{"language":[{"iso":"eng"}],"quality_controlled":"1","publisher":"ACM Press","abstract":[{"lang":"eng","text":"LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of O(1), Θ(log* n), or Θ(n), and the design of optimal algorithms can be fully automated. This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: O(1), Θ(log* n), and Θ(n). However, given an LCL problem it is undecidable whether its complexity is Θ(log* n) or Θ(n) in 2-dimensional grids.\r\nNevertheless, if we correctly guess that the complexity of a problem is Θ(log* n), we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form A' o Sk, where A' is a finite function, Sk is an algorithm for finding a maximal independent set in kth power of the grid, and k is a constant.\r\nFinally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations."}],"year":"2017","publication_identifier":{"isbn":["9781450349925"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"101-110","oa_version":"None","status":"public","extern":"1","date_published":"2017-07-01T00:00:00Z","publication_status":"published","day":"01","date_created":"2019-10-08T12:47:46Z","citation":{"chicago":"Brandt, Sebastian, Juho Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, Patric R.J. Östergård, Christopher Purcell, Joel Rybicki, Jukka Suomela, and Przemysław Uznański. “LCL Problems on Grids,” 101–10. ACM Press, 2017. https://doi.org/10.1145/3087801.3087833.","apa":"Brandt, S., Hirvonen, J., Korhonen, J. H., Lempiäinen, T., Östergård, P. R. J., Purcell, C., … Uznański, P. (2017). LCL problems on grids (pp. 101–110). Presented at the PODC: Principles of Distributed Computing, Washington, DC, United States: ACM Press. https://doi.org/10.1145/3087801.3087833","ama":"Brandt S, Hirvonen J, Korhonen JH, et al. LCL problems on grids. In: ACM Press; 2017:101-110. doi:10.1145/3087801.3087833","ieee":"S. Brandt et al., “LCL problems on grids,” presented at the PODC: Principles of Distributed Computing, Washington, DC, United States, 2017, pp. 101–110.","ista":"Brandt S, Hirvonen J, Korhonen JH, Lempiäinen T, Östergård PRJ, Purcell C, Rybicki J, Suomela J, Uznański P. 2017. LCL problems on grids. PODC: Principles of Distributed Computing, 101–110.","short":"S. Brandt, J. Hirvonen, J.H. Korhonen, T. Lempiäinen, P.R.J. Östergård, C. Purcell, J. Rybicki, J. Suomela, P. Uznański, in:, ACM Press, 2017, pp. 101–110.","mla":"Brandt, Sebastian, et al. LCL Problems on Grids. ACM Press, 2017, pp. 101–10, doi:10.1145/3087801.3087833."},"conference":{"location":"Washington, DC, United States","end_date":"2017-07-27","start_date":"2017-07-25","name":"PODC: Principles of Distributed Computing"},"type":"conference","month":"07","author":[{"first_name":"Sebastian","last_name":"Brandt","full_name":"Brandt, Sebastian"},{"last_name":"Hirvonen","full_name":"Hirvonen, Juho","first_name":"Juho"},{"last_name":"Korhonen","full_name":"Korhonen, Janne H.","first_name":"Janne H."},{"full_name":"Lempiäinen, Tuomo","last_name":"Lempiäinen","first_name":"Tuomo"},{"first_name":"Patric R.J.","last_name":"Östergård","full_name":"Östergård, Patric R.J."},{"first_name":"Christopher","last_name":"Purcell","full_name":"Purcell, Christopher"},{"first_name":"Joel","full_name":"Rybicki, Joel","last_name":"Rybicki","orcid":"0000-0002-6432-6646","id":"334EFD2E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Suomela, Jukka","last_name":"Suomela","first_name":"Jukka"},{"full_name":"Uznański, Przemysław","last_name":"Uznański","first_name":"Przemysław"}],"date_updated":"2021-01-12T08:09:39Z","doi":"10.1145/3087801.3087833","title":"LCL problems on grids","_id":"6932","article_processing_charge":"No"}