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 @math , @math , or @math , 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: @math , @math , and @math . However, given an LCL problem it is undecidable whether its complexity is @math or @math in 2-dimensional grids. Nevertheless, if we correctly guess that the complexity of a problem is @math , we can completely automate the design of optimal algorithms. For any problem we can find an algorithm that is of a normal form @math , where @math is a finite function, @math is an algorithm for finding a maximal independent set in @math th power of the grid, and @math is a constant. Finally, partially with the help of automated design tools, we classify the complexity of several concrete LCL problems related to colourings and orientations.
problems. problems were formally introduced by Naor and Stockmeyer @cite_30 . They showed that if there exists a constant-time algorithm for solving an problem @math , then there exists an order-invariant constant-time algorithm for @math , such that the algorithm only uses the relative order of unique identifiers given to the nodes. Their argument works for any time @math : a time- @math distributed algorithm implies a constant-time order-invariant algorithm; hence there are no problems with complexities strictly between @math and @math .
@cite_30
2017345786
The purpose of this paper is a study of computation that can be done locally in a distributed network, where "locally" means within time (or distance) independent of the size of the network. Locally checkable labeling (LCL) problems are considered, where the legality of a labeling can be checked locally (e.g., coloring). The results include the following: There are nontrivial LCL problems that have local algorithms. There is a variant of the dining philosophers problem that can be solved locally. Randomization cannot make an LCL problem local; i.e., if a problem has a local randomized algorithm then it has a local deterministic algorithm. It is undecidable, in general, whether a given LCL has a local algorithm. However, it is decidable whether a given LCL has an algorithm that operates in a given time @math . Any LCL problem that has a local algorithm has one that is order-invariant (the algorithm depends only on the order of the processor IDs).