{"abstract":[{"text":"In this paper, we solve the local gathering problem of a swarm of $n$\r\nindistinguishable, point-shaped robots on a two dimensional grid in\r\nasymptotically optimal time $\\mathcal{O}(n)$ in the fully synchronous\r\n$\\mathcal{FSYNC}$ time model. Given an arbitrarily distributed (yet connected)\r\nswarm of robots, the gathering problem on the grid is to locate all robots\r\nwithin a $2\\times 2$-sized area that is not known beforehand. Two robots are\r\nconnected if they are vertical or horizontal neighbors on the grid. The\r\nlocality constraint means that no global control, no compass, no global\r\ncommunication and only local vision is available; hence, a robot can only see\r\nits grid neighbors up to a constant $L_1$-distance, which also limits its\r\nmovements. A robot can move to one of its eight neighboring grid cells and if\r\ntwo or more robots move to the same location they are \\emph{merged} to be only\r\none robot. The locality constraint is the significant challenging issue here,\r\nsince robot movements must not harm the (only globally checkable) swarm\r\nconnectivity. For solving the gathering problem, we provide a synchronous\r\nalgorithm -- executed by every robot -- which ensures that robots merge without\r\nbreaking the swarm connectivity. In our model, robots can obtain a special\r\nstate, which marks such a robot to be performing specific connectivity\r\npreserving movements in order to allow later merge operations of the swarm.\r\nCompared to the grid, for gathering in the Euclidean plane for the same robot\r\nand time model the best known upper bound is $\\mathcal{O}(n^2)$.","lang":"eng"}],"author":[{"full_name":"Cord-Landwehr, Andreas ","last_name":"Cord-Landwehr","first_name":"Andreas "},{"id":"146","last_name":"Fischer","full_name":"Fischer, Matthias","first_name":"Matthias"},{"first_name":"Daniel","last_name":"Jung","id":"37827","full_name":"Jung, Daniel"},{"first_name":"Friedhelm","last_name":"Meyer auf der Heide","id":"15523","full_name":"Meyer auf der Heide, Friedhelm"}],"year":"2016","status":"public","date_created":"2020-04-07T07:20:47Z","citation":{"apa":"Cord-Landwehr, A., Fischer, M., Jung, D., &#38; Meyer auf der Heide, F. (2016). Asymptotically Optimal Gathering on a Grid. <i>ArXiv:1602.03303</i>.","ama":"Cord-Landwehr A, Fischer M, Jung D, Meyer auf der Heide F. Asymptotically Optimal Gathering on a Grid. <i>arXiv:160203303</i>. 2016.","mla":"Cord-Landwehr, Andreas, et al. “Asymptotically Optimal Gathering on a Grid.” <i>ArXiv:1602.03303</i>, 2016.","ieee":"A. Cord-Landwehr, M. Fischer, D. Jung, and F. Meyer auf der Heide, “Asymptotically Optimal Gathering on a Grid,” <i>arXiv:1602.03303</i>. 2016.","short":"A. Cord-Landwehr, M. Fischer, D. Jung, F. Meyer auf der Heide, ArXiv:1602.03303 (2016).","chicago":"Cord-Landwehr, Andreas , Matthias Fischer, Daniel Jung, and Friedhelm Meyer auf der Heide. “Asymptotically Optimal Gathering on a Grid.” <i>ArXiv:1602.03303</i>, 2016.","bibtex":"@article{Cord-Landwehr_Fischer_Jung_Meyer auf der Heide_2016, title={Asymptotically Optimal Gathering on a Grid}, journal={arXiv:1602.03303}, author={Cord-Landwehr, Andreas  and Fischer, Matthias and Jung, Daniel and Meyer auf der Heide, Friedhelm}, year={2016} }"},"external_id":{"arxiv":["1602.03303"]},"user_id":"15415","department":[{"_id":"63"}],"title":"Asymptotically Optimal Gathering on a Grid","_id":"16450","language":[{"iso":"eng"}],"publication":"arXiv:1602.03303","type":"preprint","date_updated":"2022-01-06T06:52:50Z"}