vi At ≤ 1 and so At ≤ 1/n

• Then the distance of each station to the optimal lineis at most 1

consider a random walk on a line with reflecting barriers

½ ½ ½ ½ ½ ½ ½ ½

½ ½




each element of line is a state

probability distribution to be in a particular state at beginning

w = (w1,…,wn)

the same probability distribution after t steps of random walk

w’ = w Bt

there are results stating that Bt < 1/n for t=c n2 log n

(elementary Markov Chain theory)

½ ½ ½ ½ ½ ½ ½ ½

½ ½




½ ½

½ ½

½ ½

½ ½

½ ½

½ ½

matrix B

random walk on a line and GTM have common background

in t=c n2 log n we have At<1/n




• what is the number of Go-To-The-Middle rounds necessary to get the relays next to the optimal line?

• quite a lot ≈ n2 log n

maybe such bad configurations do not come up in practice?

analysis of Go-To-The-Middle in the dynamic case



Algorithms and ComplexityGo-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




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




Experimental evaluation• explorer performs a (bayesian) random walk on plane

– ratio R remains constant




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



Algorithms and ComplexityFurther 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



Algorithms and Complexity

Heinz Nixdorf Institute& Computer Science InstituteUniversity of PaderbornFürstenallee 1133102 Paderborn, Germany

Tel.: +49 (0) 52 51/60 64 66Fax: +49 (0) 52 51/62 64 82E-Mail: [email protected]://wwwhni.upb.de/alg

Thank you for your attention!Thank you for your attention!

maintaining communication between an explorer and a base station

Documents