The problem of coloring the edges of an $n$-node graph of maximum degree $Δ$ with $2Δ- 1$ colors is one of the key symmetry breaking problems in the area of distributed graph algorithms. While there has been a lot of progress towards the understanding of this problem, the dependency of the running time on $Δ$ has been a long-standing open question. Very recently, Kuhn [SODA '20] showed that the problem can be solved in time $2^{O(\sqrt{\logΔ})}+O(\log^* n)$.
In this paper, we study the edge coloring problem in the distributed LOCAL model. We show that the $(\mathit{degree}+1)$-list edge coloring problem, and thus also the $(2Δ-1)$-edge coloring problem, can be solved deterministically in time $\log^{O(\log\logΔ)}Δ+ O(\log^* n)$. This is a significant improvement over the result of Kuhn [SODA '20].
We present a deterministic distributed algorithm, in the LOCAL model, that computes a $(1+o(1))Δ$-edge-coloring in polylogarithmic-time, so long as the maximum degree $Δ=\tildeΩ(\log n)$. For smaller $Δ$, we give a polylogarithmic-time $3Δ/2$-edge-coloring. These are the first deterministic algorithms to go below the natural barrier of $2Δ-1$ colors, and they improve significantly on the recent polylogarithmic-time $(2Δ-1)(1+o(1))$-edge-coloring of Ghaffari and Su [SODA'17] and the $(2Δ-1)$-edge-coloring of Fischer, Ghaffari, and Kuhn [FOCS'17], positively answering the main open question of t…
We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a $(2Δ-1)$-edge coloring can be computed in time $\mathrm{poly}\logΔ+ O(\log^* n)$ in the LOCAL model. This improves a result of Balliu, Kuhn, and Olivetti [PODC '20], who gave an algorithm with a quasi-polylogarithmic dependency on $Δ$. We further show that in the CONGEST model, an $(8+\varepsilon)Δ$-edge coloring can be computed in $\mathrm{poly}\logΔ+ O(\log^* n)$ rounds. The best previous $O(Δ)$-…
We study the problem of bi-chromatic coloring of hypergraphs in the LOCAL distributed model of computation. This problem can easily be solved by a randomized local algorithm with no communication. However, it is not known how to solve it deterministically with only a polylogarithmic number of communication rounds. In this paper we indeed design such a deterministic algorithm that solves this problem with polylogarithmic number of communication rounds. This is an almost exponential improvement on the previously known deterministic local algorithms for this problem. Because the bi-chromatic colo…
Consider an n-vertex graph G = (V,E) of maximum degree Delta, and suppose that each vertex v \in V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete rounds. In the distributed vertex coloring problem the objective is to color G with Delta + 1, or slightly more than Delta + 1, colors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth referred to as running time.) Efficient randomized algorithms for this problem are known for more tha…
We consider coloring problems in the distributed message-passing setting. The previously-known deterministic algorithms for edge-coloring employed at least (2Delta - 1) colors, even though any graph admits an edge-coloring with Delta + 1 colors [V64]. Moreover, the previously-known deterministic algorithms that employed at most O(Delta) colors required superlogarithmic time [B15,BE10,BE11,FHK15]. In the current paper we devise deterministic edge-coloring algorithms that employ only Delta + o(Delta) colors, for a very wide family of graphs. Specifically, as long as the arboricity is a = O(Delta…
In wireless ad hoc or sensor networks, distributed node coloring is a fundamental problem closely related to establishing efficient communication through TDMA schedules. For networks with maximum degree Delta, a Delta + 1 coloring is the ultimate goal in the distributed setting as this is always possible. In this work we propose Delta + 1 coloring algorithms for the synchronous and asynchronous setting. All algorithms have a runtime of O(Delta log n) time slots. This improves on the previous algorithms for the SINR model either in terms of the number of required colors or the runtime and match…
We study the {edge-coloring} problem in the message-passing model of distributed computing. This is one of the most fundamental and well-studied problems in this area. Currently, the best-known deterministic algorithms for (2Delta -1)-edge-coloring requires O(Delta) + log-star n time \cite{PR01}, where Delta is the maximum degree of the input graph. Also, recent results of \cite{BE10} for vertex-coloring imply that one can get an O(Delta)-edge-coloring in O(Delta^{epsilon} \cdot \log n) time, and an O(Delta^{1 + epsilon})-edge-coloring in O(log Delta log n) time, for an arbitrarily small const…
We consider graph coloring and related problems in the distributed message-passing model. {Locally-iterative algorithms} are especially important in this setting. These are algorithms in which each vertex decides about its next color only as a function of the current colors in its 1-hop neighborhood. In STOC'93 Szegedy and Vishwanathan showed that any locally-iterative (Delta+1)-coloring algorithm requires Omega(Delta log Delta + log^* n) rounds, unless there is "a very special type of coloring that can be very efficiently reduced" \cite{SV93}. In this paper we obtain this special type of colo…
We present a simple polylogarithmic-time deterministic distributed algorithm for network decomposition. This improves on a celebrated $2^{O(\sqrt{\log n})}$-time algorithm of Panconesi and Srinivasan [STOC'92] and settles a central and long-standing question in distributed graph algorithms. It also leads to the first polylogarithmic-time deterministic distributed algorithms for numerous other problems, hence resolving several well-known and decades-old open problems, including Linial's question about the deterministic complexity of maximal independent set [FOCS'87; SICOMP'92]---which had been …
The distributed (Delta + 1)-coloring problem is one of most fundamental and well-studied problems of Distributed Algorithms. Starting with the work of Cole and Vishkin in 86, there was a long line of gradually improving algorithms published. The current state-of-the-art running time is O(Delta log Delta + log^* n), due to Kuhn and Wattenhofer, PODC'06. Linial (FOCS'87) has proved a lower bound of 1/2 \log^* n for the problem, and Szegedy and Vishwanathan (STOC'93) provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve…
We show that the $(degree+1)$-list coloring problem can be solved deterministically in $O(D \cdot \log n \cdot\log^2Δ)$ rounds in the \CONGEST model, where $D$ is the diameter of the graph, $n$ the number of nodes, and $Δ$ the maximum degree. Using the recent polylogarithmic-time deterministic network decomposition algorithm by Rozhoň and Ghaffari [STOC 2020], this implies the first efficient (i.e., $\poly\log n$-time) deterministic \CONGEST algorithm for the $(Δ+1)$-coloring and the $(\mathit{degree}+1)$-list coloring problem. Previously the best known algorithm required $2^{O(\sqrt{\log n})}…
We present a simple deterministic distributed algorithm that computes a $(Δ+1)$-vertex coloring in $O(\log^2 Δ\cdot \log n)$ rounds. The algorithm can be implemented with $O(\log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(Δ+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozhoň and Ghaffari [STOC'20]. The current state of the art is based on an impr…
We develop the first parallel graph coloring heuristics with strong theoretical guarantees on work and depth and coloring quality. The key idea is to design a relaxation of the vertex degeneracy order, a well-known graph theory concept, and to color vertices in the order dictated by this relaxation. This introduces a tunable amount of parallelism into the degeneracy ordering that is otherwise hard to parallelize. This simple idea enables significant benefits in several key aspects of graph coloring. For example, one of our algorithms ensures polylogarithmic depth and a bound on the number of u…
The paper presents techniques to derive upper bounds for the mean time to recover from a single fault for self-stabilizing algorithms in the message passing model. For a new Delta+1-coloring algorithm we analytically derive a bound for the mean time to recover and show that the variance is bounded by a small constant independent of the network's size. For the class of bounded-independence graphs (e.g. unit disc graphs) all containment metrics are in O(1).
We prove new bounds on the distributed fractional coloring problem in the LOCAL model. Fractional $c$-colorings can be understood as multicolorings as follows. For some natural numbers $p$ and $q$ such that $p/q\leq c$, each node $v$ is assigned a set of at least $q$ colors from $\{1,\dots,p\}$ such that adjacent nodes are assigned disjoint sets of colors. The minimum $c$ for which a fractional $c$-coloring of a graph $G$ exists is called the fractional chromatic number $χ_f(G)$ of $G$. Recently, [Bousquet, Esperet, and Pirot; SIROCCO '21] showed that for any constant $ε>0$, a fractional $(Δ+ε…
A partition $(C_1,C_2,...,C_q)$ of $G = (V,E)$ into clusters of strong (respectively, weak) diameter $d$, such that the supergraph obtained by contracting each $C_i$ is $\ell$-colorable is called a strong (resp., weak) $(d, \ell)$-network-decomposition. Network-decompositions were introduced in a seminal paper by Awerbuch, Goldberg, Luby and Plotkin in 1989. Awerbuch et al. showed that strong $(exp\{O(\sqrt{ \log n \log \log n})\}, exp\{O(\sqrt{ \log n \log \log n})\})$-network-decompositions can be computed in distributed deterministic time $exp\{O(\sqrt{ \log n \log \log n})\}$. The result o…
We consider the distributed message-passing {LOCAL} model. In this model a communication network is represented by a graph where vertices host processors, and communication is performed over the edges. Computation proceeds in synchronous rounds. The running time of an algorithm is the number of rounds from the beginning until all vertices terminate. Local computation is free. An algorithm is called {local} if it terminates within a constant number of rounds. The question of what problems can be computed locally was raised by Naor and Stockmayer \cite{NS93} in their seminal paper in STOC'93. Si…
This paper addresses the cornerstone family of \emph{local problems} in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the $n$ nodes can only send $O(\log n)$-bit messages in each round of communication. We combine bounded independence, which we show to be sufficient for some algorithms, with the method of conditional expectations and with additional machinery, to obtain the following results. Our techniques give a…
The present paper studies local distributed graph problems in highly dynamic networks. Communication and changes of the graph happen in synchronous rounds and our algorithms always, i.e., in every round, satisfy non-trivial guarantees, no matter how dynamic the network is. We define a (in our view) natural generalization of static graph problems to the dynamic graph setting. Throughout the execution of an algorithm we consider a sliding window over the last $T$, e.g., polylogarithmic, rounds. Then, in some round, the feasibility of an output only depends on the topology of the graphs in the cu…
In classic distributed graph problems, each instance on a graph specifies a space of feasible solutions (e.g. all proper ($Δ+1$)-list-colorings of the graph), and the task of distributed algorithm is to construct a feasible solution using local information. We study distributed sampling and counting problems, in which each instance specifies a joint distribution of feasible solutions. The task of distributed algorithm is to sample from this joint distribution, or to locally measure the volume of the probability space via the marginal probabilities. The latter task is also known as inference, w…
In this paper, we study distributed graph algorithms in networks in which the nodes have a limited communication capacity. Many distributed systems are built on top of an underlying networking infrastructure, for example by using a virtual communication topology known as an overlay network. Although this underlying network might allow each node to directly communicate with a large number of other nodes, the amount of communication that a node can perform in a fixed amount of time is typically much more limited. We introduce the Node-Capacitated Clique model as an abstract communication model, …
