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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 Friedhelm Meyer auf der Heide University of Paderborn Joint work with Bastian Degener Barbara Kempkes

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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

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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

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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.

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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.

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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

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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

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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

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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

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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

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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

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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? -………

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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

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