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Joint work with Bastian Degener Barbara KempkesPowerPoint Presentation

Joint work with Bastian Degener Barbara Kempkes

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### Local Strategies for Building Geometric FormationsFriedhelm Meyer auf der Heide University of Paderborn

### Thank you for your attention!

Joint work with

Bastian Degener

Barbara Kempkes

Geometric formation problems

Gathering problem:Robots gather in one point

Sparse network formation problem:

Robots form a sparse network connecting stations

Circle formation problem:Robots form a circle

Relay chain problem:Robots minimize the length of a chain between two stations

The model

In a step,

- a robot senses its neighborhood (robots in distance one),

- decides where to move solely based on the relative positions of its neighbors,
- moves.
A round finishes as soon as each robot was active at least once. We assume an initial random order of the robots.

Asynchronous, random order sense-compute-move model

Related work

- Ando, Suzuki, Yamashita (95), Cohen, Peleg (04,05,06) gathering, focus on asynchronous setting

- Kempkes, MadH (08) sparse network formation, synchronous and asynchronous setting

- Efrima, Peleg (07) Extension to other formations
- Kutylowski, MadH (08,09) relay chain problem, asymptotically optimal local strategies
- Empirical and experimental work in Biology
and Computer Graphics

- No local gathering strategies with runtime bound known.
Our contribution: (to appear SPAA 2010)

A local algorithm for the asynchronous, random order sense-compute-move model which needs O(n²) rounds in expectation.

A simple gathering stategy

„Go-To-The-Center“

- A random relay walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.

A simple gathering stategy

„Go-To-The-Center“

- A random relay walks to the center of its neighbors,
i.e. to the center of their smallest enclosing ball.

- If it moves to a position of

another relay, they fuse

correct, terminates in finite #rounds,

no runtime bound

The new algorithm

- Algorithm for robot r at time t:
- Sense positions of robots within distance 2.
- If all detected robots are in distance 1 of r, gather them at r’s position.
- Else compute convex hull of robots in distance 2.
- If r forms a vertex of the convex hull:
- If angle of convex hull at r smaller than ¼/3, move two or more robots to the same position (“fuse” them)
- Else see picture

r

2

- Start situation:
- n robots with positions in the plane
- Unit Disk Graph of robots w.r.t. distance 1 connected
- One robot active at a time

Correctness and runtime bound

Correctness:

- UDG stays connected

- Convex hull shrinks

- Two fused robots are never splitted again

Runtime:

In a round

- Some robots are fused (at most n rounds) or

- The expected area of the convex hull is reduced by at least a constant
expected O(n2) rounds

Runtime analysis

The area of the convex hull is decreased by at least½ - 1/(2¼) ¯iin a time step

- If no robot is fused in this round, ¯i¸¼/3
- Area of red triangle¸½ cos(¯i/2)

-2/¼x + 1

¯i

ri

¸ ½ - 1/(2¼) ¯i

·¼

¸ 0

Runtime analysis

Area ofredtriangle¸ ½ - 1/(2¼)¯i

Weknow: Atthebeginningof a round:

mi=0¯i*· (m-2)¼

Thus: Area of all redtriangles

¸mi=0(½ - 1/(2¼)¯i)

¸ 1

Problem: ¯icanchangebefore

riisactive

¯i

ri

Runtime analysis

More than a constant number c ofneighbors robots are fused

Prob(ri is first active robot in its neighborhood) ¸ 1/c

E(area truncated when ri is active) ¸ - 1/c ¢1/(2¼) ¯i*+1/(2c)

Thus: convex hull is reduced by at least 1/c in expectation

- Expected O(n2) rounds without fusion

¯i

ri

Future work

- Is the bound tight?

- Do we need the randomized round model for the runtime bound?

- Is it necessary that robots can move neighbors?

- Is the double visibility range crucial?

- Lower bounds? For our algorithm, general (model!!)

- Extension to sparse network formation?
- With mobile stations?
- ………

Friedhelm Meyer auf der Heide

Heinz Nixdorf Institute & Computer Science Department

University of Paderborn

Fürstenallee 11

33102 Paderborn, Germany

Tel.: +49 (0) 52 51/60 64 80

Fax: +49 (0) 52 51/60 64 82

Mailto: [email protected]

http://wwwhni.upb.de/en/alg

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