The issue of identifiers is crucial in distributed computing. Informally, identities are used for tackling two of the fundamental difficulties that are inherent to deterministic distributed computing, namely: (1) symmetry breaking, and (2) topological information gathering. In the context of local computation, i.e., when nodes can gather information only from nodes at bounded distances, some insight regarding the role of identities has been established. For instance, it was shown that, for large classes of construction problems, the role of the identities can be rather small. However, for the identities to play no role, some other kinds of mechanisms for breaking symmetry must be employed, such as edge-labeling or sense of direction. When it comes to local distributed decision problems, the specification of the decision task does not seem to involve symmetry breaking. Therefore, it is expected that, assuming nodes can gather sufficient information about their neighborhood, one could get rid of the identities, without employing extra mechanisms for breaking symmetry. We tackle this question in the framework of the ( LOCAL ) model.
Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and (Δ+1)-coloring algorithms by Barenboim and Elkin [6], by Kuhn [22], and by Panconesi and Srinivasan [33], as well as the OΔ2-coloring algorithm by Linial [27]. Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, they assume that all nodes know good estimations of one or more global parameters of the network, e.g., the maximum degree Δ or the number of nodes n. This paper provides a rather general method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all of the state of the art non-uniform algorithms regarding MIS and Maximal Matching, as well as to many results concerning the coloring problem. (In particular, it applies to all aforementioned algorithms.) To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest. 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.