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Joint work with Bastian Degener Barbara Kempkes

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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Joint work with Bastian Degener Barbara Kempkes

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  1. Local Strategies for Building Geometric FormationsFriedhelm Meyer auf der Heide University of Paderborn Joint work with Bastian Degener Barbara Kempkes

  2. 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

  3. 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. 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. 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. 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. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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. 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

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