Reaching approximate agreement with mixed-mode faults

Abstract

In a fault-tolerant distributed system, different nonfaulty processes may arrive at different values for a given system parameter. To resolve this disagreement, processes must exchange and vote upon their respective local values. Faulty processes may attempt to inhibit agreement by acting in a malicious or "Byzantine" manner. Approximate Agreement defines one form of agreement in which the voted values obtained by the nonfaulty processes need not be identical. Instead, they need only agree to within a predefined tolerance. Approximate Agreement can be achieved by a sequence of convergent voting rounds, in which the range of values held by nonfaulty processes is reduced in each round.Historically, each new convergent voting algorithm has been accompanied by &-hoc proofs of its convergence rate and faulttolerance, using an overly conservative fault model in which aU faults exhibit worst-case Byzantine behavior. This paper presents a general method to quickly determine convergence rate and fault-tolerance for any member of a broad family of convergent voting algorithms. This method is developed under a realistic mixed-mode fault model comprised of asymmetric, symmetric, and benign fault modes. These results are employed to more accurately analyze the properties of several existing voting algorithms, to derive a sub-family of optimal mixed-mode voting algorithms, and to quickly determine the properties of proposed new voting algorithms.

Resilient Asymptotic Consensus in Robust Networks

“…There are a few key differences between the operations used in the W-MSR algorithm and the MSR algorithm of [30]. First, W-MSR does not always remove the largest and smallest F values as in the MSR algorithm [30]. Instead, only the extreme values that are strictly larger or strictly smaller than the given node's value are removed.…”

“…Besides Byzantine faults, some works also consider other threat models [30]. However, few papers have addressed the convergence of MSR algorithms in less restrictive (noncomplete) networks.…”

“…These approximation functions are used in the family of Mean-SubsequenceReduced (MSR) algorithms [30]. There are a few key differences between the operations used in the W-MSR algorithm and the MSR algorithm of [30]. First, W-MSR does not always remove the largest and smallest F values as in the MSR algorithm [30].…”

“…They consider only complete networks (where there is a direct connection between every pair of nodes), and they propose the following algorithm: each node disregards the largest and smallest F values received from its neighbors and updates its state to be the average of a carefully chosen subset of the remaining values. This algorithm was extended to a family of algorithms, named the Mean-SubsequenceReduced (MSR) algorithms, in [30]. Although the research on Approximate Byzantine Consensus for complete networks is mature, there are few papers that have attempted to analyze this algorithm in more general topologies [31], and even then, only certain special networks have been investigated.…”

“…In [29], functions that perform this type of operation are referred to as approximation functions, and both synchronous and asynchronous algorithms are studied that use these approximation functions in complete networks for resilience to F -total Byzantine faults. These approximation functions are used in the family of Mean-SubsequenceReduced (MSR) algorithms [30]. There are a few key differences between the operations used in the W-MSR algorithm and the MSR algorithm of [30].…”