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Distributed Graph Coloring, Fundamentals and Recent Developments
... Therefore, one can control the degree of the dependency graph to be constant as the system scales up, such that the computation and communication overhead produced by each sweep are scale-invariant. Furthermore, when parallelization is enabled with certain choice of graph-coloring algorithms [37], the number of colors can be easily controlled constant as the system's scale increases. In such a case, the sequential solution complexity per cycle of BCD, which is linearly related to the number of colors, can be bounded, and thus the computation time per cycle in BCD is approximately independent of the system's scale (see Fig. 8). ...
... Metrics. We use the (∆ + 1)-coloring algorithm [37], where ∆ is the maximum degree of the dependency graph, for parallelization by default, so that the number of colors can be controlled constant as systems scaling up with a constant ∆. We evaluate the total time to solve subproblems in parallel, abbreviated as PT (Parallel Time). ...
... Analogously, the last row of (37) gives that ...
... As an application of our derandomization framework, we consider the task of designing fast deterministic MPC algorithms for graph coloring -one of the most fundamental algorithmic primitives, extensively studied in various settings for several decades. Graph coloring problems have been playing a prominent role in distributed and parallel computing, not only because of their numerous applications, but also since some variants of coloring problems naturally model typical symmetry breaking problems, as frequently encountered in decentralized systems (see, e.g., [BE13] for an overview of early advances). Parallel graph coloring has been studied since the 1980s [BE13,Kar85], and nowadays (∆ + 1)-coloring and (2∆ − 1)-edge-coloring 1 are considered among the most fundamental graph problems -benchmark problems in the area (throughout the paper, ∆ refers to the maximum degree of the input graph). ...
... Graph coloring problems have been playing a prominent role in distributed and parallel computing, not only because of their numerous applications, but also since some variants of coloring problems naturally model typical symmetry breaking problems, as frequently encountered in decentralized systems (see, e.g., [BE13] for an overview of early advances). Parallel graph coloring has been studied since the 1980s [BE13,Kar85], and nowadays (∆ + 1)-coloring and (2∆ − 1)-edge-coloring 1 are considered among the most fundamental graph problems -benchmark problems in the area (throughout the paper, ∆ refers to the maximum degree of the input graph). ...
... Our application to D1LC continues a long line of research studying the parallel and distributed complexity of graph coloring problems. For the references to earlier work on distributed coloring algorithms we refer to the monograph by Barenboim and Elkin [BE13] (see also the influential papers by Linial [Lin87,Lin92]). We will discuss here only more recent advances (and final results) for the four most relevant coloring problems, (∆ + 1)-coloring, (∆ + 1)-list-coloring, D1LC, and ∆-coloring. ...
Graph coloring problems are among the most fundamental problems in parallel and distributed computing, and have been studied extensively in both settings. In this context, designing efficient deterministic algorithms for these problems has been found particularly challenging. In this work we consider this challenge, and design a novel framework for derandomizing algorithms for coloring-type problems in the Massively Parallel Computation (MPC) model with sublinear space. We give an application of this framework by showing that a recent (degree+1)-list coloring algorithm by Halldorsson et al. (STOC'22) in the LOCAL model of distributed computation can be translated to the MPC model and efficiently derandomized. Our algorithm runs in rounds, which matches the complexity of the state of the art algorithm for the -coloring problem.
... Therefore, one can control the degree of the dependency graph to be constant as the system scales up, such that the computation and communication overhead produced by each sweep are scale-invariant. Furthermore, when parallelization is enabled with certain choice of graph-coloring algorithms [37], the number of colors can be easily controlled constant as the system's scale increases. In such a case, the sequential solution complexity per cycle of BCD, which is linearly related to the number of colors, can be bounded, and thus the computation time per cycle in BCD is approximately independent of the system's scale (see Fig. 8). ...
... Metrics. We use the (∆ + 1)-coloring algorithm [37], where ∆ is the maximum degree of the dependency graph, for parallelization by default, so that the number of colors can be controlled constant as systems scaling up with a constant ∆. We evaluate the total time to solve subproblems in parallel, abbreviated as PT (Parallel Time). ...
... Analogously, the last row of (37) gives that ...
This paper explores the distance-based relative state estimation problem in large-scale systems, which is hard to solve effectively due to its high-dimensionality and non-convexity. In this paper, we alleviate this inherent hardness to simultaneously achieve scalability and robustness of inference on this problem. Our idea is launched from a universal geometric formulation, called \emph{generalized graph realization}, for the distance-based relative state estimation problem. Based on this formulation, we introduce two collaborative optimization models, one of which is convex and thus globally solvable, and the other enables fast searching on non-convex landscapes to refine the solution offered by the convex one. Importantly, both models enjoy \emph{multiconvex} and \emph{decomposable} structures, allowing efficient and safe solutions using \emph{block coordinate descent} that enjoys scalability and a distributed nature. The proposed algorithms collaborate to demonstrate superior or comparable solution precision to the current centralized convex relaxation-based methods, which are known for their high optimality. Distinctly, the proposed methods demonstrate scalability beyond the reach of previous convex relaxation-based methods. We also demonstrate that the combination of the two proposed algorithms achieves a more robust pipeline than deploying the local search method alone in a continuous-time scenario.
... Finding or ruling out a lower bound for the ∆ + 1-coloring problem is considered one of the most central open questions in distributed algorithms [34,10,30]. Understanding the complexity of the ∆-coloring problem Downloaded 01/17/23 to 178.171.66.156 . ...
... We illustrate some typical difficult cases, which also motivate our design choices. For the sake of this exposition, it's best to think of an AC as a clique on approximately ∆ nodes, possibly with a single edge missing, also see Figure 3 on page 10. Then, in order to reduce to the (deg + 1)-list coloring problem it suffices to give just one node of an AC slack, i.e., a toehold. ...
... There is a vast amount of randomized and deterministic distributed vertex coloring algorithms that color with more than ∆ colors. The excellent monograph [10] covers many of these results until 2013. We only cover a few selected newer results, and, e.g., completely spare the large body of work on edge colorings or colorings of special graph classes. ...
... Given a coloring of the vertices of a graph with the set of colors {1, . . . , c}, there is a simple greedy algorithm [5] that produces a maximal independent set in c rounds: In round i, all active nodes with color i send a message to their neighbors, output 1, and terminate. All active processes that receive a message in round i output 0 and terminate. ...
... There is a (∆+1)-Vertex Coloring algorithm [28,5,6] that is fault tolerant and has round complexity O(∆ + log * d), but is not uniform with respect to ∆. As a baseline algorithm, consider this algorithm as phase 1, and the simple greedy algorithm that produces a maximal independent set from the coloring in ∆ rounds as phase 2. When ∆ is known to be constant, the Parallel Template with the Distributed Greedy MIS Algorithm as the component-size sensitive algorithm gives a Maximal Independent Set algorithm with predictions that is consistent, robust and η-degrading. ...
We initiate the study of deterministic distributed graph algorithms with predictions in synchronous message passing systems. The process at each node in the graph is given a prediction, which is some extra information about the problem instance that may be incorrect. The processes may use the predictions to help them solve the problem. The overall goal is to develop algorithms that both work faster when predictions are good and do not work much worse than algorithms without predictions when predictions are bad. Concepts from the more general area of algorithms with predictions, such as error measures, consistency, robustness, and smoothness, are adapted to distributed graph algorithms with predictions. We consider algorithms with predictions for four distributed graph problems, Maximal Independent Set, Maximal Matching, -Vertex Coloring, and -Edge Coloring, where denotes the degree of the graph. For each, we define an appropriate error measure. We present generic templates that can be used to design deterministic distributed graph algorithms with predictions from existing algorithms without predictions. Using these templates, we develop algorithms with predictions for Maximal Independent Set. Alternative error measures for the Maximal Independent Set problem are also considered. We obtain algorithms with predictions for general graphs and for rooted trees and analyze them using two of these error measures.
... Graph coloring-assigning colors to the vertices of the graph such that no two adjacent vertices have the same color-has been a central problem in the study of distributed graph algorithms. We refer to the Distributed Graph Coloring book by Barenboim and Elkin [BE13]. ...
... Only afterwards we add the removed nodes to the graph and let them compute their output last. The idea of putting nodes away to be colored in the end has already been used in the deterministic coloring algorithm in [BE13] where graphs with bounded arboricity are colored. However, we are not aware of any randomized algorithm that uses this technique. ...
We present a randomized distributed algorithm that computes a -coloring in any non-complete graph with maximum degree in rounds, as well as a randomized algorithm that computes a -coloring in rounds when . Both these algorithms improve on an -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 round lower bound of Brandt et al.~[STOC'16].
... • In the first phase, we run a simple and elegant randomized distributed procedure directly adapted from Algorithm 19 in [1], for (log log ) iterations (and thus for (log log ) rounds). Nodes that are properly colored in Algorithm 1 terminate, while the nodes that remain uncolored throughout the whole execution of Algorithm 1 continue to the second phase. ...
... Algorithm 1: Randomized distributed procedure for (deg +1)-list-coloring directly adapted from Algorithm 19 in [1]. Code of node parameterized by the number of iterations, and the input list ⊂ N {0} of colors given to . ...
In distributed network computing, a variant of the LOCAL model has been recently introduced, referred to as the SLEEPING model. In this model, nodes have the ability to decide on which round they are awake, and on which round they are sleeping. Two (adjacent) nodes can exchange messages in a round only if both of them are awake in that round. The SLEEPING model captures the ability of nodes to save energy when they are sleeping. In this framework, a major question is the following: is it possible to design algorithms that are energy efficient, i.e., where each node is awake for a few number of rounds only, without losing too much on the time efficiency, i.e., on the total number of rounds? This paper answers positively to this question, for one of the most fundamental problems in distributed network computing, namely -coloring networks of maximum degree . We provide a randomized algorithm with average awake-complexity constant, maximum awake-complexity in n-node networks, and round-complexity .
... The above definition is equivalent to the well-known notion of Independent set in Graph Theory [29]. From the perspective of the i-th agent, the following vector becomes relevant ...
... new sequence of Turns, denoted as S ′ := T ′ k q ′ k=1 , , q ′ < q, is used in Algorithm 1. For the second step, the parallel iterative greedy algorithm presented in [29] can be used, which guarantees a correct coloring with at most ∆(Γ) + 1 colors. Since this algorithm may require several time steps to complete, it is designed to run in the background without affecting the activities at the TB Supervision level. ...
This paper presents a novel distributed constrained monitoring strategy for cyber-physical systems that focuses on enforcing point-in-time set-membership coordination constraints among the subsystem evolutions. The proposed scheme extends previous solutions by allowing the handling of time-varying dynamic interconnections, including the online addition/removal of subsystems (plug-and-play) during normal system operations. This approach ensures global coordination of the subsystems while effectively handling corresponding changes in the underlying constraint topology. The coordination is achieved by determining a distributed feasible set-point for each subsystem when the nominal set-point becomes unfeasible due to topological changes in the system, resulting in constraints violation. In order to address this challenge, the paper generalizes the distributed command governor (CG) theory to handle online requests for structural changes in the system and/or in the coordination constraints. An add-on module is provided for each local CG to process the coordination constraints and appropriately schedule the topology changes when certain formal conditions are satisfied. This ensures that the coordination constraints are satisfied even during the topological changes. Case studies are presented by evaluating the proposed strategy on the coordination of a formation of moving vehicles and a water distribution network. These simulations are used to evaluate the effectiveness and performance of the approach.
... For = log ω (1) n, this improves on a recent randomized min , log log n log log log n lower bound for the same problem [5]. The lower bound also almost matches the best known randomized MIS algorithm in trees, which has a worst-case complexity of O min √ log n, log + log log n log log log n [23], and we thus nearly resolve Open Problem 11.15 in the book by Barenboim and Elkin [6]. ...
... We comment that the above statement is also implicit in the classical maximal matching result by Israeli and Itai [29]. 6 Theorem 5 There is a deterministic CONGEST algorithm that computes a maximal matching and has edge-averaged complexity O(log 2 + log * n), node-averaged complexity O(log 3 + log * n), and worst-case complexity O(log 2 · log n). ...
We continue the recently started line of work on the distributed node-averaged complexity of distributed graph algorithms. The node-averaged complexity of a distributed algorithm running on a graph G=(V,E)G=(V,E) is the average over the times at which the nodes V of G finish their computation and commit to their outputs. We study the node-averaged complexity for some of the central distributed symmetry breaking problems and provide the following results (among others). As our main result, we show that the randomized node-averaged complexity of computing a maximal independent set (MIS) in n-node graphs of maximum degree Δ is at least Ω(min{logΔloglogΔ,lognloglogn}). This bound is obtained by a novel adaptation of the well-known lower bound by Kuhn, Moscibroda, and Wattenhofer [JACM’16]. As a side result, we obtain that the worst-case randomized round complexity for computing an MIS in trees is also Ω(min{logΔloglogΔ,lognloglogn})—this essentially answers open problem 11.15 in the book by Barenboim and Elkin and resolves the complexity of MIS on trees up to an O(loglogn) factor. We also show that, perhaps surprisingly, a minimal relaxation of MIS, which is the same as (2, 1)-ruling set, to the (2, 2)-ruling set problem drops the randomized node-averaged complexity to O(1). For maximal matching, we show that while the randomized node-averaged complexity is Ω(min{logΔloglogΔ,lognloglogn}), the randomized edge-averaged complexity is O(1). Further, we show that the deterministic edge-averaged complexity of maximal matching is O(log2Δ+log∗n) and the deterministic node-averaged complexity of maximal matching is O(log3Δ+log∗n). Finally, we consider the problem of computing a sinkless orientation of a graph. The deterministic worst-case complexity of the problem is known to be Θ(logn), even on bounded-degree graphs. We show that the problem can be solved deterministically with node-averaged complexity O(log∗n), while keeping the worst-case complexity in O(logn).
... Finding or ruling out a lower bound for the ∆ + 1-coloring problem is considered one of the most central open questions in distributed algorithms [Lin92,BE13,HMKS16]. Understanding the complexity of the ∆-coloring problem may help shed a light on this [BBKO22]. ...
... There is a vast amount of randomized and deterministic distributed vertex coloring algorithms that color with more than ∆ colors. The excellent monograph [BE13] covers many of these results until 2013. We only cover a few selected newer results, and, e.g., completely spare the large body of work on edge colorings or colorings of special graph classes. ...
We give a randomized -coloring algorithm in the LOCAL model that runs in rounds, where n is the number of nodes of the input graph and is its maximum degree. This means that randomized -coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, , given the known lower bound [Brandt et al., STOC '16]. Our main technical contribution is a constant time reduction to a constant number of -list coloring instances, for , resulting in a -round CONGEST algorithm for such graphs. This reduction is of independent interest for other settings, including providing a new proof of Brooks' theorem for high degree graphs, and leading to a constant-round Congested Clique algorithm in such graphs. When , our algorithm even runs in rounds, showing that the base in the lower bound is unavoidable. Previously, the best LOCAL algorithm for all considered settings used a logarithmic number of rounds. Our result is the first CONGEST algorithm for -coloring non-constant degree graphs.
... Distributed graph coloring is a fruitful and fast evolving area, here we briefly discuss some most relevant work, interested readers can refer to, e.g., the monograph by Barenboim and Elkin [5] for more details. ...
... It has been shown that r * ≤ log * n + O(1). (See, e.g., Section 3.10 of [5].) During the Linial phase, vertices will reduce the number of colors used to n i after i rounds, thus within r * ≤ log * n + O(1) rounds the algorithm produces a proper O(∆ 2 )-coloring. ...
Distributed graph coloring is one of the most extensively studied problems in distributed computing. There is a canonical family of distributed graph coloring algorithms known as the locally-iterative coloring algorithms, first formalized in the seminal work of [Szegedy and Vishwanathan, STOC'93]. In such algorithms, every vertex iteratively updates its own color according to a predetermined function of the current coloring of its local neighborhood. Due to the simplicity and naturalness of its framework, locally-iterative coloring algorithms are of great significance both in theory and practice. In this paper, we give a locally-iterative -coloring algorithm with running time. This is the first locally-iterative -coloring algorithm with sublinear-in- running time, and answers the main open question raised in a recent breakthrough [Barenboim, Elkin, and Goldberg, JACM'21]. A key component of our algorithm is a locally-iterative procedure that transforms an -coloring to a -coloring in time. Inside this procedure we work on special proper colorings that encode (arb)defective colorings, and reduce the number of used colors quadratically in a locally-iterative fashion. As a main application of our result, we also give a self-stabilizing distributed algorithm for -coloring with stabilization time. To the best of our knowledge, this is the first self-stabilizing algorithm for -coloring with sublinear-in- stabilization time.
... Distributed solutions of combinatorial optimization problems on graphs have been intensely studied in the last two decades. Research was aimed at fast vertex coloring [2,3], fast construction of maximal independent sets [1,7,12], of dominating and k-dominating sets [8,9], and of minimum weight spanning trees [4,9]. Various communication models have been used, ranging from the LOCAL model used in this paper, to the CON GEST model in which messages must be of logarithmic size [9], the radio network model [12], and to the highly contrived beeping model [1] in which a node can transmit only one beep in each round. ...
A k-dominating set is a set D of nodes of a graph such that, for each node v, there exists a node at distance at most k from v. Our aim is the deterministic distributed construction of small T-dominating sets in time T in networks modeled as undirected n-node graphs and under the communication model. For any positive integer T, if b is the size of a pairwise disjoint collection of balls of radii at least T in a graph, then b is an obvious lower bound on the size of a T-dominating set. Our first result shows that, even on rings, it is impossible to construct a T-dominating set of size s asymptotically b (i.e., such that ) in time T. In the range of time , the size of a T-dominating set turns out to be very sensitive to multiplicative constants in running time. Indeed, it follows from \cite{KP}, that for time with large constant , it is possible to construct a T-dominating set whose size is a small fraction of n. By contrast, we show that, for time for small constant , the size of a T-dominating set must be a large fraction of n. Finally, when , the above lower bound implies that, for any constant , it is impossible to construct a T-dominating set of size smaller than xn, even on rings. On the positive side, we provide an algorithm that constructs a T-dominating set of size on all graphs.
... Barenboim and Elkin [BE13] provide a thorough tour on coloring algorithms (though there are some additional recent results). An excellent survey on local problems is given by Suomela [Suo13]. ...
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 -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 deterministic maximal independent set (MIS) algorithm in the CONGEST model, where the communication graph is identical to the input graph, in rounds, where D is the diameter of the graph. The best known running time in terms of n alone is , which is super-polylogarithmic, and requires large messages. For the CONGEST model, the only known previous solution is a coloring-based -round algorithm, where is the maximal degree in the graph. On the way to obtaining the above, we show that in the \emph{Congested Clique} model, which allows all-to-all communication, there is a deterministic MIS algorithm that runs in rounds.%, where is the maximum degree. When , the bound improves to and holds also for -coloring. In addition, we deterministically construct a -spanner with edges in rounds. For comparison, in the more stringent CONGEST model, the best deterministic algorithm for constructing a -spanner with edges runs in rounds.
... Rand4DColor is based on a very simple randomized algorithm well-known for message-passing models since decades [16,Chapter 10]. It has recently been proven to be efficient in the SINR model by Fuchs and Prutkin [2]. ...
In this paper we evaluate distributed node coloring algorithms for wireless networks using the network simulator Sinalgo [by DCG@ETHZ]. All considered algorithms operate in the realistic signal-to-interference-and-noise-ratio (SINR) model of interference. We evaluate two recent coloring algorithms, Rand4DColor and ColorReduction (in the following ColorRed), proposed by Fuchs and Prutkin in [SIROCCO'15], the MW-Coloring algorithm introduced by Moscibroda and Wattenhofer [DC'08] and transferred to the SINR model by Derbel and Talbi [ICDCS'10], and a variant of the coloring algorithm of Yu et al. [TCS'14]. We additionally consider several practical improvements to the algorithms and evaluate their performance in both static and dynamic scenarios. Our experiments show that Rand4DColor is very fast, computing a valid (4Degree)-coloring in less than one third of the time slots required for local broadcasting, where Degree is the maximum node degree in the network. Regarding other O(Degree)-coloring algorithms Rand4DColor is at least 4 to 5 times faster. Additionally, the algorithm is robust even in networks with mobile nodes and an additional listening phase at the start of the algorithm makes Rand4DColor robust against the late wake-up of large parts of the network. Regarding (Degree+1)-coloring algorithms, we observe that ColorRed it is significantly faster than the considered variant of the Yu et al. coloring algorithm, which is the only other (Degree+1)-coloring algorithm for the SINR model. Further improvement can be made with an error-correcting variant that increases the runtime by allowing some uncertainty in the communication and afterwards correcting the introduced conflicts.
... There are already many resources on various aspects of local algorithms: the classical book of Peleg [Pel00], survey of Suomela [Suo13], book of Barenboim and Elkin [BE13], introductory text of Suomela [Suo20], or a recent book by Hirvonen and Suomela [HS20]. Unlike other texts, this one primarily explores the field's conceptual framework and complexity-theoretical aspects, rather than delving into individual problems. ...
This text provides an introduction to the field of distributed local algorithms -- an area at the intersection of theoretical computer science and discrete mathematics. We collect many recent results in the area and demonstrate how they lead to a clean theory. We also discuss many connections of local algorithms to areas such as parallel, distributed, and sublinear algorithms, or descriptive combinatorics.
... There is an abundance of efficient deterministic and randomized ∆ + 1-coloring algorithms in LOCAL and CONGEST for various settings, e.g., [Bar15, HSS18, FHK16, CLP18, BKM20, RG20, MT20, HKMT21, HN21, HKNT22, FK23]. The excellent monograph on distributed graph coloring by Barenboim and Elkin is still a great resource for older results [BE13]. ...
We consider the problem of coloring graphs of maximum degree with colors in the distributed setting with limited bandwidth. Specifically, we give a -round randomized algorithm in the CONGEST model. This is close to the lower bound of rounds from [Brandt et al., STOC '16], which holds also in the more powerful LOCAL model. The core of our algorithm is a reduction to several special instances of the constructive Lov\'asz local lemma (LLL) and the deg+1-list coloring problem.
... Distributed algorithms for GCP are extensively analyzed and discussed in [11], focusing on both deterministic and randomized instances. A representative example of a distributed deterministic algorithm for the GCP is presented in [12], managing to conclude a graph coloring in linear (Δ ) time. ...
In conjunction with the traffic overload of next-generation wireless communication and computer networks, their resource-constrained nature calls for effective methods to deal with the fundamental resource allocation problem. In this context, the Minimum Collisions Assignment (MCA) problem in an interdependent networked system refers to the assignment of a finite set of resources over the nodes of the network, such that the number of collisions, i.e., the number of interdependent nodes receiving the same resource, is minimized. Given the interdependent networked system's organization in the form of a graph, there already exists a randomized algorithm that converges with high probability to an assignment of resources having zero collisions when the number of resources is larger than the maximum degree of the underlying graph. In this article, differing from the prevailing literature, we investigate the case of a resource-constrained networked system, where the number of resources is less than or equal to the maximum degree of the underlying graph. We introduce two distributed, randomized algorithms that converge in a logarithmic number of rounds to an assignment of resources over the network for which every node has at most a certain number of collisions. The proposed algorithms apply to settings where the available resources at each node are equal to three and two, respectively, while they are executed in a fully-distributed manner without requiring information exchange between the networked nodes.
... Coloring problems are amongst the most intensively studied problems in the distributed graph literature for they capture the main challenges of symmetry breaking (see, e.g., [BE13]). Symmetry breaking on power graphs appears naturally in numerous settings [KMR01, BEPS16, Gha16, Gha19, EM19, FGG + 23]. ...
We provide a -round randomized algorithm for distance-2 coloring in CONGEST with colors. For , this improves exponentially on the algorithm of [Halld\'orsson, Kuhn, Maus, Nolin, DISC'20]. Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lov\'asz Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.
... Since the (∆ + 1)-coloring problem can be solved by a simple sequential greedy algorithm, but it is challenging to be solved efficiently in distributed (and parallel) setting, the (∆ + 1)-coloring problem became a benchmark problem for distributed computing and a significant amount of research has been devoted to the study of these problems in all main distributed models: LOCAL, CONGEST, and CongestedClique. The monograph [BE13] gives a comprehensive description of many of the earlier results. ...
We consider the distributed complexity of the (degree+1)-list coloring problem, in which each node u of degree d(u) is assigned a palette of d(u)+1 colors, and the goal is to find a proper coloring using these color palettes. The (degree+1)-list coloring problem is a natural generalization of the classical -coloring and -list coloring problems, both being benchmark problems extensively studied in distributed and parallel computing. In this paper we settle the complexity of the (degree+1)-list coloring problem in the Congested Clique model by showing that it can be solved deterministically in a constant number of rounds.
... For matching approximation, the aforementioned work of Bar-Yehuda et al. [BYCHGS17] achieves a complexity of O(log ∆/ log log ∆) for (1 + ε)-approximation, which is the optimal round complexity for ∆ = O( √ log n), matching a lower bound of Kuhn et al. [KMW04,KMW16]. There are many other algorithmic results for MIS and matching for other restricted graph families, e.g., graphs of bounded arboricity [BE10,BE13], growth bounded graphs [SW08], etc. ...
Chatterjee, Gmyr, and Pandurangan [PODC 2020] recently introduced the notion of awake complexity for distributed algorithms, which measures the number of rounds in which a node is awake. In the other rounds, the node is sleeping and performs no computation or communication. Measuring the number of awake rounds can be of significance in many settings of distributed computing, e.g., in sensor networks where energy consumption is of concern. In that paper, Chatterjee et al. provide an elegant randomized algorithm for the Maximal Independent Set (MIS) problem that achieves an O(1) node-averaged awake complexity. That is, the average awake time among the nodes is O(1) rounds. However, to achieve that, the algorithm sacrifices the more standard round complexity measure from the well-known bound of MIS, due to Luby [STOC'85], to rounds. Our first contribution is to present a simple randomized distributed MIS algorithm that, with high probability, has O(1) node-averaged awake complexity and worst-case round complexity. Our second, and more technical contribution, is to show algorithms with the same O(1) node-averaged awake complexity and worst-case round complexity for -approximation of maximum matching and -approximation of minimum vertex cover, where denotes an arbitrary small positive constant.
... It however remained open for over three decades whether there also is a deterministic poly log n-time algorithm for MIS, and this came to be known as Linial's open question, first raised in [Lin87,Lin92]. See also the book of Barenboim and Elkin [BE13] for further discussions on the significance of this and other related problems. ...
We develop a general deterministic distributed method for locally rounding fractional solutions of graph problems for which the analysis can be broken down into analyzing pairs of vertices. Roughly speaking, the method can transform fractional/probabilistic label assignments of the vertices into integral/deterministic label assignments for the vertices, while approximately preserving a potential function that is a linear combination of functions, each of which depends on at most two vertices (subject to some conditions usually satisfied in pairwise analyses). The method unifies and significantly generalizes prior work on deterministic local rounding techniques [Ghaffari, Kuhn FOCS'21; Harris FOCS'19; Fischer, Ghaffari, Kuhn FOCS'17; Fischer DISC'17] to obtain polylogarithmic-time deterministic distributed solutions for combinatorial graph problems. Our general rounding result enables us to locally and efficiently derandomize a range of distributed algorithms for local graph problems, including maximal independent set (MIS), maximum-weight independent set approximation, and minimum-cost set cover approximation. As a highlight, we in particular obtain a deterministic -round algorithm for computing an MIS in the LOCAL model and an almost as efficient -round deterministic MIS algorithm in the CONGEST model. As a result, the best known deterministic distributed time complexity of the four most widely studied distributed symmetry breaking problems (MIS, maximal matching, -vertex coloring, and -edge coloring) is now . Our new MIS algorithm is also the first direct polylogarithmic-time deterministic distributed MIS algorithm, which is not based on network decomposition.
... Graph coloring has a rich history in both graph theory and computer science. We refer the interested reader to excellent texts by Molloy and Reed [205] and by Barenboim and Elkin [36] for an extensive background on this problem. ...
As the scale of the problems we want to solve in real life becomes larger, the input sizes of the problems we want to solve could be much larger than the memory of a single computer. In these cases, the classical algorithms may no longer be feasible options, even when they run in linear time and linear space, as the input size is too large. In this thesis, we study various combinatorial problems in different computation models that process large input sizes using limited resources. In particular, we consider the query model, streaming model, and massively parallel computation model. In addition, we also study the tradeoffs between the adaptivity and performance of algorithms in these models.We first consider two graph problems, vertex coloring problem and metric traveling salesman problem (TSP). The main results are structure results for these problems, which give frameworks for achieving sublinear algorithms of these problems in different models. We also show that the sublinear algorithms for (∆ + 1)-coloring problem are tight. We then consider the graph sparsification problem, which is an important technique for designing sublinear algorithms. We give proof of the existence of a linear size hypergraph cut sparsifier, along with a polynomial algorithm that calculates one. We also consider sublinear algorithms for this problem in the streaming and query models. Finally, we study the round complexity of submodular function minimization (SFM). In particular, we give a polynomial lower bound on the number of rounds we need to compute s − t max flow - a special case of SFM - in the streaming model. We also prove a polynomial lower bound on the number of rounds we need to solve the general SFM problem in polynomial queries.
This paper investigates the application of graph theory and variants of greedy graph coloring algorithms for the optimization of distributed peer-to-peer networks, with a special focus on private blockchain networks. The graph coloring problem, as an NP-hard problem, presents a challenge in determining the minimum number of colors needed to efficiently allocate resources within the network. The paper deals with the influence of different graph density, i.e., the number of links, on the efficiency of greedy algorithms such as DSATUR, Descending, and Ascending. Experimental results show that increasing the number of links in the network contributes to a more uniform distribution of colors and increases the resistance of the network, whereby the DSATUR algorithm achieves the most uniform color saturation. The optimal configuration for a 100-node network has been identified at around 2000 to 2500 links, which achieves stability without excessive redundancy. These results are applied in the context of a private blockchain network that uses optimal connectivity to achieve high resilience and efficient resource allocation. The research findings suggest that adapting network configuration using greedy algorithms can contribute to the optimization of distributed systems, making them more stable and resilient to loads.
By prior work, there is a distributed algorithm that finds a maximal fractional matching (maximal edge packing) in rounds, where is the maximum degree of the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in rounds. Our work gives the first linear-in- lower bound for a natural graph problem in the standard model of distributed computing---prior lower bounds for a wide range of graph problems have been at best logarithmic in .
We show that every Borel graph G G of subexponential growth has a Borel proper edge-coloring with Δ ( G ) + 1 \Delta (G) + 1 colors. We deduce this from a stronger result, namely that an n n -vertex (finite) graph G G of subexponential growth can be properly edge-colored using Δ ( G ) + 1 \Delta (G) + 1 colors by an O ( log ∗ n ) O(\log ^\ast n) -round deterministic distributed algorithm in the LOCAL model, where the implied constants in the O ( ⋅ ) O(\cdot ) notation are determined by a bound on the growth rate of G G .
Graph coloring is fundamental to distributed computing. We give the first general treatment of the coloring of virtual graphs, where the graph H to be colored is locally embedded within the communication graph G. Besides generalizing classical distributed graph coloring (where H=G), this captures other previously studied settings, including cluster graphs and power graphs. We find that the complexity of coloring a virtual graph depends on the edge congestion of its embedding. The main question of interest is how fast we can color virtual graphs of constant congestion. We find that, surprisingly, these graphs can be colored nearly as fast as ordinary graphs. Namely, we give a -round algorithm for the deg+1-coloring problem, where each node is assigned more colors than its degree. This can be viewed as a case where a distributed graph problem can be solved even when the operation of each node is decentralized.
The Graph Coloring Using Bacterial Foraging Optimization (BFO) algorithm efficiently addresses the graph coloring problem by ensuring adjacent nodes are assigned distinct colors. Utilizing BFO, we conducted experiments on graphs ranging from 10 to 100 nodes, demonstrating its effectiveness and scalability. Results show that the algorithm consistently achieved valid colorings with minimal chromatic numbers: 3 colors for 10 and 25 nodes, 6 colors for 50 and 75 nodes, and 7 colors for 100 nodes. Execution times were notably short, with the fastest at 0.004464 seconds for 10 nodes and 0.007576 seconds for 100 nodes. These findings underscore BFO's computational efficiency and suitability for graph coloring tasks across various graph sizes, suggesting its potential for broader applications in optimization and graph theory.
Fault-tolerance is a central theme in distributed computing. Self-stabilization is a key property that guarantees a distributed system starting from an arbitrary state eventually converges to a desired behavior. Such strong level of fault-tolerance is often desirable due to the error-prone nature of distributed systems. Developing fast and robust coloring algorithms has been a central topic in the study of distributed graph algorithms. In this paper, we give a -coloring algorithm with stabilization time, only using messages of size , on input graphs of n vertices and maximum degree . This is the first self-stabilizing -coloring algorithm with sublinear-in- stabilization time. The key building block of our algorithm is a new locally-iterative -coloring algorithm with runtime. To the best of our knowledge, this is the first locally-iterative -coloring algorithm with sublinear-in- runtime. This answers an open question raised in [Barenboim, Elkin, and Goldberg, JACM ’21].
In this paper, we present a deterministic 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 k ≥ 1 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, Elkin, Goldenberg, PODC’18], and various algorithms to compute defective and arbdefective colorings. Our algorithm can smoothly scale between several of these previous results and also simplifies the state of the art ( Δ + 1)-coloring algorithm. At the cost of losing some of the algorithm’s simplicity we also provide a O ( kΔ )-coloring algorithm in rounds. We also provide improved deterministic algorithms for ruling sets, and, additionally, we provide a tight characterization for 1-round color reduction algorithms.
In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made well-behaved in the sense of topology or measure. To this end, we show that such well-behaved colorings can be produced using certain powerful techniques from finite combinatorics and computer science. First, we prove that efficient distributed coloring algorithms (on finite graphs) yield well-behaved colorings of Borel graphs of bounded degree; roughly speaking, deterministic algorithms produce Borel colorings, while randomized algorithms give measurable and Baire-measurable colorings. Second, we establish measurable and Baire-measurable versions of the Symmetric Lovász Local Lemma (under the assumption p(d+1)8⩽2−15, which is stronger than the standard LLL assumption p(d+1)⩽e−1 but still sufficient for many applications). From these general results, we derive a number of consequences in descriptive combinatorics and ergodic theory.
Graph edge coloring is a fundamental problem in graph theory and has been widely used in a variety of applications. Existing solutions for edge coloring mainly focus on static graphs. However, many graphs in real world are highly dynamic. Motivated by this, we study the dynamic edge coloring problem in this paper. Since edge coloring is NP-Complete, to obtain an effective dynamic edge coloring, we aim to incrementally maintain the edge coloring in a way such that the coloring result is consistent with one of the best approximate static edge coloring algorithms when the graph is dynamically updated. Unfortunately, our theoretical result shows that the problem of finding such dynamic graph edge coloring is unbounded. Despite this, we propose an efficient dynamic edge coloring algorithm that only explores the edges with color change and their 2-hop incident edges to maintain the coloring. Moreover, we propose some early pruning rules to further reduce the unnecessary computation. Experimental results on real graphs demonstrate the efficiency of our approach.
Combinatorial test generation, also called t t ‐way testing, is the process of generating sets of input parameters for a system under test, by considering interactions between values of multiple parameters. In order to decrease total testing time, there is an interest in techniques that generate smaller test suites. In our previous work, we used graph techniques to produce high‐quality test suites. However, these techniques require a lot of computing power and memory, which is why this paper investigates distributed computing for t t ‐way testing. We first introduce our distributed graph colouring method, with new algorithms for building the graph and for colouring it. Second, we present our distributed hypergraph vertex covering method and a new heuristic. Third, we show how to build a distributed IPOG algorithm by leveraging either graph colouring or hypergraph vertex covering as vertical growth algorithms. Finally, we test these new methods on a computer cluster and compare them to existing t t ‐way testing tools. Combinatorial test generation is the process of generating sets of input parameters for a system under test, by considering interactions between t values of multiple parameters; the paper investigates the use of distributed algorithms to generate such test suites. It proposes reductions of the t t ‐way test suite generation to two problems on graphs and provides distributed algorithms to solve them; these algorithms are then used as vertical growth algorithms to build a distributed version of IPOG. Finally, we test these new methods on a computer cluster and compare them to existing t t ‐way testing tools.
Kabupaten Kediri adalah kabupaten yang cukup berkembang, bahkan saat ini perkembangan kabupaten Kediri tergolong pesat. Kabupaten Kediri memiliki 26 kecamatan, kecamatan tersebut saling terhubung. Hubunngan antar kecamatan ini yang akan mempengaruhi perkembanganya. Pada saat ini kabupaten Kediri melakukan pembangunan yang pesat. Pembangunan kabupaten Kediri dilaksanakan pada kecamatan yang memiliki letak yang strategis, mudah diakses oleh kecamatan disekitarnya. Kecamatan yang memliki degree banyak maka kecamatan tersebut bisa dijadikan proritas pengembngan kabupaten Kediri. Karena kecamatan yang memiliki degree banyak pasti sangat mudah diakses oleh kecamatan disekitarnya. Dengan memanfaatkan metode graph coloring jumlah vertek dan edgee yang ada pemerintah kabupaten Kediri dapat mengetahui posisi setiap kecamatan dengan mudah dan mengetahui ke strategisan setiap kecamatan. Setelah dilakukan pencarian degree terbanyak maka pemerintah kabupaten Kediri dapat melakukan pembangunan pada kecamatan tersebut. Dengan melakukan pembangunan pada kecamatan yang memiliki degree terbanyak (strategis), pembangunan pusat ekonomi, kesehatan dan pendidikan akan dilakukan pada kecamatan tersebut akan berimbas pada kecamatan disekitarnya, selain itu kecamatan yang memiliki degree banyak sangat mudah diakses oleh kecamatan disekitarnya.
Kata kunci: edgee , graph coloring, pemetaan, vertex.
Given a network property or a data structure, a local certification is a labeling that allows to efficiently check that the property is satisfied, or that the structure is correct. The quality of a certification is measured by the size of its labels: the smaller, the better. This notion plays a central role in self-stabilization, because the size of the certification is a lower bound (and often an upper bound) on the memory needed for silent self-stabilizing construction of distributed data structures.When it comes to the size of the certification labels, one can identify three important regimes: the properties for which the optimal size is polynomial in the number of vertices of the graph, the ones that require only polylogarithmic size, and the ones that can be certified with a constant number of bits. The first two regimes are well studied, with several upper and lower bounds, specific techniques, and active research questions. On the other hand, the constant regime has never been really explored.The main contribution of this paper is the first non-trivial lower bound for this low regime. More precisely, we show that by using certification on just one bit (a binary certification), one cannot certify k-colorability for . To do so, we develop a new technique, based on the notion of score, and both local symmetry arguments and a global parity argument. We hope that this technique will be useful for establishing stronger results.We complement this result with an upper bound for a related problem, illustrating that in some cases one can do better than the natural upper bound.
This paper presents significantly improved deterministic algorithms for some of the key problems in the area of distributed graph algorithms, including network decomposition, hitting sets, and spanners. As the main ingredient in these results, we develop novel randomized distributed algorithms that we can analyze using only pairwise independence, and we can thus derandomize efficiently. As our most prominent end-result, we obtain a deterministic construction for -color -strong diameter network decomposition in rounds. This is the first construction that achieves almost in both parameters, and it improves on a recent line of exciting progress on deterministic distributed network decompositions [Rozho\v{n}, Ghaffari STOC'20; Ghaffari, Grunau, Rozho\v{n} SODA'21; Chang, Ghaffari PODC'21; Elkin, Haeupler, Rozho\v{n}, Grunau FOCS'22].
We study algorithms in the LOCAL model that produce secured output. Specifically, each vertex computes its part in the output, the entire output is correct, but each vertex cannot discover the output of other vertices, with a certain probability. As the extensive research in the distributed algorithms field yielded efficient decentralized algorithms, the discussion about the security of distributed algorithms was somewhat neglected. Nevertheless, many protocols and algorithms were devised in the research area of secure multi-party computation problem. However, the focus in those protocols was to work for every function f at the expense of increasing the round complexity, or the necessity of several computational assumptions. We present a novel approach, which identifies and develops those algorithms that are inherently secure (which means they do not require any further constructions). This approach yields efficient secure algorithms for various labeling and decomposition problems without requiring any hardness assumption, but only a private randomness generator in each vertex.
The rapid development of Internet of Things (IoT) technologies has not only enabled new applications, but also presented new challenges for reliable communication with limited resources. In this work, we define a novel problem that can arise in these scenarios, in which a set of sensors need to communicate a joint observation. This observation is shared by a random subset of the nodes, which need to propagate it to the rest of the network, but coordination is complex: as signaling constraints require the use of random access schemes over shared channels, sensors need to implicitly coordinate, so that at least one transmission gets through without collisions. Unlike the majority of existing medium access schemes, the goal is to make sure that the shared message gets through, regardless of the sender. We analyze this coordination problem theoretically and provide low-complexity solutions. While a clustering-based approach is near-optimal if the sensors have prior knowledge, we provide a distributed multi-armed bandit (MAB) solution for the more general case and validate it by simulation.
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