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Local Strategies for Building Geometric Formations Friedhelm Meyer auf der Heide University of Paderborn. Joint work with Bastian Degener Barbara Kempkes. Geometric formation problems. Gathering problem: Robots gather in one point Sparse network formation problem:

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

Joint work with

Bastian Degener

Barbara Kempkes


Geometric formation problems
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
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
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
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 stategy1
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
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 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
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 analysis1
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 analysis2
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
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?

  • ………


Thank you for your attention

Thank you for your attention!

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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