Tight Bounds for Connectivity and Set Agreement in Byzantine Synchronous Systems

In this paper, we show that the protocol complex of a Byzantine synchronous system can remain $(k - 1)$-connected for up to $\lceil t/k \rceil$ rounds, where $t$ is the maximum number of Byzantine processes, and $t \ge k \ge 1$. This topological property implies that $\lceil t/k \rceil + 1$ rounds are necessary to solve $k$-set agreement in Byzantine synchronous systems, compared to $\lfloor t/k \rfloor + 1$ rounds in synchronous crash-failure systems. We also show that our connectivity bound is tight as we indicate solutions to Byzantine $k$-set agreement in exactly $\lceil t/k \rceil + 1$ synchronous rounds, at least when $n$ is suitably large compared to $t$. In conclusion, we see how Byzantine failures can potentially require one extra round to solve $k$-set agreement, and, for $n$ suitably large compared to $t$, at most that

Some New Results With k-set agreement

In this article, we investigate the solvability of $k$-set agreement among $n$ processes in distributed systems prone to different types of process failures. Specifically, we explore two scenarios: synchronous message-passing systems prone to up to $t$ Byzantine failures of processes. And asynchronous shared memory systems prone to up to $t$ crash failures of processes. Our goal is to address the gaps left by previous works\cite{SSS,AsynchKset} in these areas. For Byzantine failures case we consider systems with authentication where processes have unforgeable signatures. For synchronous message-passing systems, we present an authenticated algorithm that achieves $k$-set agreement in only two rounds, with no constraints on the number of faults $t$, with $k$ determined as $k \geq \lfloor \frac{n}{n-t} \rfloor + 1$. In fact the lower bound for $k$ is $k \geq \lfloor \frac{n}{n-t} \rfloor$ that is obtained by an algorithm based on traditional consensus with $t+1$ rounds. In asynchronous shared memory systems, we introduce an algorithm that accomplishes $k$-set agreement for values of $k$ greater than $\lfloor \frac{n-t}{n-2t} \rfloor +1$. This algorithm uses a snapshot primitive to handle crash failures and enable effective set agreement

A Topological Treatment of Early-Deciding Set-Agreement

This paper considers the k-set-agreement problem in a synchronous message passing distributed system where up to t processes can fail by crashing. We determine the number of communication rounds needed for all correct processes to reach a decision in a given run, as a function of k, the degree of coordination, and f <= t the number of processes that actually fail in the run. We prove a lower bound of min(\floor{f/k}+2,\floor{t/k}+1) rounds. Our proof uses simple topological tools to reason about runs of a full information set-agreement protocol. In particular, we introduce a new topological operator, which we call the early deciding operator, to capture rounds where k processes fail but correct processes see only k-1 failures

From a Static Impossibility to an Adaptive Lower Bound: the Complexity of Early Deciding Set Agreement

In the year 2007 a reform was established that allowed private citizens in Sweden to make tax deductions on companies providing services pertaining to the household (called RUT-deduction). The year later another reform was introduced granting citizens additional tax deductions but this time concerning household renovation, reconstruction and extensive construction (called ROT-deduction). These tax deductions resulted in higher employment and more jobs being executed legally. The purpose of this paper is to examine and analyze what kind of effects these types of tax deductions would have on workers’ real wages and to look at to what extent these effects differ within a female dominated occupation and a male dominated occupation, from a gender perspective. The two professions that are chosen to be researched in this paper are the cleaning and painting professions. Furthermore, the purpose with this study is to examine whether this effect differs within the two separate professions. The study is executed with the use of econometric models, point estimation, economic theory and empirical studies. The result indicates that the RUT-deduction has the biggest positive impact on cleaners’ real wages. This paper shows that one underlying reason to this outcome could be that the cleaning service is a more price sensitive service and that the RUT-deduction might therefore have generated an excess in demand for that service. Nonetheless, to establish an equilibrium in the labor market the wages are required to rise in order to attract more cleaners to enter the certain market. However, this paper is unable to eliminate the possible theory of there being a general wage increase among workers in the private sector. In addition, this study is comprised of an inadequate amount of observations which impedes any reliable conclusions from being made based on evidence

Practical scalable consensus for pseudo-synchronous distributed systems

The ability to consistently handle faults in a distributed en-vironment requires, among a small set of basic routines, an agreement algorithm allowing surviving entities to reach a consensual decision between a bounded set of volatile re-sources. This paper presents an algorithm that implements an Early Returning Agreement (ERA) in pseudo-synchronous systems, which optimistically allows a process to resume its activity while guaranteeing strong progress. We prove the correctness of our ERA algorithm, and expose its logarith-mic behavior, which is an extremely desirable property for any algorithm which targets future exascale platforms. We detail a practical implementation of this consensus algorithm in the context of an MPI library, and evaluate both its effi-ciency and scalability through a set of benchmarks and two fault tolerant scientific applications. CCS Concepts •Computing methodologies → Distributed algorithms; •Computer systems organization→Reliability; Fault-tolerant network topologies; •Software and its engi-neering → Software fault tolerance

Conditions for Set Agreement with an Application to Synchronous Systems

The $k$-set agreement problem is a generalization of the consensus problem: considering a system made up of $n$ processes where each process proposes a value, each non-faulty process has to decide a value such that a decided value is a proposed value, and no more than $k$ different values are decided. While this problem cannot be solved in an asynchronous system prone to $t$ process crashes when $t \geq k$, it can always be solved in a synchronous system; $\lfloor \frac{t}{k} \rfloor +1$ is then a lower bound on the number of rounds (consecutive communication steps) for the non-faulty processes to decide. The {\it condition-based} approach has been introduced in the consensus context. Its aim was to both circumvent the consensus impossibility in asynchronous systems, and allow for more efficient consensus algorithms in synchronous systems. This paper addresses the condition-based approach in the context of the $k$-set agreement problem. It has two main contributions. The first is the definition of a framework that allows defining conditions suited to the $\ell$-set agreement problem. More precisely, a condition is defined as a set of input vectors such that each of its input vectors can be seen as ``encoding'' $\ell$ values, namely, the values that can be decided from that vector. A condition is characterized by the parameters $t$, $\ell$, and a parameter denoted $d$ such that the greater $d+\ell$, the least constraining the condition (i.e., it includes more and more input vectors when $d+\ell$ increases, and there is a condition that includes all the input vectors when $d+\ell>t$). The conditions characterized by the triple of parameters $t$, $d$ and $\ell$ define the class of conditions denoted ${\cal S}_t^{d,\ell}$, $0\leq d\leq t$, $1\leq \ell \leq n-1$. The properties of the sets ${\cal S}_t^{d,\ell}$are investigated, and it is shown that they have a lattice structure. The second contribution is a generic synchronous $k$-set agreement algorithm based on a condition $C\in {\cal S}_t^{d,\ell}$, i.e., a condition suitedto the $\ell$-set agreement problem, for $\ell \leq k$. This algorithm requires at most $\left\lfloor \frac{d-1+\ell}{k} \right\rfloor +1$ rounds when the input vector belongs to $C$, and $\left\lfloor \frac{t}{k} \right\rfloor +1$ rounds otherwise. (Interestingly, this algorithm includes as particular cases the classical synchronous $k$-set agreement algorithm that requires $\left\lfloor \frac{t}{k} \right\rfloor+1$ rounds (case $d=t$ and $\ell=1$), and the synchronous consensus condition-based algorithm that terminates in $d+1$ rounds when the input vector belongs to the condition, and in $t+1$ rounds otherwise (case $k=\ell=1$).

Tight bounds for k-set agreement

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