The LOCAL model is among the main models for studying locality in the framework of distributed network computing. This model is however subject to pertinent criticisms, including the facts that all nodes wake up simultaneously, perform in lock steps, and are failure-free. We show that relaxing these hypotheses to some extent does not hurt local computing. In particular, we show that, for any construction task @math associated to a locally checkable labeling (LCL), if @math is solvable in @math rounds in the LOCAL model, then @math remains solvable in @math rounds in the asynchronous LOCAL model. This improves the result by [SSS 2016], which was restricted to 3-coloring the rings. More generally, the main contribution of this paper is to show that, perhaps surprisingly, asynchrony and failures in the computations do not restrict the power of the LOCAL model, as long as the communications remain synchronous and failure-free.
This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and (Δ+1)-vertex coloring), the randomized complexity is at most polylogarithmic in the size n of the network, while the best deterministic complexity is typically 2O(√logn). Understanding and potentially narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in logn rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and (Δ+1)-coloring) in logn rounds in the LOCAL model. Perhaps most surprisingly, we show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values. In addition, our formal framework also allows us to develop polylogarithmic-time randomized distributed algorithms in a simpler way. As a result, we provide a polylog-time distributed approximation scheme for arbitrary distributed covering and packing integer linear programs. Abstract We consider two models of computation: centralized local algorithms and local distributed algorithms. Algorithms in one model are adapted to the other model to obtain improved algorithms. Distributed vertex coloring is employed to design improved centralized local algorithms for: maximal independent set, maximal matching, and an approximation scheme for maximum (weighted) matching over bounded degree graphs. The improvement is threefold: the algorithms are deterministic, stateless, and the number of probes grows polynomially in log ⁎ ⁡ n , where n is the number of vertices of the input graph. The recursive centralized local improvement technique by Nguyen and Onak (FOCS 2008) is employed to obtain a distributed approximation scheme for maximum (weighted) matching. This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree @math d receives messages through @math d input ports and sends messages through @math d output ports, both numbered with @math 1 , 2 , ? , d . In this work, @math VV c is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of @math VV c : Now we have many trivial containment relations, such as @math SB ⊆ MB ⊆ VB ⊆ VV ⊆ VV c , but it is not obvious if, for example, either of @math VB ⊆ SV or @math SV ⊆ VB should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that @math SB ? MB = VB ? SV = MV = VV ? VV c . The same holds for the constant-time versions of these classes. We also show that the constant-time variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, one can use tools from modal logic to study these classes. We show an ( ( ^ 1 3 - 3 ) ) lower bound on the runtime of any deterministic distributed ( O ( ^ 1+ ) )-graph coloring algorithm in a weak variant of the ( LOCAL ) model.