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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
Friedhelm Meyer auf der Heide 2
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
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
Friedhelm Meyer auf der Heide 3
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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
Friedhelm Meyer auf der Heide 4
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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.
Friedhelm Meyer auf der Heide 5
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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.
Friedhelm Meyer auf der Heide 6
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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
Friedhelm Meyer auf der Heide 7
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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
Friedhelm Meyer auf der Heide 8
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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
Friedhelm Meyer auf der Heide 9
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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
Friedhelm Meyer auf der Heide 10
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
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
Friedhelm Meyer auf der Heide 11
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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
Friedhelm Meyer auf der Heide 12
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
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?- ………
Friedhelm Meyer auf der Heide 13
HEINZ NIXDORF INSTITUTEUniversity of Paderborn
Algorithms and Complexity
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