Distrib Comput Research Articles

Voting-based probabilistic consensuses and their applications in distributed ledgers

We review probabilistic models known as majority dynamics (also known as threshold voter models) and discuss their possible applications for achieving consensus in cryptocurrency systems. In particular, we show that using this approach in a straightforward way for practical consensus in a Byzantine setting can be problematic and requires extensive further research. We then discuss the Fast Probabilistic Consensus (FPC) protocol (Popov and Buchanan, J Parallel Distrib Comput 147:77–86, 2021), which circumvents the problems mentioned above by using external randomness.

Optimal distributed covering algorithms

We present a time-optimal deterministic distributed algorithm for approximating a minimum weight vertex cover in hypergraphs of rank f. This problem is equivalent to the Minimum Weight Set Cover problem in which the frequency of every element is bounded by f. The approximation factor of our algorithm is (f+varepsilon ). Let varDelta denote the maximum degree in the hypergraph. Our algorithm runs in the congest model and requires O(log {varDelta } / log log varDelta ) rounds, for constants varepsilon in (0,1] and fin {mathbb {N}}^+. This is the first distributed algorithm for this problem whose running time does not depend on the vertex weights nor the number of vertices. Thus adding another member to the exclusive family of provably optimal distributed algorithms. For constant values of f and varepsilon , our algorithm improves over the (f+varepsilon )-approximation algorithm of Kuhn et al. (SODA, 2006)whose running time is O(log varDelta + log W), where W is the ratio between the largest and smallest vertex weights in the graph. Our algorithm also achieves an f-approximation for the problem in O(flog n) rounds, improving over the classical result of Khuller et al. (J Algorithms, 1994) that achieves a running time of O(flog ^2 n). Finally, for weighted vertex cover (f=2) our algorithm achieves a deterministic running time of O(log n), matching the randomized previously best result of Koufogiannakis and Young (Distrib Comput, 2011). We also show that integer covering-programs can be reduced to the Minimum Weight Set Cover problem in the distributed setting. This allows us to achieve an (flceil log _2(M)+1 rceil +varepsilon )-approximate integral solution in O(1+f/logn)·logΔloglogΔ+(f·logM)1.01·logε-1·(logΔ)0.01\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} O\\left( (1+f/\\log n)\\cdot \\left( {\\frac{\\log \\varDelta }{ \\log \\log \\varDelta } + ({f\\cdot \\log M})^{1.01}\\cdot \\log \\varepsilon ^{-1}\\cdot (\\log \\varDelta )^{0.01}}\\right) \\right) \\end{aligned}$$\\end{document}rounds, where f bounds the number of variables in a constraint, varDelta bounds the number of constraints a variable appears in, and M=max left{ 1, lceil 1/a_{min } rceil right} , where a_{min } is the smallest normalized constraint coefficient.

The complexity of verifying population protocols

Population protocols (Angluin et al. in PODC, 2004) are a model of distributed computation in which indistinguishable, finite-state agents interact in pairs to decide if their initial configuration, i.e., the initial number of agents in each state, satisfies a given property. In a seminal paper Angluin et al. classified population protocols according to their communication mechanism, and conducted an exhaustive study of the expressive power of each class, that is, of the properties they can decide (Angluin et al. in Distrib Comput 20(4):279–304, 2007). In this paper we study the correctness problem for population protocols, i.e., whether a given protocol decides a given property. A previous paper (Esparza et al. in Acta Inform 54(2):191–215, 2017) has shown that the problem is decidable for the main population protocol model, but at least as hard as the reachability problem for Petri nets, which has recently been proved to have non-elementary complexity. Motivated by this result, we study the computational complexity of the correctness problem for all other classes introduced by Angluin et al., some of which are less powerful than the main model. Our main results show that for the class of observation models the complexity of the problem is much lower, ranging from varPi _2^p to PSPACE.

Parallel Algorithms for Successive Convolution

The development of modern computing architectures with ever-increasing amounts of parallelism has allowed for the solution of previously intractable problems across a variety of scientific disciplines. Despite these advances, multiscale computing problems continue to pose an incredible challenge to modern architectures because they require resolving scales that often vary by orders of magnitude in both space and time. Such complications have led us to consider alternative discretizations for partial differential equations (PDEs) which use expansions involving integral operators to approximate spatial derivatives (Christlieb et al. in J Comput Phys 379:214–236, 2019; Christlieb et al. J Sci Comput 82:52(3):1–29, 2020; Christlieb et al. J Comput Phys 415:1–25, 2020). These constructions use explicit information within the integral terms, but treat boundary data implicitly, which contributes to the overall speed of the method. This approach is provably unconditionally stable for linear problems and stability has been demonstrated experimentally for nonlinear problems. Additionally, it is matrix-free in the sense that it is not necessary to invert linear systems and iteration is not required for nonlinear terms. Moreover, the scheme employs a fast summation algorithm that yields a method with a computational complexity of $${\mathcal {O}}(N)$$ , where N is the number of mesh points along a coordinate direction. While much work has been done to explore the theory behind these methods, their practicality in large scale computing environments is a largely unexplored topic. In this work, we explore the performance of these methods by developing a domain decomposition algorithm suitable for distributed memory systems along with shared memory algorithms. As a first pass, we derive an artificial Courant–Friedrichs–Lewy condition that enforces a nearest-neighbor (N-N) communication pattern and briefly discuss possible generalizations. We also analyze several approaches for implementing the parallel algorithms by optimizing predominant loop structures and maximizing data reuse. Using a hybrid design that employs MPI and Kokkos (Edwards and Trott in J Parallel Distrib Comput 74:3202–3216, 2014) for the distributed and shared memory components of the algorithms, respectively, we show that our methods are efficient and can sustain an update rate $$> 1\times 10^8$$ DOF/node/s. We provide results that demonstrate the scalability and versatility of our algorithms using several different PDE test problems, including a nonlinear example, which employs an adaptive time-stepping rule.

Ordered and delayed adversaries and how to work against them on a shared channel

An execution of a distributed algorithm is often seen as a game between the algorithm and a conceptual adversary causing specific distractions to the computation. In this work we define a class of ordered adaptive adversaries, which cause distractions—in particular crashes—online according to some partial order of the participating stations, which is fixed by the adversary before the execution. We distinguish: Linearly-Ordered adversary, restricted by some pre-defined linear order of (potentially) crashing stations; Anti-Chain-Ordered adversary, previously known as the Weakly-Adaptive adversary, which is restricted by some pre-defined set of crash-prone stations (it can be seen as an ordered adversary with the order being an anti-chain, i.e., a collection of incomparable elements, consisting of these stations); k-Thick-Ordered adversary restricted by partial orders of stations with a maximum anti-chain of size k. We initiate a study of how they affect performance of algorithms. For this purpose, we focus on the well-known Do-All problem of performing t tasks by p synchronous crash-prone stations communicating on a shared channel. The channel restricts communication by the fact that no message is delivered to the operational stations if more than one station transmits at the same time. The question addressed in this work is how the ordered adversaries controlling crashes of stations influence work performance, defined as the total number of available processor steps during the whole execution and introduced by Kanellakis and Shvartsman (Distrib Comput 5(4):201–217, 1992) in the context of Write-All algorithms. The first presented algorithm solves the Do-All problem with work {mathcal {O}}(t+p sqrt{t}log p) against the Linearly-Ordered adversary. Surprisingly, the upper bound on performance of this algorithm does not depend on the number of crashes f and is close to the absolute lower bound varOmega (t+psqrt{t}) proved in Chlebus et al. (Distrib Comput 18(6):435–451, 2006). Another algorithm is developed against the Weakly-Adaptive adversary. Work done by this algorithm is mathcal {O}(t + psqrt{t} + pmin left{ p/(p-f),tright} log p ), which is close to the lower bound varOmega (t + psqrt{t} + pmin left{ p/(p-f),tright} ) proved in [11] and answers the open questions posed there. We generalize this result to the class of k-Thick-Ordered adversaries, in which case the work of the algorithm is bounded by mathcal {O}(t + psqrt{t} + pmin left{ p/(p-f),k,tright} log p ). We complement this result by proving the almost matching lower bound Ω(t+pt+pminp/(p-f),k,t).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\varOmega (t + p\\sqrt{t} + p\\min \\left\\{ p/(p-f),k,t\\right\\} ). \\end{aligned}$$\\end{document}Independently from the results for the ordered adversaries, we consider a class of delayed adaptive adversaries, which could see random choices with some delay. We present an algorithm that works efficiently against the 1-RD adversary, which could see random choices of stations with one round delay, achieving close to optimal {mathcal {O}}(t+p sqrt{t}log ^{2} p) work complexity. This shows that restricting the adversary by not allowing it to react on random decisions immediately makes it significantly weaker, in the sense that there is an algorithm achieving (almost) optimal work performance.

Randomized proof-labeling schemes

Proof-labeling schemes, introduced by Korman et al. (Distrib Comput 22(4):215–233, 2010. https://doi.org/10.1007/s00446-010-0095-3 ), are a mechanism to certify that a network configuration satisfies a given boolean predicate. Such mechanisms find applications in many contexts, e.g., the design of fault-tolerant distributed algorithms. In a proof-labeling scheme, predicate verification consists of neighbors exchanging labels, whose contents depends on the predicate. In this paper, we introduce the notion of randomized proof-labeling schemes where messages are randomized and correctness is probabilistic. We show that randomization reduces verification complexity exponentially while guaranteeing probability of correctness arbitrarily close to one. We also present a novel message-size lower bound technique that applies to deterministic as well as randomized proof-labeling schemes. Using this technique, we establish several tight bounds on the verification complexity of MST, acyclicity, connectivity, and longest cycle size.

Upper and lower bounds for deterministic broadcast in powerline communication networks

Powerline communication networks assume an interesting position in the communication network space: similarly to wireless networks, powerline networks are based on a shared broadcast medium; unlike wireless networks, however, the signal propagation is constrained to the power lines of the electrical infrastructure, which is essentially a graph. This article presents an algorithmic model to study the design of communication services over powerline communication networks. As a case study, we focus on the fundamental broadcast problem, and present and analyze a distributed algorithm $$\textsc {Color}\textsc {Cast}$$COLORCAST which terminates in at most n communication rounds, where n denotes the network size, even in a model where link qualities are unpredictable and time-varying. For comparison, the achieved broadcast time is lower than what can be achieved by any unknown-topology algorithm (lower bounds $$\varOmega (n\log n / \log (n/D))$$Ω(nlogn/log(n/D)) and $$\varOmega (n \log D)$$Ω(nlogD) are proved in Kowalski and Pelc (Distrib Comput 18(1):43---57, 2005) resp. Clementi et al. (2001) where D is the network diameter). Moreover, existing known-topology broadcast algorithms often fail to deliver the broadcast message entirely in this model. This article also presents a general broadcast lower bound for the powerline model.

Towards an integrated formal method for verification of liveness properties in distributed systems: with application to population protocols

State-based formal methods [e.g. Event-B/RODIN (Abrial in Modeling in Event-B—system and software engineering. Cambridge University Press, Cambridge, 2010; Abrial et al. in Int J Softw Tools Technol Transf (STTT) 12(6):447–466, 2010)] for critical system development and verification are now well established, with track records including tool support and industrial applications. The focus of proof-based verification, in particular, is on safety properties. Liveness properties, which guarantee eventual, or converging computations of some requirements, are less well dealt with. Inductive reasoning about liveness is not explicitly supported. Liveness proofs are often complex and expensive, requiring high-skill levels on the part of the verification engineer. Fairness-based temporal logic approaches have been proposed to address this, e.g. TLA Lamport (ACM Trans Program Lang Syst 16(3):872–923, 1994) and that of Manna and Pnueli (Temporal verification of reactive systems—safety. Springer, New York, 1995). We contribute to this technology need by proposing a fairness-based method integrating temporal and first-order logic, proof and tools for modelling and verification of safety and liveness properties. The method is based on an integration of Event-B and TLA. Building on our previous work (Mery and Poppleton in Integrated formal methods, 10th international conference, IFM 2013, Turku, Finland, pp 208–222, 2013. doi: 10.1007/978-3-642-38613-8_15 ), we present the method via three example population protocols Angluin et al. (Distrib Comput 18(4):235–253, 2006). These were proposed as a theoretical framework for computability reasoning about Wireless Sensor Network and Mobile Ad-Hoc Network algorithms. Our examples present typical liveness and convergence requirements. We prove convergence results for the examples by integrated modelling and proof with Event-B/RODIN and TLA. We exploit existing proof rules, define and apply three new proof rules; soundness proofs are also provided. During the process we observe certain repeating patterns in the proofs. These are easily identified and reused because of the explicit nature of the reasoning.

Minimal output unstable configurations in chemical reaction networks and deciders

We study the set of output stable configurations of chemical reaction deciders (CRDs). It turns out that CRDs with only bimolecular reactions (which are almost equivalent to population protocols) have a special structure that allows for an algorithm to efficiently compute their finite set of minimal output unstable configurations. As a consequence, a relatively large set of configurations may be efficiently checked for output stability. We also provide a number of observations regarding the semilinearity result of Angluin et al. (Distrib Comput 20(4):279–304, 2007) from the context of population protocols (which is a central result for output stable CRDs). In particular, we observe that the computation-friendly class of totally stable CRDs has equal expressive power as the larger class of output stable CRDs.

An optimal boundary fair scheduling

Nowadays, many real-time operating systems discretize the time relying on a system time unit. To take this behavior into account, real-time scheduling algorithms must adopt a discrete-time model in which both timing requirements of tasks and their time allocations have to be integer multiples of the system time unit. That is, tasks cannot be executed for less than one time unit, which implies that they always have to achieve a minimum amount of work before they can be preempted. Assuming such a discrete-time model, the authors of Zhu et al. (Proceedings of the 24th IEEE international real-time systems symposium (RTSS 2003), 2003, J Parallel Distrib Comput 71(10):1411–1425, 2011) proposed an efficient “boundary fair” algorithm (named BF) and proved its optimality for the scheduling of periodic tasks while achieving full system utilization. However, BF cannot handle sporadic tasks due to their inherent irregular and unpredictable job release patterns. In this paper, we propose an optimal boundary-fair scheduling algorithm for sporadic tasks (named BF\(^2\)), which follows the same principle as BF by making scheduling decisions only at the job arrival times and (expected) task deadlines. This new algorithm was implemented in Linux and we show through experiments conducted upon a multicore machine that BF\(^2\) outperforms the state-of-the-art discrete-time optimal scheduler (PD\(^2\)), benefiting from much less scheduling overheads. Furthermore, it appears from these experimental results that BF\(^2\) is barely dependent on the length of the system time unit while PD\(^2\)—the only other existing solution for the scheduling of sporadic tasks in discrete-time systems—sees its number of preemptions, migrations and the time spent to take scheduling decisions increasing linearly when improving the time resolution of the system.

Structuring unreliable radio networks

In this paper we study the problem of building a constant-degree connected dominating set (CCDS), a network structure that can be used as a communication backbone, in the dual graph radio network model (Clementi et al. in J Parallel Distrib Comput 64:89–96, 2004; Kuhn et al. in Proceedings of the international symposium on principles of distributed computing 2009, Distrib Comput 24(3–4):187–206 2011, Proceedings of the international symposium on principles of distributed computing 2010). This model includes two types of links: reliable, which always deliver messages, and unreliable, which sometimes fail to deliver messages. Real networks compensate for this differing quality by deploying low-layer detection protocols to filter unreliable from reliable links. With this in mind, we begin by presenting an algorithm that solves the CCDS problem in the dual graph model under the assumption that every process \(u\) is provided with a local link detector set consisting of every neighbor connected to \(u\) by a reliable link. The algorithm solves the CCDS problem in \(O\left( \frac{\varDelta \log ^2{n}}{b} + \log ^3{n}\right) \) rounds, with high probability, where \(\varDelta \) is the maximum degree in the reliable link graph, \(n\) is the network size, and \(b\) is an upper bound in bits on the message size. The algorithm works by first building a Maximal Independent Set (MIS) in \(\log ^3{n}\) time, and then leveraging the local topology knowledge to efficiently connect nearby MIS processes. A natural follow-up question is whether the link detector must be perfectly reliable to solve the CCDS problem. With this in mind, we first describe an algorithm that builds a CCDS in \(O(\varDelta \)polylog\((n))\) time under the assumption of \(O(1)\) unreliable links included in each link detector set. We then prove this algorithm to be (almost) tight by showing that the possible inclusion of only a single unreliable link in each process’s local link detector set is sufficient to require \(\varOmega (\varDelta )\) rounds to solve the CCDS problem, regardless of message size. We conclude by discussing how to apply our algorithm in the setting where the topology of reliable and unreliable links can change over time.

A linear system form solution to compute the local space average color

In this document, we present an alternative to the method introduced by Ebner (Pattern Recognit 60---67, 2003; J Parallel Distrib Comput 64(1):79---88, 2004; Color constancy using local color shifts, pp 276---287, 2004; Color Constancy, 2007; Mach Vis Appl 20(5):283---301, 2009) for computing the local space average color. We show that when the problem is framed as a linear system and the resulting series is solved, there is a solution based on LU decomposition that reduces the computing time by at least an order of magnitude.

Toward more localized local algorithms: removing assumptions concerning global knowledge

Numerous sophisticated local algorithm were suggested in the literature for various fundamental problems. Notable examples are the MIS and $$(\Delta +1)$$ -coloring algorithms by Barenboim and Elkin (Distrib Comput 22(5–6):363–379, 2010), by Kuhn (2009), and by Panconesi and Srinivasan (J Algorithms 20(2):356–374, 1996), as well as the $$O\mathopen {}(\Delta ^2)$$ -coloring algorithm by Linial (J Comput 21:193, 1992). Unfortunately, most known local algorithms (including, in particular, the aforementioned algorithms) are non-uniform, that is, local algorithms generally use good estimations of one or more global parameters of the network, e.g., the maximum degree $$\Delta $$ or the number of nodes $$n$$ . This paper provides a method for transforming a non-uniform local algorithm into a uniform one. Furthermore, the resulting algorithm enjoys the same asymptotic running time as the original non-uniform algorithm. Our method applies to a wide family of both deterministic and randomized algorithms. Specifically, it applies to almost all state of the art non-uniform algorithms for MIS and Maximal Matching, as well as to many results concerning the coloring problem (In particular, it applies to all aforementioned algorithms). To obtain our transformations we introduce a new distributed tool called pruning algorithms, which we believe may be of independent interest.

Distributed deterministic edge coloring using bounded neighborhood independence

We study the edge-coloring problem in the message-passing model of distributed computing. This is one of the most fundamental problems in this area. Currently, the best-known deterministic algorithms for (2Δ −1)-edge-coloring requires O(Δ) + log* n time (Panconesi and Rizzi in Distrib Comput 14(2):97–100, 2001), where Δ is the maximum degree of the input graph. Also, recent results of Barenboim and Elkin (2010) for vertex-coloring imply that one can get an O(Δ)-edge-coloring in $${O(\Delta^{\epsilon}\cdot \log n)}$$ time, and an $${O(\Delta^{1 + \epsilon})}$$ -edge-coloring in O(log Δ log n) time, for an arbitrarily small constant $${\epsilon > 0}$$ . In this paper we devise a significantly faster deterministic edge-coloring algorithm. Specifically, our algorithm computes an O(Δ)-edge-coloring in $${O(\Delta^{\epsilon}) + \log* n}$$ time, and an $${O(\Delta^{1 + \epsilon})}$$ -edge-coloring in O(log Δ) + log* n time. This result improves the state-of-the-art running time for deterministic edge-coloring with this number of colors in almost the entire range of maximum degree Δ. Moreover, it improves it exponentially in a wide range of Δ, specifically, for 2 Ω(log*n) ≤ Δ ≤ polylog(n). In addition, for small values of Δ (up to log1 - δ n, for some fixed δ > 0) our deterministic algorithm outperforms all the existing randomized algorithms for this problem. Also, our algorithm is the first O(Δ)-edge-coloring algorithm that has running time o(Δ) + log* n, for the entire range of Δ. All previous (deterministic and randomized) O(Δ)-edge-coloring algorithms require $${\Omega(\min \{\Delta, \sqrt{\log n}\ \})}$$ time. On our way to these results we study the vertex-coloring problem on graphs with bounded neighborhood independence. This is a large family of graphs, which strictly includes line graphs of r-hypergraphs (i.e., hypergraphs in which each hyperedge contains r or less vertices) for r = O(1), and graphs of bounded growth. We devise a very fast deterministic algorithm for vertex-coloring graphs with bounded neighborhood independence. This algorithm directly gives rise to our edge-coloring algorithms, which apply to general graphs. Our main technical contribution is a subroutine that computes an O(Δ/p)-defective p-vertex coloring of graphs with bounded neighborhood independence in O(p 2) + log* n time, for a parameter p, 1 ≤ p ≤ Δ. In all previous efficient distributed routines for m-defective p-coloring the product m· p is super-linear in Δ. In our routine this product is linear in Δ, and this enables us to speed up the algorithm drastically.

Detecting the phenomenon of strange non-chaotic attractors

I comment on how chaos might be defined. A sample of dynamical systems that have quasi-periodic forcing functions is then considered. The normal approach found in the literature is to start with an ordinary differential equation, change to a difference equation, and then plot a graph. The question of how to detect a strange non-chaotic attractor without the underlying ordinary differential equation is posed and some pointers are given as to a possible method of solution using statistical analysis. References Banks J., Brooks J., Cairns G., Davis G. and Stacey P., On Devaney's definition of chaos, American Mathematics Monthly . 99 (1992), 332--334. Box G. E. P, Jenkins G. M. and Reinsel G. C., Time series analysis John Wiley and Sons, New Jersy, 2008. Feitelson D. G., Workload modeling for performance evaluation, In: Lecture notes in computer science , vol. 2459, Springer, Berlin, 2002, 114--141. Glendinning P., The nonsmooth pitchfork bifurcation, Discrete and Continuous Dynamical Systems B . 4 (2004), 457--464. Glendinning P., T. H. Jager and G. Keller, How chaotic are strange non-chaotic attractors?, Nonlinearity . 19 (2006), 2005--2022. Glendinning P., View from the Pennines: strange nonchaotic attractors, Mathematics Today . 44 (2008), 119--120. Keller G., A note on strange nonchaotic attractors, Fundamenta Mathematicae . 151 (1996), 139--148. Li H., Workload dynamics on clusters and grids, Journal of Supercomputing . 47) (2009), 1--20. Lublin U., Feitelson D. G., The workload of parallel supercomputers: modeling the characteristics of rigid jobs, J. Parallel Distrib Comput . 63(11) (2003), 1105--1122. Squillante M. S., Yao D. D and Zhang L. The impact of job arrival patterns on parallel scheduling, ACM SIGMETRICS Perform Eval Rev. 26(4) (1999), 52--59.

A knowledge-based analysis of global function computation

Consider a distributed system N in which each agent has an input value and each communication link has a weight. Given a global function, that is, a function f whose value depends on the whole network, the goal is for every agent to eventually compute the value f (N). We call this problem global function computation. Various solutions for instances of this problem, such as Boolean function computation, leader election, (minimum) spanning tree construction, and network determination, have been proposed, each under particular assumptions about what processors know about the system and how this knowledge can be acquired. We give a necessary and sufficient condition for the problem to be solvable that generalizes a number of well-known results (Attyia et al. in J ACM 35(4):845–875, 1988; Yamashita and Kameda in IEEE Trans Parallel Distrib Syst 7(1):69–89, 1996; Yamashita and Kameda in IEEE Trans Parallel Distrib Syst 10(9):878–887, 1999). We then provide a knowledge-based (kb) program (like those of Fagin et al. (Reasoning about knowledge, MIT Press, Cambridge, 1995, Distrib Comput 10(4):199–225, 1997)) that solves global function computation whenever possible. Finally, we improve the message overhead inherent in our initial kb program by giving a counterfactual belief-based program (Halpern and Moses in Distrib Comput 17(2):91–106, 2004) that also solves the global function computation whenever possible, but where agents send messages only when they believe it is necessary to do so. The latter program is shown to be implemented by a number of well-known algorithms for solving leader election.

On the performance of Dijkstra’s third self-stabilizing algorithm for mutual exclusion and related algorithms

In Dijkstra (Commun ACM 17(11):643–644, 1974) introduced the notion of self-stabilizing algorithms and presented three such algorithms for the problem of mutual exclusion on a ring of n processors. The third algorithm is the most interesting of these three but is rather non intuitive. In Dijkstra (Distrib Comput 1:5–6, 1986) a proof of its correctness was presented, but the question of determining its worst case complexity—that is, providing an upper bound on the number of moves of this algorithm until it stabilizes—remained open. In this paper we solve this question and prove an upper bound of \({3\frac{13}{18} n^2 + O(n)}\) for the complexity of this algorithm. We also show a lower bound of \({1\frac{5}{6} n^2 - O(n)}\) for the worst case complexity. For computing the upper bound, we use two techniques: potential functions and amortized analysis. We also present a new-three state self-stabilizing algorithm for mutual exclusion and show a tight bound of \({\frac{5}{6} n^2 + O(n)}\) for the worst case complexity of this algorithm. In Beauquier and Debas (Proceedings of the second workshop on self-stabilizing systems, pp 17.1–17.13, 1995) presented a similar three-state algorithm, with an upper bound of \({5\frac{3}{4}n^2+O(n)}\) and a lower bound of \({\frac{1}{8}n^2-O(n)}\) for its stabilization time. For this algorithm we prove an upper bound of \({1\frac{1}{2}n^2 + O(n)}\) and show a lower bound of n2−O(n). As far as the worst case performance is considered, the algorithm in Beauquier and Debas (Proceedings of the second workshop on self-stabilizing systems, pp 17.1–17.13, 1995) is better than the one in Dijkstra (Commun ACM 17(11):643–644, 1974) and our algorithm is better than both.

The impact of randomization in smoothing networks

We revisit randomized smoothing networks (Herlihy and Tirthapura in J Parallel Distrib Comput 66(5):626–632, 2006), which are made up of balancers and wires. We assume that balancers are oriented independently and uniformly at random. Tokens arrive arbitrarily on w input wires and propagate asynchronously through the network; each token is served on the output wire it arrives at. The smoothness is the maximum discrepancy among the numbers of tokens arriving at the w output wires. We present a collection of lower and upper bounds on smoothness, which are to some extent surprising: (1) The smoothness of a single block network is lg lg w+Θ(1) (with high probability), where the additive constant is between −2 and 3. This tight bound improves vastly over the upper bound of \({\mathcal{O}(\sqrt{lg\, w})}\) from Herlihy and Tirthapura (2006), and it essentially settles our understanding of the smoothing properties of the block network. Further, this lg lg w+Θ(1) bound cannot be extended to universal smoothing a property stronger than smoothing that we introduce. (2) Most significantly, the smoothness of the cascade of two block networks is no more than 17 (with high probability); this is the first known randomized network with so small depth (2 lg w) and so good (constant) smoothness. The proof introduces some combinatorial and probabilistic structures and techniques which may be further applicable; the result demonstrates the full power of randomization in smoothing networks. (3) There is no randomized 1-smoothing network of width w and depth d that achieves 1-smoothness with probability better than \({\frac{\mathsf{d}} {w-1} }\) . In view of the deterministic, Θ(lg w)-depth, 1-smoothing network in Klugerman and Plaxton (Proceedings of the 24th Annual ACM Symposium on Theory of Computing, 1992), this result implies the first separation between deterministic and randomized smoothing networks. These results demonstrate an unexpected limitation on the power of randomization for smoothing networks: although it yields constant smoothness using small depth, going down to smoothness of 1 requires linear depth.