1. 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 University of Paderborn Joint work with Bastian Degener Barbara Kempkes

2. Friedhelm Meyer auf der Heide 2 HEINZ NIXDORF INSTITUTE University 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

3. Friedhelm Meyer auf der Heide 3 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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

4. Friedhelm Meyer auf der Heide 4 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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.

5. Friedhelm Meyer auf der Heide 5 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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.

6. Friedhelm Meyer auf der Heide 6 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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

7. Friedhelm Meyer auf der Heide 7 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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 rs 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

8. Friedhelm Meyer auf der Heide 8 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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(n 2 ) rounds

9. Friedhelm Meyer auf der Heide 9 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Runtime analysis The area of the convex hull is decreased by at least ½ - 1/(2¼) ¯ i in a time step riri ¯i¯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· ¼ ¸ 0

10. Friedhelm Meyer auf der Heide 10 HEINZ NIXDORF INSTITUTE University 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 r i is active riri ¯i¯i Runtime analysis

11. Friedhelm Meyer auf der Heide 11 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity Runtime analysis More than a constant number c of neighbors robots are fused Prob(r i is first active robot in its neighborhood) ¸ 1/c E(area truncated when r i is active) ¸ - 1/c ¢ 1/(2¼) ¯ i * +1/(2c) Thus: convex hull is reduced by at least 1/c in expectation Expected O(n 2 ) rounds without fusion riri ¯i¯i

12. Friedhelm Meyer auf der Heide 12 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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? -………

13. Friedhelm Meyer auf der Heide 13 HEINZ NIXDORF INSTITUTE University of Paderborn Algorithms and Complexity 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: fmadh@upb.de http://wwwhni.upb.de/en/alg