Distributed vertex coloring is one of the classic problems and probably also the most widely studied problems in the area of distributed graph algorithms. We present a new randomized distributed vertex coloring algorithm for the standard CONGEST model, where the network is modeled as an $n$-node graph $G$, and where the nodes of $G$ operate in synchronous communication rounds in which they can exchange $O(\log n)$-bit messages over all the edges of $G$. For graphs with maximum degree $Δ$, we show that the $(Δ+1)$-list coloring problem (and therefore also the standard $(Δ+1)$-coloring problem) can be solved in $O(\log^5\log n)$ rounds. Previously such a result was only known for the significantly more powerful LOCAL model, where in each round, neighboring nodes can exchange messages of arbitrary size. The best previous $(Δ+1)$-coloring algorithm in the CONGEST model had a running time of $O(\logΔ+ \log^6\log n)$ rounds. As a function of $n$ alone, the best previous algorithm therefore had a round complexity of $O(\log n)$, which is a bound that can also be achieved by a naïve folklore algorithm. For large maximum degree $Δ$, our algorithm hence is an exponential improvement over the previous state of the art.
We give efficient randomized and deterministic distributed algorithms for computing a distance-$2$ vertex coloring of a graph $G$ in the CONGEST model. In particular, if $Δ$ is the maximum degree of $G$, we show that there is a randomized CONGEST model algorithm to compute a distance-$2$ coloring of $G$ with $Δ^2+1$ colors in $O(\logΔ\cdot\log n)$ rounds. Further if the number of colors is slightly increased to $(1+ε)Δ^2$ for some $ε>1/{\rm polylog}(n)$, we show that it is even possible to compute a distance-$2$ coloring deterministically in polylog$(n)$ time in the CONGEST model. Finally, we …
We present a procedure for efficiently sampling colors in the {\congest} model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to $Θ(\log n)$ semi-random colors unused by their neighbors in $O(1)$ rounds, even in the distance-2 setting. This yields algorithms with $O(\log^* Δ)$ complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an $O(\log^* Δ)$-round CONGEST algorithm for $(1+ε)Δ$-edge coloring when $Δ\ge \log^{1+1/\log^*n} n$, and a poly($\…
We present a new technique to efficiently sample and communicate a large number of elements from a distributed sampling space. When used in the context of a recent LOCAL algorithm for $(\operatorname{degree}+1)$-list-coloring (D1LC), this allows us to solve D1LC in $O(\log^5 \log n)$ CONGEST rounds, and in only $O(\log^* n)$ rounds when the graph has minimum degree $Ω(\log^7 n)$, w.h.p. The technique also has immediate applications in testing some graph properties locally, and for estimating the sparsity/density of local subgraphs in $O(1)$ CONGEST rounds, w.h.p.
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 settle the complexity of the $(Δ+1)$-coloring and $(Δ+1)$-list coloring problems in the CONGESTED CLIQUE model by presenting a simple deterministic algorithm for both problems running in a constant number of rounds. This matches the complexity of the recent breakthrough randomized constant-round $(Δ+1)$-list coloring algorithm due to Chang et al. (PODC'19), and significantly improves upon the state-of-the-art $O(\log Δ)$-round deterministic $(Δ+1)$-coloring bound of Parter (ICALP'18). A remarkable property of our algorithm is its simplicity. Whereas the state-of-the-art randomized algorithm…
The main results of this paper are (I) a simulation algorithm which, under quite general constraints, transforms algorithms running on the Congested Clique into algorithms running in the MapReduce model, and (II) a distributed $O(Δ)$-coloring algorithm running on the Congested Clique which has an expected running time of (i) $O(1)$ rounds, if $Δ\geq Θ(\log^4 n)$; and (ii) $O(\log \log n)$ rounds otherwise. Applying the simulation theorem to the Congested-Clique $O(Δ)$-coloring algorithm yields an $O(1)$-round $O(Δ)$-coloring algorithm in the MapReduce model. Our simulation algorithm illustrate…
In this paper, we present improved algorithms for the $(Δ+1)$ (vertex) coloring problem in the Congested-Clique model of distributed computing. In this model, the input is a graph on $n$ nodes, initially each node knows only its incident edges, and per round each two nodes can exchange $O(\log n)$ bits of information. Our key result is a randomized $(Δ+1)$ vertex coloring algorithm that works in $O(\log\log Δ\cdot \log^* Δ)$-rounds. This is achieved by combining the recent breakthrough result of [Chang-Li-Pettie, STOC'18] in the \local\ model and a degree reduction technique. We also get the f…
We give a new randomized distributed algorithm for the $Δ+1$-list coloring problem. The algorithm and its analysis dramatically simplify the previous best result known of Chang, Li, and Pettie [SICOMP 2020]. This allows for numerous refinements, and in particular, we can color all $n$-node graphs of maximum degree $Δ\ge \log^{2+Ω(1)} n$ in $O(\log^* n)$ rounds. The algorithm works in the CONGEST model, i.e., it uses only $O(\log n)$ bits per message for communication. On low-degree graphs, the algorithm shatters the graph into components of size $\operatorname{poly}(\log n)$ in $O(\log^* Δ)$ r…
We give an improved randomized CONGEST algorithm for distance-$2$ coloring that uses $Δ^2+1$ colors and runs in $O(\log n)$ rounds, improving the recent $O(\log Δ\cdot \log n)$-round algorithm in [Halldórsson, Kuhn, Maus; PODC '20]. We then improve the time complexity to $O(\log Δ) + 2^{O(\sqrt{\log\log n})}$.
We give a randomized $Δ$-coloring algorithm in the LOCAL model that runs in $\text{poly} \log \log n$ rounds, where $n$ is the number of nodes of the input graph and $Δ$ is its maximum degree. This means that randomized $Δ$-coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, $\text{poly}\log\log n$, given the known $Ω(\log_Δ\log n)$ lower bound [Brandt et al., STOC '16]. Our main technical contribution is a constant time reduction to a constant number of $(\text{deg}+1)$-list coloring instances, for $Δ= ω(\log^4 n)$, resulting in a $\text{poly} \…
Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for $(Δ+1)$-list coloring in the randomized ${\sf LOCAL}$ model running in $O(\mathsf{Det}_{\scriptscriptstyle d}(\text{poly} \log n))$ time, where $\mathsf{Det}_{\scriptscriptstyle d}(n')$ is the deterministic complexity of $(\text{deg}+1)$-list coloring on $n'$-vertex graphs. (In this problem, each $v$ has a palette of size $\text{deg}(v)+1$.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [STOC'16, JACM'18] with complexi…
We present a randomized distributed algorithm that computes a $Δ$-coloring in any non-complete graph with maximum degree $Δ\geq 4$ in $O(\log Δ) + 2^{O(\sqrt{\log\log n})}$ rounds, as well as a randomized algorithm that computes a $Δ$-coloring in $O((\log \log n)^2)$ rounds when $Δ\in [3, O(1)]$. Both these algorithms improve on an $O(\log^3 n/\log Δ)$-round algorithm of Panconesi and Srinivasan~[STOC'1993], which has remained the state of the art for the past 25 years. Moreover, the latter algorithm gets (exponentially) closer to an $Ω(\log\log n)$ round lower bound of Brandt et al.~[STOC'16]…
We provide a deterministic CONGEST algorithm to constant factor approximate the minimum dominating set on graphs of bounded arboricity in $O(\log n)$ rounds. This improves over the well-known randomized algorithm of Lenzen and Wattenhofer[DISC2010] by making it a deterministic algorithm.
Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast $Δ$-coloring of trees requires random bits: Building on the recent lower b…
The {Congested Clique} is a distributed-computing model for single-hop networks with restricted bandwidth that has been very intensively studied recently. It models a network by an $n$-vertex graph in which any pair of vertices can communicate one with another by transmitting $O(\log n )$ bits in each round. Various problems have been studied in this setting, but for some of them the best-known results are those for general networks. In this paper we devise significantly improved algorithms for various symmetry-breaking problems, such as forests-decompositions, vertex-colorings, and maximal in…
We present a new approach to randomized distributed graph coloring that is simpler and more efficient than previous ones. In particular, it allows us to tackle the $(\operatorname{deg}+1)$-list-coloring (D1LC) problem, where each node $v$ of degree $d_v$ is assigned a palette of $d_v+1$ colors, and the objective is to find a proper coloring using these palettes. While for $(Δ+1)$-coloring (where $Δ$ is the maximum degree), there is a fast randomized distributed $O(\log^3\log n)$-round algorithm (Chang, Li, and Pettie [SIAM J. Comp. 2020]), no $o(\log n)$-round algorithms are known for the D1LC…
We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size $n$ can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized $(2+o(1))$-approximation algorithm to compute all pairs shortest paths (APSP) in time $\tilde{O}(\sqrt{n})$ on weighted graphs, where we use the convention that $\tildeΩ(f(n))$ is essentially $Ω(f(n)/$polylog$f(n))$ and $\tilde{O}(f(n))$ is essentially $O(f(n) $polylog$f(n))$. We complement this res…
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…
This paper explores the application of a new algebraic method of color exchanges to the edge coloring of simple graphs. Vizing's theorem states that the edge coloring of a simple graph $G$ requires either $Δ$ or $Δ+1$ colors, where $Δ$ is the maximum vertex degree of $G$. Holyer proved that it is {\bf NP}-complete to decide whether $G$ is $Δ$-edge-colorable even for cubic graphs. By introducing the concept of complex colors, we show that the color-exchange operation follows the same multiplication rules as quaternion. An initially $Δ$-edge-colored graph $G$ allows variable-colored edges, which…
We present new randomized algorithms that improve the complexity of the classic $(Δ+1)$-coloring problem, and its generalization $(Δ+1)$-list-coloring, in three well-studied models of distributed, parallel, and centralized computation: Distributed Congested Clique: We present an $O(1)$-round randomized algorithm for $(Δ+1)$-list coloring in the congested clique model of distributed computing. This settles the asymptotic complexity of this problem. It moreover improves upon the $O(\log^\ast Δ)$-round randomized algorithms of Parter and Su [DISC'18] and $O((\log\log Δ)\cdot \log^\ast Δ)$-round r…
We present a deterministic $O(\log \log \log n)$-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of $(Δ+1)$-coloring on $n$-vertex graphs. In this model, every machine has a sublinear local memory of size $n^ϕ$ for any arbitrary constant $ϕ\in (0,1)$. Our algorithm works under the relaxed setting where each machine is allowed to perform exponential (in $n^ϕ$) local computation, while respecting the $n^ϕ$ space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious $(Δ+1)$-coloring LOCAL algorithm by Chang-L…
In this paper we present a deterministic CONGEST algorithm to compute an $O(kΔ)$-vertex coloring in $O(Δ/k)+\log^* n$ rounds, where $Δ$ is the maximum degree of the network graph and $1\leq k\leq O(Δ)$ can be freely chosen. The algorithm is extremely simple: Each node locally computes a sequence of colors and then it "tries colors" from the sequence in batches of size $k$. Our algorithm subsumes many important results in the history of distributed graph coloring as special cases, including Linial's color reduction [Linial, FOCS'87], the celebrated locally iterative algorithm from [Barenboim, E…
We present ${\rm poly\log\log n}$-round randomized distributed algorithms to compute vertex splittings, a partition of the vertices of a graph into $k$ parts such that a node of degree $d(u)$ has $\approx d(u)/k$ neighbors in each part. Our techniques can be seen as the first progress towards general ${\rm poly\log\log n}$-round algorithms for the Lovász Local Lemma. As the main application of our result, we obtain a randomized ${\rm poly\log\log n}$-round CONGEST algorithm for $(1+ε)Δ$-edge coloring $n$-node graphs of sufficiently large constant maximum degree $Δ$, for any $ε>0$. Further, our…
We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. …
We give efficient distributed algorithms for the minimum vertex cover problem in bipartite graphs in the CONGEST model. From Kőnig's theorem, it is well known that in bipartite graphs the size of a minimum vertex cover is equal to the size of a maximum matching. We first show that together with an existing $O(n\log n)$-round algorithm for computing a maximum matching, the constructive proof of Kőnig's theorem directly leads to a deterministic $O(n\log n)$-round CONGEST algorithm for computing a minimum vertex cover. We then show that by adapting the construction, we can also convert an \emph{a…
We present $O(\log^2 \log n)$ time 3-coloring, maximal independent set and maximal matching algorithms for trees in the Massively Parallel Computation (MPC) model. Our algorithms are deterministic, apply to arbitrary-degree trees and work in the low-space MPC model, where local memory is $O(n^δ)$ for $δ\in (0,1)$ and global memory is $O(m)$. Our main result is the 3-coloring algorithm, which contrasts the randomized, state-of-the-art 4-coloring algorithm of Ghaffari, Grunau and Jin [DISC'20]. The maximal independent set and maximal matching algorithms follow in $O(1)$ time after obtaining the …
We study the maximum cardinality matching problem in a standard distributed setting, where the nodes $V$ of a given $n$-node network graph $G=(V,E)$ communicate over the edges $E$ in synchronous rounds. More specifically, we consider the distributed CONGEST model, where in each round, each node of $G$ can send an $O(\log n)$-bit message to each of its neighbors. We show that for every graph $G$ and a matching $M$ of $G$, there is a randomized CONGEST algorithm to verify $M$ being a maximum matching of $G$ in time $O(|M|)$ and disprove it in time $O(D + \ell)$, where $D$ is the diameter of $G$ …
We present improved results for approximating maximum-weight independent set ($\MaxIS$) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let $n$ and $Δ$ be the number of nodes and maximum degree, respectively, and let $\MIS(n,Δ)$ be the the running time of finding a \emph{maximal} independent set ($\MIS$) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a $Δ$-approximation for $\MaxIS$ in $O(\MIS(n,Δ)\log W)$ rounds, where $W$ is the maximum weight of a node in the graph, which can be as high as…
In this paper, we initiate the study of the vertex coloring problem of a graph in the semi streaming model. In this model, the input graph is defined by a stream of edges, arriving in adversarial order and any algorithm must process the edges in the order of arrival using space linear (up to polylogarithmic factors) in the number of vertices of the graph. In the offline settings, there is a simple greedy algorithm for $(Δ+1)$-vertex coloring of a graph with maximum degree $Δ$. We design a one pass randomized streaming algorithm for $(1+\varepsilon)Δ$-vertex coloring problem for any constant $\…
In this paper we study the problem of testing graph isomorphism (GI) in the CONGEST distributed model. In this setting we test whether the distributive network, $G_U$, is isomorphic to $G_K$ which is given as an input to all the nodes in the network, or alternatively, only to a single node. We first consider the decision variant of the problem in which the algorithm distinguishes $G_U$ and $G_K$ which are isomorphic from $G_U$ and $G_K$ which are not isomorphic. We provide a randomized algorithm with $O(n)$ rounds for the setting in which $G_K$ is given only to a single node. We prove that for…
We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires $Ω(n^2/\log^2{n})$ rounds in the worst case in the CONGEST model, as well as any algorithm for $χ$-coloring a graph, where $χ$ is the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of round…
We study local symmetry breaking problems in the CONGEST model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. A $β$-ruling set is an independent set such that every node in the graph is at most $β$ hops from a node in the independent set. Our work is motivated by the following central question: can we break the $Θ(\log n)$ time complexity barrier and the $Θ(m)$ message complexity barrier in the CONGEST model for MIS or closely-related symmetry breaking problems? We present the following results: - Time Complexity: We show that we can b…
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…
Triangle-free graphs play a central role in graph theory, and triangle detection (or triangle finding) as well as triangle enumeration (triangle listing) play central roles in the field of graph algorithms. In distributed computing, algorithms with sublinear round complexity for triangle finding and listing have recently been developed in the powerful CONGEST clique model, where communication is allowed between any two nodes of the network. In this paper we present the first algorithms with sublinear complexity for triangle finding and triangle listing in the standard CONGEST model, where the …
Censor-Hillel et al. [PODC'15] recently showed how to efficiently implement centralized algebraic algorithms for matrix multiplication in the congested clique model, a model of distributed computing that has received increasing attention in the past few years. This paper develops further algebraic techniques for designing algorithms in this model. We present deterministic and randomized algorithms, in the congested clique model, for efficiently computing multiple independent instances of matrix products, computing the determinant, the rank and the inverse of a matrix, and solving systems of li…
Following previous theoretical work by Srinivasan (FOCS 2001) and the first author (STACS 2006) and a first experimental evaluation on random instances (ALENEX 2009), we investigate how the recently developed different approaches to generate randomized roundings satisfying disjoint cardinality constraints behave when used in two classical algorithmic problems, namely low-congestion routing in networks and max-coverage problems in hypergraphs. We generally find that all randomized rounding algorithms work well, much better than what is guaranteed by existing theoretical work. The derandomized v…
We study a family of closely-related distributed graph problems, which we call degree splitting, where roughly speaking the objective is to partition (or orient) the edges such that each node's degree is split almost uniformly. Our findings lead to answers for a number of problems, a sampling of which includes: -- We present a $poly(\log n)$ round deterministic algorithm for $(2Δ-1)\cdot (1+o(1))$-edge-coloring, where $Δ$ denotes the maximum degree. Modulo the $1+o(1)$ factor, this settles one of the long-standing open problems of the area from the 1990's (see e.g. Panconesi and Srinivasan [PO…
The problem of (vertex) $(Δ+1)$-coloring a graph of maximum degree $Δ$ has been extremely well-studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA'18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized data structure for maintaining a $(Δ+1)$-coloring with $O(\log Δ)$ expected amortized update time. In this paper, we present a $(Δ+1)$-coloring data structure that achieves a constant amortized update time and show that this time bound holds not only in expectation but also with high p…
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…
Network decomposition is a central tool in distributed graph algorithms. We present two improvements on the state of the art for network decomposition, which thus lead to improvements in the (deterministic and randomized) complexity of several well-studied graph problems. - We provide a deterministic distributed network decomposition algorithm with $O(\log^5 n)$ round complexity, using $O(\log n)$-bit messages. This improves on the $O(\log^7 n)$-round algorithm of Rozhoň and Ghaffari [STOC'20], which used large messages, and their $O(\log^8 n)$-round algorithm with $O(\log n)$-bit messages. Th…
