Asymptotically Optimal Gathering on a Grid

A. Cord-Landwehr, M. Fischer, D. Jung, F. Meyer auf der Heide, in: Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), ACM, 2016, pp. 301–312.

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In this paper, we solve the local gathering problem of a swarm of n indistinguishable, point-shaped robots on a two dimensional grid in asymptotically optimal time O(n) in the fully synchronous FSYNC time model. Given an arbitrarily distributed (yet connected) swarm of robots, the gathering problem on the grid is to locate all robots within a 2x2- sized area that is not known beforehand. Two robots are connected if they are vertical or horizontal neighbors on the grid. The locality constraint means that no global control, no compass, no global communication and only local vision is available; hence, a robot can only see its grid neighbors up to a constant L1-distance, which also limits its movements. A robot can move to one of its eight neighboring grid cells and if two or more robots move to the same location they are merged to be only one robot. The locality constraint is the significant challenging issue here, since robot move- ments must not harm the (only globally checkable) swarm connectivity. For solving the gathering problem, we provide a synchronous algorithm { executed by every robot { which ensures that robots merge without breaking the swarm con- nectivity. In our model, robots can obtain a special state, which marks such a robot to be performing specific connec- tivity preserving movements in order to allow later merge operations of the swarm. Compared to the grid, for gath- ering in the Euclidean plane for the same robot and time model the best known upper bound is O(n^2).
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Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)
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301-312
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Cord-Landwehr A, Fischer M, Jung D, Meyer auf der Heide F. Asymptotically Optimal Gathering on a Grid. In: Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA). ACM; 2016:301-312. doi:10.1145/2935764.2935789
Cord-Landwehr, A., Fischer, M., Jung, D., & Meyer auf der Heide, F. (2016). Asymptotically Optimal Gathering on a Grid. In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA) (pp. 301–312). ACM. https://doi.org/10.1145/2935764.2935789
@inproceedings{Cord-Landwehr_Fischer_Jung_Meyer auf der Heide_2016, title={Asymptotically Optimal Gathering on a Grid}, DOI={10.1145/2935764.2935789}, booktitle={Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA)}, publisher={ACM}, author={Cord-Landwehr, Andreas and Fischer, Matthias and Jung, Daniel and Meyer auf der Heide, Friedhelm}, year={2016}, pages={301–312} }
Cord-Landwehr, Andreas, Matthias Fischer, Daniel Jung, and Friedhelm Meyer auf der Heide. “Asymptotically Optimal Gathering on a Grid.” In Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 301–12. ACM, 2016. https://doi.org/10.1145/2935764.2935789.
A. Cord-Landwehr, M. Fischer, D. Jung, and F. Meyer auf der Heide, “Asymptotically Optimal Gathering on a Grid,” in Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2016, pp. 301–312.
Cord-Landwehr, Andreas, et al. “Asymptotically Optimal Gathering on a Grid.” Proceedings of the 28th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), ACM, 2016, pp. 301–12, doi:10.1145/2935764.2935789.

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