Consider a synchronous point-to-point network of n nodes connected by directed links, wherein each node has a binary input. This paper proves a tight necessary and sucient condition on the underlying communication topology for achieving Byzantine consensus among these nodes in the presence of up to f Byzantine faults. We derive a necessary condition, and then we provide a constructive proof of suciency by presenting a Byzantine consensus algorithm for directed graphs that satisfy the necessary condition.
Our recent work @cite_5 @cite_2 @cite_18 has considered a restricted class of iterative algorithms for achieving approximate Byzantine consensus in directed graphs, where fault-free nodes must agree on values that are approximately equal to each other using iterative algorithms with limited memory. The conditions developed in such prior work are not necessary when no such restrictions are imposed. Independently, @cite_16 @cite_20 , and Zhang and Sundaram @cite_11 @cite_19 have developed results for iterative algorithms for approximate consensus under a weaker fault model, where a faulty node must send identical messages to all the neighbors. In this work, we consider the problem of exact consensus (i.e., the outputs at fault-free nodes must be exactly identical), and we do not impose any restriction on the algorithms or faulty nodes.
@cite_18 @cite_19 @cite_2 @cite_5 @cite_16 @cite_20 @cite_11
2025375132 1760770890 2147056869 1510375514
This paper proves a necessary and sufficient condition for the existence of iterative, algorithms that achieve approximate Byzantine consensus in arbitrary directed graphs, where each directed edge represents a communication channel between a pair of nodes. The class of iterative algorithms considered in this paper ensures that, after each iteration of the algorithm, the state of each fault-free node remains in the convex hull of the states of the fault-free nodes at the end of the previous iteration. The following convergence requirement is imposed: for any e > 0, after a sufficiently large number of iterations, the states of the fault-free nodes are guaranteed to be within e of each other. To the best of our knowledge, tight necessary and sufficient conditions for the existence of such iterative consensus algorithms in synchronous arbitrary point-to-point networks in presence of Byzantine faults, have not been developed previously. The methodology and results presented in this paper can also be extended to asynchronous systems. This work addresses Byzantine vector consensus, wherein the input at each process is a d-dimensional vector of reals, and each process is required to decide on a decision vector that is in the convex hull of the input vectors at the fault-free processes [9,12]. The input vector at each process may also be viewed as a point in the d-dimensional Euclidean space R d , where di¾?>i¾?0 is a finite integer. Recent work [9,12] has addressed Byzantine vector consensus, and presented algorithms with optimal fault tolerance in complete graphs. This paper considers Byzantine vector consensus in incomplete graphs using a restricted class of iterative algorithms that maintain only a small amount of memory across iterations. For such algorithms, we prove a necessary condition, and a sufficient condition, for the graphs to be able to solve the vector consensus problem iteratively. We present an iterative Byzantine vector consensus algorithm, and prove it correct under the sufficient condition. The necessary condition presented in this paper for vector consensus does not match with the sufficient condition for di¾?>i¾?1; thus, a weaker condition may potentially suffice for Byzantine vector consensus. Much of the past work on asynchronous approximate Byzantine consensus has assumed scalar inputs at the nodes [4, 8]. Recent work has yielded approximate Byzantine consensus algorithms for the case when the input at each node is a d-dimensional vector, and the nodes must reach consensus on a vector in the convex hull of the input vectors at the fault-free nodes [9, 13]. The d-dimensional vectors can be equivalently viewed as points in the d-dimensional Euclidean space. Thus, the algorithms in [9, 13] require the fault-free nodes to decide on a point in the d-dimensional space. In our recent work [12], we proposed a generalization of the consensus problem, namely Byzantine convex consensus (BCC), which allows the decision to be a convex polytope in the d-dimensional space, such that the decided polytope is within the convex hull of the input vectors at the fault-free nodes. We also presented an asynchronous approximate BCC algorithm. In this paper, we propose a new BCC algorithm with optimal fault-tolerance that also agrees on a convex polytope that is as large as possible under adversarial conditions. Our prior work [12] does not guarantee the optimality of the output polytope. In this work, we consider a generalized fault model [7,9,5] that can be used to represent a wide range of failure scenarios, including correlated failures and non-uniform node reliabilities. Under the generalized fault model, we explore iterative approximate Byzantine consensus (IABC) algorithms [15] in arbitrary directed networks. We prove a tight necessary and sufficient condition on the underlying communication graph for the existence of IABC algorithms.