What Can Be Computed Locally ?

1 Summary The paper [1] deals with locality on several levels: It introduces a model for distributed computations and the notion of locally checkable labelings (LCLs). It proves properties about algorithms for locally checkable labeling problems. Two algorithms are analyzed in detail: One for the Weak Coloring problem and one for the Formal Dining Philosophers problem. The paper shows fundamental properties of local algorithms for LCL problems: • Undecidability: In general, it's impossible to decide if a certain LCL problem has a local algorithm. However, it is possible to decide if the time bound is fixed. Unfortunately, this proof doesn't give instructions on how to find an algorithm. • Randomization: If there's a randomized local algorithm for a LCL problem, then a deterministic algorithm with the same time bound can be found. This proof is important: We can concentrate on deterministic algorithms if we want to find one for a particular problem. The Weak Coloring problem is quite artificial and has only few applications. It can be used where two types of resources are needed to perform an operation, but since every node only gets one type of resource, they have to cooperate. The first phase creates a coloring with 2 d) 1 d (d + + ⋅ colors. If d ≥ 8, this expression is bigger than 2 32. In cases where we have less than 2 32 IDs, we could take the ID as color number (example: IP addresses). The authors don't give instructions on how to assign a different subset to every color number. Of course, in the model used (any computation on a single processor can be carried out in one time step) this isn't a problem, but in practice it is.