Introduction to coding theory (2nd edition), by J.H. van Lint. Pp.183 DM86. 1992. ISBN 0-387-54894-7 (Springer)

FPC-BI: Fast Probabilistic Consensus within Byzantine Infrastructures

On coalescence time in graphs: When is coalescing as fast as meeting?: Extended Abstract.

We present improved analyses for two of the well-studied randomized dynamics of stabilizing consensus, namely 2-choice and 3-majority. The resulting bounds are tight up to logarithmic factors. In the latter case, this answers an open question of Becchetti et al. [SODA'16]. Consider a distributed system of n nodes, each initially holding an opinion in \1, 2, ...., k\ . The system should converge to a setting where all (non-corrupted) nodes hold the same opinion. This consensus opinion should be valid, meaning that it should be among the initially supported opinions, and the (fast) convergence should happen even in the presence of a malicious adversary who can corrupt a bounded number of nodes per round and in particular modify their opinions. Two of the well-studied distributed algorithms for this problem work as follows: In both of these dynamics, each node gathers the opinion of three nodes and sets its new opinion equal to the majority of this set. In the 2-choice dynamics, the three nodes are the node itself and two random nodes and ties are broken towards the node's own opinion. In the 3-majority dynamics, the three nodes are selected at random and ties are broken randomly. Becchetti et al. [SODA'16] showed that the 3-majority dynamics converges to consensus in O((k^2\sq√rtlog n + klog n)(k+log n)) rounds, even in the presence of a limited adversary, for k bounded by some polynomial of n. We prove that, even with a stronger adversary, the convergence happens within O(klog n) rounds, both for 3-majority and 2-choice. For 3-majority, this bound is known to be optimal for k=\tildeO (√ ). More generally, we prove that 3-majority converges always in O (n^2/3 ) time, improving on a O (n^3/4 ) upper bound of Berenbrink et al. [PODC'17].

Nearly-Tight Analysis for 2-Choice and 3-Majority Consensus Dynamics

Plurality consensus considers a network of n nodes, each having one of k opinions. Nodes execute a (randomized) distributed protocol with the goal that all nodes adopt the plurality (the opinion initially supported by the most nodes). Communication is realized via the Gossip (or random phone call) model. A major open question has been whether there is a protocol for the complete graph that converges (w.h.p.) in polylogarithmic time and uses only polylogarithmic memory per node (local memory). We answer this question affirmatively. We propose two protocols that need only mild assumptions on the bias in favor of the plurality. As an example of our results, consider the complete graph and an arbitrarily small constant multiplicative bias in favor of the plurality. Our first protocol achieves plurality consensus in O(logk·loglogn) rounds using logk + Θ(loglogk) bits of local memory. Our second protocol achieves plurality consensus in O(logn·loglogn) rounds using only logk + 4 bits of local memory. This disproves a conjecture by Becchetti et al. (SODA'15) implying that any protocol with local memory logk+O(1) has worst-case runtime Ω(k). We provide similar bounds for much weaker bias assumptions. At the heart of our protocols lies an undecided state, an idea introduced by Angluin et al. (Distributed Computing'08).

https://hal.archives-ouvertes.fr/hal-01353690/document

Efficient plurality consensus, or : the benefits of cleaning up from time to time.

We consider distributed plurality consensus on a complete graph of size n with k initial opinions in the following asynchronous communication model. Each node is equipped with a random Poisson clock with parameter lambda=1. Whenever a node's clock ticks, it samples some neighbors uniformly at random and adjusts its opinion according to the sample. Distributed plurality consensus has been deeply studied in the synchronous communication model. A prominent example is the so-called Two-Choices protocol, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient when k=2. If k=O(ne) for some small epsilon, we show that it converges to the initial plurality opinion within O(k log n) rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is Omega(sqrt(n log n)). On the negative side, we show that there are cases in which Omega(k) rounds are needed, w.h.p. To beat this lower bound, we combine the Two-Choices protocol with push-pull broadcasting. We divide the process into several phases, where each phase consists of a two-choices round followed by several broadcasting rounds. Our main contribution is a non-trivial adaptation of this approach to the above asynchronous model. If the support of the most frequent opinion is at least (1+e) times that of the second-most frequent one and k=O(Exp(log n / log log n)), then our protocol achieves the best possible run time of O(log n), w.h.p. Key to our adaptation is that we relax full synchronicity by allowing o(n) nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this sufficient synchronicity by introducing a novel gadget into the protocol. Other parts of the adaptation are made to work using arguments and techniques based on a Polya urn model.

/pdf/brief-announcement-rapid-asynchronous-plurality-consensus-w2e83cl2bc.pdf

Brief Announcement: Rapid Asynchronous Plurality Consensus

We consider the problem of distributed $k$-party voting with two choices as well as a simple modification of this protocol in complete graphs. In the standard version, we are given a graph in which every node possesses one of $k$ different opinions at the beginning. In each step, every node chooses two neighbors uniformly at random. If the opinions of the two neighbors coincide, then this opinion is adopted. It is known that if $k=2$ and the difference between the two opinions is $\Omega(\sqrt{n \log n})$, then after $\mathcal{O}(\log n)$ steps, every node will possess the largest initial opinion, with high probability. 
We show that if $k =\mathcal{O}(n^\epsilon)$ for some small $\epsilon$, then this protocol converges to the initial majority within $\mathcal{O}(k\log{n})$ steps, with high probability, as long as the initial difference between the largest and second largest opinion is $\Omega(\sqrt{n \log n})$. Furthermore, there exist initial configurations where the $\Theta(k)$ bound on the run time is matched. If the initial difference is $\mathcal{O}(\sqrt{n})$, then the largest opinion may loose the vote with constant probability. To speed up our process, we consider the following variant of the two-choices protocol. The process is divided into several phases, and in the first step of a phase every node applies the two choices protocol. If a new opinion is adopted, the node remembers it by setting a certain bit to true. In the subsequent steps of that phase, each node samples one neighbor, and if the bit of this neighbor is set to true, then the node takes the opinion of this neighbor and sets its bit to true as well. At the end of the phase, the bits are reset to false. Then, the phases are repeated several times. We show that this modified protocol improves significantly over the standard two-choices protocol.

pdf/efficient-k-party-voting-with-two-choices-2tcecqvo6p.pdf

Efficient k-Party Voting with Two Choices.

We consider distributed plurality consensus in a complete graph of size $n$ with $k$ initial opinions. We design an efficient and simple protocol in the asynchronous communication model that ensures that all nodes eventually agree on the initially most frequent opinion. In this model, each node is equipped with a random Poisson clock with parameter $\lambda=1$. Whenever a node's clock ticks, it samples some neighbors, uniformly at random and with replacement, and adjusts its opinion according to the sample. 
A prominent example is the so-called two-choices algorithm in the synchronous model, where in each round, every node chooses two neighbors uniformly at random, and if the two sampled opinions coincide, then that opinion is adopted. This protocol is very efficient and well-studied when $k=2$. If $k=O(n^\varepsilon)$ for some small $\varepsilon$, we show that it converges to the initial plurality opinion within $O(k \cdot \log{n})$ rounds, w.h.p., as long as the initial difference between the largest and second largest opinion is $\Omega(\sqrt{n \log n})$. On the other side, we show that there are cases in which $\Omega(k)$ rounds are needed, w.h.p. 
One can beat this lower bound in the synchronous model by combining the two-choices protocol with randomized broadcasting. Our main contribution is a non-trivial adaptation of this approach to the asynchronous model. If the support of the most frequent opinion is at least $(1+\varepsilon)$ times that of the second-most frequent one and $k=O(\exp(\log{n}/\log \log{n}))$, then our protocol achieves the best possible run time of $O(\log n)$, w.h.p. We relax full synchronicity by allowing $o(n)$ nodes to be poorly synchronized, and the well synchronized nodes are only required to be within a certain time difference from one another. We enforce this synchronicity by introducing a novel gadget into the protocol.

Rapid Asynchronous Plurality Consensus

In (Can J Math 51(2):326---346, 1999), Martin and Stinson provide a generalized MacWilliams identity for linear ordered orthogonal arrays and linear ordered codes (introduced by Rosenbloom and Tsfasman (Prob Inform Transm 33(1):45---52, 1997) as "codes for the m-metric") using association schemes. We give an elementary proof of this generalized MacWilliams identity using group characters and use it to derive an explicit formula for the dual type distribution of a linear ordered code or orthogonal array.

https://link.springer.com/content/pdf/10.1007%2Fs10623-008-9226-2.pdf

A simple derivation of the MacWilliams identity for linear ordered codes and orthogonal arrays

The triple distribution of codes and ordered codes

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