Multidimensional agreement in Byzantine systems

Abstract

Consider a network of n processes, where each process inputs a 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 d-dimensional vector v ∈ R d , inside the convex hull of 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 d-dimensional vectors in R d , all within a fixed Euclidean distance of each other, and inside the convex hull of d-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems (M. Herlihy) Supported by NSF 0830491. (N. Vaidya and V. K. Garg) above, while tolerating up to f Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, n > max{3 f, (d + 1) f } is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, 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.

Convergence and covering on graphs for wait-free robots

“…Also, we consider only crash failures. Interesting future work may consider Byzantine failure techniques such as [28]. In contrast, we expect that the assumption that robots have identifiers is common in practice.…”

“…Roughly speaking, asynchronous communicating robots cannot coordinate to converge to a single point because consensus is impossible in the presence of even a single crash failure [17], but robots can move towards points which are arbitrarily close to each other, using a solution to approximate agreement [16]. Robots can also converge in Euclidean space; see [28] for a recent treatment of a basic multi-dimensional robot convergence task tolerating Byzantine faults, including a discussion of applications to robots, distributed voting and optimization problems, as well as further related references. Various other *Correspondence: armando.castaneda@im.unam.mx 1 Instituto de Matemáticas, UNAM, Ciudad Universitaria, 04510 Mexico city, Mexico Full list of author information is available at the end of the article applications and specific robot convergence tasks appear in, e.g., [8,22,23,27,30].…”

“…For instance, the space K could be the d-dimensional Euclidean space as in [28], and robots may be required to converge on regions spanned by the convex hull of their initial positions; if all start on the same point, they should remain there, and if all start in two points, they should converge to points close to each other along the straight line connecting the two points. Another example is the loop agreement task [23], where there is a given loop in the space K, and three given distinguished points v 1 , v 2 , v 3 on the loop.…”

Resilient Consensus Through Event-Based Communication

“…This relation shows that if an update happens at node i, then this node will move out of X 0 (k + 1, ε 0 (k + 1)). We note that inequality (19) also holds for the regular nodes that are not inside X 0 (k, ε 0 (k)) at time k. This means that such nodes cannot move in X 0 (k + 1, ε 0 (k + 1)). It is also similar with X 0 (k + 1, ε 0 (k + 1)).…”

“…Hence, it is useful to remove the F largest values as well as the F smallest values among those received from the neighbors. This class of algorithms is sometimes called the mean subsequence reduced (MSR) algorithms and has been employed in computer science (e.g., [19], [33]), control theory (e.g., [3], [16], [36]), and robotics (e.g., [6], [24], [26]). An important recent progress lies in the characterization of the necessary requirement on the topology of the agent networks.…”