Friedhelm Meyer auf der Heide 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Local Strategies for Building Geometric Formations

Friedhelm Meyer auf der Heide 1

HEINZ NIXDORF INSTITUTEUniversity of Paderborn

Algorithms and Complexity

Local Strategies for Building Geometric Formations

Friedhelm Meyer auf der Heide University of Paderborn

Joint work with

Bastian Degener

Barbara Kempkes




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

Geometric formation problems



Algorithms and ComplexityThe 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



Algorithms and ComplexityRelated 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.



Algorithms and ComplexityA 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.



Algorithms and ComplexityA 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



Algorithms and ComplexityThe 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



Algorithms and ComplexityCorrectness 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



Algorithms and ComplexityRuntime analysis

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

ri

¯i

If no robot is fused in this round, ¯i ¸ ¼/3

Area of red triangle ¸ ½ cos(¯i/2)

¸ ½ - 1/(2¼) ¯i

-2/¼ x + 1

· ¼

¸ 0




Area of red triangle ¸ ½ - 1/(2¼) ¯i

We know: At the beginning of a round: m

i=0 ¯i* · (m-2)¼

Thus: Area of all red triangles¸ m

i=0 (½ - 1/(2¼) ¯i) ¸ 1

Problem: ¯i can change before ri is active

ri

¯i

Runtime analysis



Algorithms and ComplexityRuntime 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

ri

¯i



Algorithms and ComplexityFuture 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 HeideHeinz Nixdorf Institute & Computer Science

DepartmentUniversity of Paderborn

Fürstenallee 1133102 Paderborn, Germany

Tel.: +49 (0) 52 51/60 64 80Fax: +49 (0) 52 51/60 64 82

Mailto: [email protected]://wwwhni.upb.de/en/alg

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Friedhelm Meyer auf der Heide 1 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Local Strategies for Building Geometric Formations