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1. Byzantine Consensus in Abstract MAC Layer

   https://doi.org/10.4230/LIPICS.OPODIS.2023.9  

   Tseng, Lewis; Sardina, Callie (January 2024, International Conference on Principles of Distributed Systems)

   Bessani, Alysson; Défago, Xavier; Nakamura, Junya; Wada, Koichi; Yamauchi, Yukiko (Ed.)

   This paper studies the design of Byzantine consensus algorithms in an asynchronous single-hop network equipped with the "abstract MAC layer" [DISC09], which captures core properties of modern wireless MAC protocols. Newport [PODC14], Newport and Robinson [DISC18], and Tseng and Zhang [PODC22] study crash-tolerant consensus in the model. In our setting, a Byzantine faulty node may behave arbitrarily, but it cannot break the guarantees provided by the underlying abstract MAC layer. To our knowledge, we are the first to study Byzantine faults in this model. We harness the power of the abstract MAC layer to develop a Byzantine approximate consensus algorithm and a Byzantine randomized binary consensus algorithm. Both of our algorithms require only the knowledge of the upper bound on the number of faulty nodes f, and do not require the knowledge of the number of nodes n. This demonstrates the "power" of the abstract MAC layer, as consensus algorithms in traditional message-passing models require the knowledge of both n and f. Additionally, we show that it is necessary to know f in order to reach consensus. Hence, from this perspective, our algorithms require the minimal knowledge. The lack of knowledge of n brings the challenge of identifying a quorum explicitly, which is a common technique in traditional message-passing algorithms. A key technical novelty of our algorithms is to identify "implicit quorums" which have the necessary information for reaching consensus. The quorums are implicit because nodes do not know the identity of the quorums - such notion is only used in the analysis. 

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2. Fault-Tolerant Consensus with an Abstract MAC Layer

   Newport, Calvin; Robinson, Peter (October 2018, Proceedings of the International Symposium on Distributed Computing (DISC))

   In this paper, we study fault-tolerant distributed consensus in wireless systems. In more detail, we produce two new randomized algorithms that solve this problem in the abstract MAC layer model, which captures the basic interface and communication guarantees provided by most wireless MAC layers. Our algorithms work for any number of failures, require no advance knowledge of the network participants or network size, and guarantee termination with high probability after a number of broadcasts that are polynomial in the network size. Our first algorithm satisfies the standard agreement property, while our second trades a faster termination guarantee in exchange for a looser agreement property in which most nodes agree on the same value. These are the first known fault-tolerant consensus algorithms for this model. In addition to our main upper bound results, we explore the gap between the abstract MAC layer and the standard asynchronous message passing model by proving fault-tolerant consensus is impossible in the latter in the absence of information regarding the network participants, even if we assume no faults, allow randomized solutions, and provide the algorithm a constant-factor approximation of the network size. 

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3. Consensus with Max Registers

   https://doi.org/http://dx.doi.org/10.4230/LIPIcs.DISC.2019.1  

   Aspnes, James; Er, He Yang (October 2019, Leibniz international proceedings in informatics)

   We consider the problem of implementing randomized wait-free consensus from max registers under the assumption of an oblivious adversary. We show that max registers solve m-valued consensus for arbitrary m in expected O(log^* n) steps per process, beating the Omega(log m/log log m) lower bound for ordinary registers when m is large and the best previously known O(log log n) upper bound when m is small. A simple max-register implementation based on double-collect snapshots translates this result into an O(n log n) expected step implementation of m-valued consensus from n single-writer registers, improving on the best previously-known bound of O(n log^2 n) for single-writer registers. 

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4. Sleeping is Efficient: MIS in O(1)-rounds Node-averaged Awake Complexity

   https://doi.org/10.1145/3382734.3405718  

   Chatterjee, Soumyottam; Gmyr, Robert; Pandurangan, Gopal (July 2020, PODC '20: Proceedings of the 39th Symposium on Principles of Distributed Computing)

   null (Ed.)

   Maximal Independent Set (MIS) is one of the fundamental problems in distributed computing. The round (time) complexity of distributed MIS has traditionally focused on the worst-case time for all nodes to finish. The best-known (randomized) MIS algorithms take O(log n) worst-case rounds on general graphs (where n is the number of nodes). Breaking the O(log n) worst-case bound has been a longstanding open problem, while currently the best-known lower bound is [EQUATION] rounds. Motivated by the goal to reduce total energy consumption in energy-constrained networks such as sensor and ad hoc wireless networks, we take an alternative approach to measuring performance. We focus on minimizing the total (or equivalently, the average) time for all nodes to finish. It is not clear whether the currently best-known algorithms yield constant-round (or even o(log n)) node-averaged round complexity for MIS in general graphs. We posit the sleeping model, a generalization of the traditional model, that allows nodes to enter either "sleep" or "waking" states at any round. While waking state corresponds to the default state in the traditional model, in sleeping state a node is "offline", i.e., it does not send or receive messages (and messages sent to it are dropped as well) and does not incur any time, communication, or local computation cost. Hence, in this model, only rounds in which a node is awake are counted and we are interested in minimizing the average as well as the worst-case number of rounds a node spends in the awake state, besides the traditional worst-case round complexity (i.e., the rounds for all nodes to finish including both the awake and sleeping rounds). Our main result is that we show that MIS can be solved in (expected) O(1) rounds under node-averaged awake complexity measure in the sleeping model. In particular, we present a randomized distributed algorithm for MIS that has expected O(1)-rounds node-averaged awake complexity and, with high probability1 has O(log n)-rounds worst-case awake complexity and O(log3.41 n)-rounds worst-case complexity. Our work is a step towards understanding the node-averaged complexity of MIS both in the traditional and sleeping models, as well as designing energy-efficient distributed algorithms for energy-constrained networks. 

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5. Symmetry Breaking in the CONGEST Model: Time- and Message-Efficient Algorithms for Ruling Sets

   Pai, S; Pandurangan, G; Pemmaraju, S.; Riaz, T; Robinson, P. (October 2017, 31st International Symposium on Distributed Computing (DISC), 2017)

   We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round “barrier” achieved by Luby’s algorithm, but these yield o(log n)-round complexity only when the maximum degree
   is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby’s algorithm) even for moderately small
   (i.e., for
   = (log n) and
   = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby’s algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the (log n) time complexity barrier and the (m) message complexity barrier in the Congest model for MIS or closelyrelated symmetry breaking problems? This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A -ruling set is an independent set such that every node in the graph is at most hops from a node in the independent set. We present the following results: Time Complexity: We show that we can break the O(log n) “barrier” for 2- and 3-ruling sets. We compute 3-ruling sets in O  log n log log n  rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O  log
   · (log n)1/2+" + log n log log n  rounds for any " > 0, which is o(log n) for a wide range of
   values (e.g.,
   = 2(log n)1/2−" ). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby’s algorithm in the Congest model. Message Complexity: We show an (n2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log2 n) messages and runs in O(
   log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor). 

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