Consider a distributed system with @math processors out of which @math can be Byzantine faulty. In the approximate agreement task, each processor @math receives an input value @math and has to decide on an output value @math such that - the output values are in the convex hull of the non-faulty processors' input values, - the output values are within distance @math of each other. Classically, the values are assumed to be from an @math -dimensional Euclidean space, where @math . In this work, we study the task in a discrete setting, where input values with some structure expressible as a graph. Namely, the input values are vertices of a finite graph @math and the goal is to output vertices that are within distance @math of each other in @math , but still remain in the graph-induced convex hull of the input values. For @math , the task reduces to consensus and cannot be solved with a deterministic algorithm in an asynchronous system even with a single crash fault. For any @math , we show that the task is solvable in asynchronous systems when @math is chordal and @math , where @math is the clique number of @math . In addition, we give the first Byzantine-tolerant algorithm for a variant of lattice agreement. For synchronous systems, we show tight resilience bounds for the exact variants of these and related tasks over a large class of combinatorial structures.
The seminal result of @cite_0 showed that consensus cannot be reached in asynchronous systems in the presence of crash faults. @cite_5 showed that it is however possible to reach in an asynchronous system even with arbitrary faulty behavior when the values reside on the continuous real line. Subsequently, the one-dimensional approximate agreement problem has been extensively studied @cite_5 @cite_10 @cite_36 @cite_41 . Fekete @cite_36 showed that any algorithm reducing the distance of values from @math to @math requires @math asynchronous rounds when @math ; in the discrete setting this yields the bound @math for paths of length @math . Recently, @cite_13 introduced the natural generalisation of approximate agreement and showed that the @math -dimensional problem is solvable in an asynchronous system with Byzantine faults if and only if @math holds for any given @math .
@cite_13 @cite_41 @cite_36 @cite_0 @cite_5 @cite_10
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Consider a network of @math n processes, where each process inputs a @math d-dimensional vector of reals. All processes can communicate directly with others via reliable FIFO channels. We discuss two problems. The multidimensional Byzantine consensus problem, for synchronous systems, requires processes to decide on a single @math d-dimensional vector @math v?Rd, inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. Also, the multidimensional Byzantine approximate agreement (MBAA) problem, for asynchronous systems, requires processes to decide on multiple @math d-dimensional vectors in @math Rd, all within a fixed Euclidean distance @math ∈ of each other, and inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems above, while tolerating up to @math f Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, @math n>max 3f,(d+1)f is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, @math n>(d+2)f is necessary and sufficient to solve the multidimensional approximate agreement problem. Our sufficiency proofs are constructive, giving explicit protocols for the problems. In particular, for the MBAA problem, we give two protocols with strictly different properties and applications. Consider an asynchronous system where each process begins with an arbitrary real value. Given some fixed e>0, an approximate agreement algorithm must have all non-faulty processes decide on values that are at most e from each other and are in the range of the initial values of the non-faulty processes. Previous constructions solved asynchronous approximate agreement only when there were at least 5t+1 processes, t of which may be Byzantine. In this paper we close an open problem raised by in 1983. We present a deterministic optimal resilience approximate agreement algorithm that can tolerate any t Byzantine faults while requiring only 3t+1 processes. The algorithm's rate of convergence and total message complexity are efficiently bounded as a function of the range of the initial values of the non-faulty processes. All previous asynchronous algorithms that are resilient to Byzantine failures may require arbitrarily many messages to be sent. This paper examines the Approximate Agreement Problem in an asynchronous failure-by-omission system with deterministic protocols. We give a simple algorithm, and prove that the algorithm is optimal by considering the power of the "adversary" scheduler to disrupt processors? views. We show that the adversary need not cause any omissions to achieve its purpose, and therefore no algorithm can do better than simply to operate round-by-round, as our does. We extend these results to asynchronous crash-failure systems. The resulting understanding of the adversary should be applicable to other problems in asynchronous failure-by-omission or crash-failure-systems. The consensus problem involves an asynchronous system of processes, some of which may be unreliable. The problem is for the reliable processes to agree on a binary value. In this paper, it is shown that every protocol for this problem has the possibility of nontermination, even with only one faulty process. By way of contrast, solutions are known for the synchronous case, the “Byzantine Generals” problem. This paper considers a variant of the Byzantine Generals problem, in which processes start with arbitrary real values rather than Boolean values or values from some bounded range, and in which approximate, rather than exact, agreement is the desired goal. Algorithms are presented to reach approximate agreement in asynchronous, as well as synchronous systems. The asynchronous agreement algorithm is an interesting contrast to a result of , who show that exact agreement with guaranteed termination is not attainable in an asynchronous system with as few as one faulty process. The algorithms work by successive approximation, with a provable convergence rate that depends on the ratio between the number of faulty processes and the total number of processes. Lower bounds on the convergence rate for algorithms of this form are proved, and the algorithms presented are shown to be optimal. This paper introduces some algorithms to solve crash-failure, failure-by-omission and Byzantine failure versions of the Byzantine Generals or consensus problem, where non-faulty processors need only arrive at values that are close together rather than identical. For each failure model and each value ofS, we give at-resilient algorithm usingS rounds of communication. IfS=t+1, exact agreement is obtained. In the algorithms for the failure-by-omission and Byzantine failure models, each processor attempts to identify the faulty processors and corrects values transmited by them to reduce the amount of disagreement. We also prove lower bounds for each model, to show that each of our algorithms has a convergence rate that is asymptotic to the best possible in that model as the number of processors increases.