Brief Announcement: Effects of Topology Knowledge and Relay Depth on Asynchronous Consensus
Brief Announcement: Effects of Topology Knowledge and Relay Depth on Asynchronous Consensus
Dimitris Sakavalas Boston College 0
Lewis Tseng Boston College 0
Nitin H. Vaidya 0
0 Georgetown University , USA
Consider an asynchronous incomplete directed network. We study the feasibility and efficiency of approximate crash-tolerant consensus under different restrictions on topology knowledge and relay depth, i.e., the maximum number of hops any message can be relayed. 2012 ACM Subject Classification Computer systems organization ? Fault-tolerant network topologies
and phrases Asynchrony; crash fault; consensus; topology knowledge; relay
The fault-tolerant consensus problem introduced by Lamport et al.  and its variations have
been studied extensively. The need to overcome the FLP impossibility result for consensus
in asynchronous systems has led to the study of the approximate consensus problem ,
where nodes are required to output roughly the same value. We consider a directed network of
n nodes, wherein at most f nodes are subject to crash failure. We explore the feasibility
and efficiency of achieving approximate consensus in asynchronous incomplete networks under
different restrictions on topology knowledge and relay depth (defined as the maximum number
of hops that information can be propagated). These constraints are useful in large-scale
networks to avoid memory overload and network congestion.
Our prior work  showed that exact crash-tolerant consensus is solvable in synchronous
networks with only one-hop knowledge and relay depth 1, i.e., each node only needs to know its
immediate neighbors, and no message needs to be relayed. Such a local algorithm is of practical
interest due to low deployment cost and message complexity in each round. In asynchronous
undirected networks, there exists a simple flooding-based algorithm adapted from  that
achieves approximate consensus with up to f crash faults if the network satisfies (f + 1)
node-connectivity and n > 2f , where n is the number of nodes. However, the sufficiency of the
conditions is not guaranteed if we restrict topology knowledge and relay depth. Motivated by
this observation, this work addresses the following question in asynchronous systems:
What is a tight condition on the underlying communication graph to achieve approximate
consensus if each node has only a k-hop topology knowledge and relay depth k0?
To the best of our knowledge, two prior papers [1, 6] examined a similar problem
? synchronous Byzantine consensus. In , Su and Vaidya identified the condition under
different relay depths. Alchieri et al.  studied the problem under unknown participants. The
technique developed for asynchronous consensus in this work is significantly different. Please
refer to our technical report  for more discussion on other related work.
Model and Terminology. The point-to-point message-passing network is represented by
directed graph G(V, E), where V is the set of n nodes, and E is the set of directed edges. The
communication links are assumed to be reliable. Node i can transmit messages to its outgoing
neighbors and itself. Up to f nodes may suffer crash failures in an execution, in which case they
stop taking steps. We consider asynchronous communication. i.e., a message may be delayed
arbitrarily but will eventually be delivered. Let Ni?, Ni+ denote the sets of incoming neighbors
and outgoing neighbors of node i respectively. Also, for a node i, its k-hop incoming neighbors
Ni?(k), are defined as the nodes j which can reach i using a directed path in G that has ? k
hops. The notion of k-hop outgoing neighbors Ni+(k), is defined similarly. For set B ? V, node
i is said to be an incoming neighbor of set B if i 6? B, and there exists j ? B such that (i, j) ? E.
With NB? we will denote the incoming neighbors of B.
Approximate Consensus. In the approximate consensus problem , each node i maintains a
state vi with vi[p] denoting the state of node i at the end of phase (or iteration) p. The
initial state of node i, vi, is equal to the initial input provided to node i. At the start of
asynchronous phase p (p > 0), the state of node i is vi[p?1]. Let U [p] and ?[p] be the maximum
and the minimum state at nodes that have not crashed by the end of phase p. Then, a correct
approximate consensus algorithm needs to satisfy the following two conditions for any > 0:
Validity: ?p > 0, U [p] ? U  and ?[p] ? ?; and
-Convergence: ?p, ?r ? p, U [r] ? ?[r] < .
Limited Topology Knowledge and Relay Depth
Prior works (e.g., ) assumed that each node has n-hop topology knowledge and relay
depth n, which is not realistic in large-scale networks. Hence, we are interested in the family
of algorithms (iterative k-hop algorithms) in which nodes only know their k-hop neighborhoods,
and propagate state values to nodes that are at most k-hops away for 1 ? k ? n. Note that no
exchange of topology information takes place.
Iterative k-hop algorithms. Each node i performs the following three steps in phase p:
1. Transmit: Transmit messages of the form (vi[p?1], ?) to nodes that are reachable from node
i via at most k hops away, through intermediate relays.
2. Receive: Receive messages from all k-hop incoming neighbors. Denote by Ri[p] the set of
messages that node i received at phase p.
3. Update: Update state using a transition function Zi, where Zi is a part of the specification
of the algorithm, and takes as input the set Ri[t]. i.e., vi[t] := Zi(Ri[t], vi[t?1]) at node i.
Main Results. Below, we present two definitions to facilitate the discussion.
I Definition 1 (A ?k B). Given disjoint non-empty subsets of nodes A and B, we will say
that A ?k B holds if there exists a node i in B for which there exist at least f +1 node-disjoint
paths of length at most k from distinct nodes in A to i. More formally, if PiA(k) is the family
of all sets of k-length node-disjoint paths (with i being their only common node) initiating in A
and ending in node i, A ?k B means that ?i ? B, P ?mPaiAx(k) |P | ? f + 1.
I Definition 2 (Condition k-CCA). For any partition L, C, R of V, where L and R are both
non-empty, either L ? C ?k R or R ? C ?k L.
I Theorem 3. Approximate crash-tolerant consensus in an asynchronous system using
iterative k-hop algorithms is feasible iff G satisfies Condition k-CCA.
The complete proof is presented in . We only sketch the proof here. The necessity is
proved using an indistinguishable argument inspired by [3, 7]. For sufficiency, we present
Algorithm k-LocWA. Our key contribution is to identify what are the set of messages that each
node needs to receive before updating its state value in Step 3 of the iterative k-hop algorithms.
Algorithm k-LocWA relies on Condition k-WAIT : For Fi ? Ni?(k), we denote with reachik(Fi)
the set of nodes that have paths of length l ? k to node i in GV ?Fi . That is, the set of
k-hop incoming neighbors of i that remain connected with i even when all nodes in set Fi crash.
The condition is satisfied at node i, in phase p if there exists Fi ? Ni?(k) with |Fi[p]| ? f such
that reachik(Fi[p]) ? heardi[p]. Finally, we show that if G satisfies Condition k-CCA, then
Algorithm k-LocWA correctly solves approximate consensus.
We derive an upper bound on the number of asynchronous phases needed for -convergence
of Algorithm k-LocWA in . This upper bound is naturally a function of values , k, f, n and
? = U  ? ?. As a function of k, the bound implies that for k0 ? k, Algorithm k0-LocWA
-converges faster than k-LocWA. We also prove that for values k, k0 ? N with k ? k0,
Condition k-CCA implies Condition k0-CCA and that n-CCA is equivalent to CCA from .
Topology Discovery and Unlimited Relay Depth. Even if the topology knowledge of the
nodes is restricted to their 1-hop neighborhood, we show that allowing topology information
exchange and relay depth n, one can achieve approximate consensus whenever condition
CCA  holds. This can be achieved through Algorithm LWA, presented in the full version
, which introduces a topology discovery mechanism to learn the crucial topology information
that is necessary to achieve consensus. This result implies that knowledge of topology does not
affect the feasibility of the problem if topology knowledge can be relayed.
Eduardo A. P. Alchieri , Alysson Neves Bessani, Joni da Silva Fraga, and Fab?ola Greve. Byzantine consensus with unknown participants . In OPODIS 2008 , volume 5401 of LNCS , pages 22 - 40 . Springer, 2008 . doi: 10 .1007/978-3- 540 -92221- 6 _ 4 .
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M. Pease , R. Shostak , and L. Lamport . Reaching agreement in the presence of faults . J. ACM , 27 ( 2 ): 228 - 234 , 1980 . doi: 10 .1145/322186.322188.
Dimitris Sakavalas , Lewis Tseng , and Nitin H. Vaidya . Effects of topology knowledge and relay depth on asynchronous consensus . CoRR , abs/ 1803 .04513, 2018 . arXiv: 1803 .04513.