Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G has property P by having each node run a constant-time distributed decision algorithm. In a yes-instance all nodes should output yes, while in a no-instance at least one node should output no. Recently, (OPODIS 2012) gave several conditions under which identifiers are not needed, and they conjectured that identifiers are not needed in any decision problem. In the present work, we disprove the conjecture. More than that, we analyse two critical variations of the underlying model of distributed computing: (B): the size of the identifiers is bounded by a function of the size of the input network, (¬B): the identifiers are unbounded, (C): the nodes run a computable algorithm, (¬C): the nodes can compute any, possibly uncomputable function. While it is easy to see that under (¬B, ¬C) identifiers are not needed, we show that under all other combinations there are properties that can be decided locally if and only if identifiers are present.
A central theme in distributed network algorithms concerns understanding and coping with the issue of locality . Despite considerable progress, research efforts in this direction have not yet resulted in a solid basis in the form of a fundamental computational complexity theory for locality. Inspired by sequential complexity theory, we focus on a complexity theory for . 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. We consider the standard @math model of computation and define @math (for local decision ) as the class of decision problems that can be solved in @math 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 @math , containing all languages for which there exists a randomized algorithm that runs in @math rounds, accepts correct instances with probability at least @math and rejects incorrect ones with probability at least @math . We show that @math is a threshold for the containment of @math in @math . More precisely, we show that there exists a language that does not belong to @math for any @math but does belong to @math for any @math such that @math . On the other hand, we show that, restricted to hereditary languages, @math , for any function @math and any @math such that @math . In addition, we investigate the impact of non-determinism on local decision, and establish some structural results inspired by classical computational complexity theory. Specifically, we show that non-determinism does help, but that this help is limited, as there exist languages that cannot be decided non-deterministically. Perhaps surprisingly, it turns out that it is the combination of randomization with non-determinism that enables to decide languages . Finally, we introduce the notion of local reduction, and establish some completeness results.