41 research outputs found
Byzantine Approximate Agreement on Graphs
Consider a distributed system with n processors out of which f can be Byzantine faulty. In the approximate agreement task, each processor i receives an input value x_i and has to decide on an output value y_i such that
1) the output values are in the convex hull of the non-faulty processors\u27 input values,
2) the output values are within distance d of each other.
Classically, the values are assumed to be from an m-dimensional Euclidean space, where m >= 1.
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 G and the goal is to output vertices that are within distance d of each other in G, but still remain in the graph-induced convex hull of the input values. For d=0, 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 d >= 1, we show that the task is solvable in asynchronous systems when G is chordal and n > (omega+1)f, where omega is the clique number of G. 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
Tight Bounds for Asymptotic and Approximate Consensus
We study the performance of asymptotic and approximate consensus algorithms
under harsh environmental conditions. The asymptotic consensus problem requires
a set of agents to repeatedly set their outputs such that the outputs converge
to a common value within the convex hull of initial values. This problem, and
the related approximate consensus problem, are fundamental building blocks in
distributed systems where exact consensus among agents is not required or
possible, e.g., man-made distributed control systems, and have applications in
the analysis of natural distributed systems, such as flocking and opinion
dynamics. We prove tight lower bounds on the contraction rates of asymptotic
consensus algorithms in dynamic networks, from which we deduce bounds on the
time complexity of approximate consensus algorithms. In particular, the
obtained bounds show optimality of asymptotic and approximate consensus
algorithms presented in [Charron-Bost et al., ICALP'16] for certain dynamic
networks, including the weakest dynamic network model in which asymptotic and
approximate consensus are solvable. As a corollary we also obtain
asymptotically tight bounds for asymptotic consensus in the classical
asynchronous model with crashes.
Central to our lower bound proofs is an extended notion of valency, the set
of reachable limits of an asymptotic consensus algorithm starting from a given
configuration. We further relate topological properties of valencies to the
solvability of exact consensus, shedding some light on the relation of these
three fundamental problems in dynamic networks
Matrix Representation of Iterative Approximate Byzantine Consensus in Directed Graphs
This paper presents a proof of correctness of an iterative approximate
Byzantine consensus (IABC) algorithm for directed graphs. The iterative
algorithm allows fault- free nodes to reach approximate conensus despite the
presence of up to f Byzantine faults. Necessary conditions on the underlying
network graph for the existence of a correct IABC algorithm were shown in our
recent work [15, 16]. [15] also analyzed a specific IABC algorithm and showed
that it performs correctly in any network graph that satisfies the necessary
condition, proving that the necessary condition is also sufficient. In this
paper, we present an alternate proof of correctness of the IABC algorithm,
using a familiar technique based on transition matrices [9, 3, 17, 19].
The key contribution of this paper is to exploit the following observation:
for a given evolution of the state vector corresponding to the state of the
fault-free nodes, many alternate state transition matrices may be chosen to
model that evolution cor- rectly. For a given state evolution, we identify one
approach to suitably "design" the transition matrices so that the standard
tools for proving convergence can be applied to the Byzantine fault-tolerant
algorithm as well. In particular, the transition matrix for each iteration is
designed such that each row of the matrix contains a large enough number of
elements that are bounded away from 0
An Improved Approximate Consensus Algorithm in the Presence of Mobile Faults
This paper explores the problem of reaching approximate consensus in
synchronous point-to-point networks, where each pair of nodes is able to
communicate with each other directly and reliably. We consider the mobile
Byzantine fault model proposed by Garay '94 -- in the model, an omniscient
adversary can corrupt up to nodes in each round, and at the beginning of
each round, faults may "move" in the system (i.e., different sets of nodes may
become faulty in different rounds). Recent work by Bonomi et al. '16 proposed a
simple iterative approximate consensus algorithm which requires at least
nodes. This paper proposes a novel technique of using "confession" (a mechanism
to allow others to ignore past behavior) and a variant of reliable broadcast to
improve the fault-tolerance level. In particular, we present an approximate
consensus algorithm that requires only nodes, an
improvement over the state-of-the-art algorithms.
Moreover, we also show that the proposed algorithm is optimal within a family
of round-based algorithms
Byzantine Vector Consensus in Complete Graphs
Consider a network of n processes each of which has a d-dimensional vector of
reals as its input. Each process can communicate directly with all the
processes in the system; thus the communication network is a complete graph.
All the communication channels are reliable and FIFO (first-in-first-out). The
problem of Byzantine vector consensus (BVC) requires agreement on a
d-dimensional vector that is in the convex hull of the d-dimensional input
vectors at the non-faulty processes. We obtain the following results for
Byzantine vector consensus in complete graphs while tolerating up to f
Byzantine failures:
* We prove that in a synchronous system, n >= max(3f+1, (d+1)f+1) is
necessary and sufficient for achieving Byzantine vector consensus.
* In an asynchronous system, it is known that exact consensus is impossible
in presence of faulty processes. For an asynchronous system, we prove that n >=
(d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector
consensus.
Our sufficiency proofs are constructive. We show sufficiency by providing
explicit algorithms that solve exact BVC in synchronous systems, and
approximate BVC in asynchronous systems.
We also obtain tight bounds on the number of processes for achieving BVC
using algorithms that are restricted to a simpler communication pattern
Iterative Approximate Consensus in the presence of Byzantine Link Failures
This paper explores the problem of reaching approximate consensus in
synchronous point-to-point networks, where each directed link of the underlying
communication graph represents a communication channel between a pair of nodes.
We adopt the transient Byzantine link failure model [15, 16], where an
omniscient adversary controls a subset of the directed communication links, but
the nodes are assumed to be fault-free.
Recent work has addressed the problem of reaching approximate consen- sus in
incomplete graphs with Byzantine nodes using a restricted class of iterative
algorithms that maintain only a small amount of memory across iterations [22,
21, 23, 12]. However, to the best of our knowledge, we are the first to
consider approximate consensus in the presence of Byzan- tine links. We extend
our past work that provided exact characterization of graphs in which the
iterative approximate consensus problem in the presence of Byzantine node
failures is solvable [22, 21]. In particular, we prove a tight necessary and
sufficient condition on the underlying com- munication graph for the existence
of iterative approximate consensus algorithms under transient Byzantine link
model. The condition answers (part of) the open problem stated in [16].Comment: arXiv admin note: text overlap with arXiv:1202.609
Reaching Approximate Byzantine Consensus in Partially-Connected Mobile Networks
We consider the problem of approximate consensus in mobile networks
containing Byzantine nodes. We assume that each correct node can communicate
only with its neighbors and has no knowledge of the global topology. As all
nodes have moving ability, the topology is dynamic. The number of Byzantine
nodes is bounded by f and known by all correct nodes. We first introduce an
approximate Byzantine consensus protocol which is based on the linear iteration
method. As nodes are allowed to collect information during several consecutive
rounds, moving gives them the opportunity to gather more values. We propose a
novel sufficient and necessary condition to guarantee the final convergence of
the consensus protocol. The requirement expressed by our condition is not
"universal": in each phase it affects only a single correct node. More
precisely, at least one correct node among those that propose either the
minimum or the maximum value which is present in the network, has to receive
enough messages (quantity constraint) with either higher or lower values
(quality constraint). Of course, nodes' motion should not prevent this
requirement to be fulfilled. Our conclusion shows that the proposed condition
can be satisfied if the total number of nodes is greater than 3f+1.Comment: No. RR-7985 (2012