A central theme in distributed network algorithms concerns understanding and coping with the issue of locality. Inspired by sequential complexity theory, we focus on a complexity theory for distributed decision problems. In the context of locality, solving a decision problem requires the processors to independently inspect their local neighborhoods and then collectively decide whether a given global input instance belongs to some specified language. This paper introduces several classes of distributed decision problems, proves separation among them and presents some complete problems. More specifically, we consider the standard LOCAL model of computation and define LD (for local decision) as the class of decision problems that can be solved in constant number of communication rounds. We first study the intriguing question of whether randomization helps in local distributed computing, and to what extent. Specifically, we define the corresponding randomized class BPLD, and ask whether LD=BPLD. We provide a partial answer to this question by showing that in many cases, randomization does not help for deciding hereditary languages. In addition, we define the notion of local many-one reductions, and introduce the (nondeterministic) class NLD of decision problems for which there exists a certificate that can be verified in constant number of communication rounds. We prove that there exists an NLD-complete problem. We also show that there exist problems not in NLD. On the other hand, we prove that the class NLD#n, which is NLD assuming that each processor can access an oracle that provides the number of nodes in the network, contains all (decidable) languages. For this class we provide a natural complete problem as well.
The question of what can be computed in a constant number of communication rounds was investigated in the seminal work of Naor and Stockmeyer @cite_16 . In particular, that paper considers a subclass of @math , called LCL, which is essentially @math restricted to languages involving graphs of constant maximum degree, and involving processor inputs taken from a set of constant size, and studies the question of how to compute in @math rounds the constructive versions of decision problems in LCL. The paper provides some beautiful general results. In particular, the authors show that if there exists a randomized algorithm that constructs a solution for a problem in LCL in @math rounds, then there is also a deterministic algorithm constructing a solution for this problem in @math rounds. Unfortunately, the proof of this result relies heavily on the definition of LCL. Indeed, the constant bound constraints on the degrees and input sizes allow the authors to cleverly use Ramsey theory. It is thus not clear whether it is possible to extend this result to all languages in @math .
@cite_16
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We show that any randomised Monte Carlo distributed algorithm for the Lov 'asz local lemma requires @math communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of @math , where @math is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lov 'asz local lemma with a running time of @math rounds in bounded-degree graphs, and the best lower bound before our work was @math rounds [ 2014].