Agreement Tasks in Fault-Prone Synchronous Networks of Arbitrary Structure

TL;DR

The paper resolves the optimality of the radius bound for oblivious consensus in finite graphs under the synchronous $t$-resilient model, proving that no oblivious algorithm can solve consensus in fewer than $radius(G,t)$ rounds for any $0\le t<\kappa(G)$ and extending the framework to $k$-set agreement with $radius(G,t,k)$. It introduces a robust information-flow graph and corrects previous definitions to characterize solvability via domination of components, then presents a generic, oblivious $k$-set agreement algorithm that floods from a center set of size at most $k$ and achieves $radius(G,t,k)$ rounds. Beyond the connectivity threshold, the work defines local consensus and local set agreement to handle disconnected graphs, providing radius-based algorithms and proofs of correctness. Together, these results create a unified, graph-agnostic theory linking radius-like graph parameters to the round complexity of fundamental agreement tasks in fault-prone synchronous networks, with clear implications for generic protocol design on arbitrary networks.

Abstract

Consensus is arguably the most studied problem in distributed computing as a whole, and particularly in the distributed message-passing setting. In this latter framework, research on consensus has considered various hypotheses regarding the failure types, the memory constraints, the algorithmic performances (e.g., early stopping and obliviousness), etc. Surprisingly, almost all of this work assumes that messages are passed in a \emph{complete} network, i.e., each process has a direct link to every other process. A noticeable exception is the recent work of Castañeda et al. (Inf. Comput. 2023) who designed a generic oblivious algorithm for consensus running in $\radius(G,t)$ rounds in every graph~$G$, when up to $t$ nodes can crash by irrevocably stopping, where $t$ is smaller than the node-connectivity $κ$ of~$G$. Here, $\radius(G,t)$ denotes a graph parameter called the \emph{radius of~$G$ whenever up to $t$ nodes can crash}. For $t=0$, this parameter coincides with $\radius(G)$, the standard radius of a graph, and, for $G=K_n$, the running time $\radius(K_n,t)=t +1$ of the algorithm exactly matches the known round-complexity of consensus in the clique~$K_n$. Our main result is a proof that $\radius(G,t)$ rounds are necessary for oblivious algorithms solving consensus in $G$ when up to $t$ nodes can crash, thus validating a conjecture of Castañeda et al., and demonstrating that their consensus algorithm is optimal for any graph~$G$. We also extend the result of Castañeda et al. to two different settings: First, to the case where the number $t$ of failures is not necessarily smaller than the connectivity $κ$ of the considered graph; Second, to the $k$-set agreement problem for which agreement is not restricted to be on a single value as in consensus, but on up to $k$ different values.

Figures (5)

• Figure 1: Input configurations $I_0,\ldots,I_n$ of a graph $G=(V,E)$, where $V=\{v_1,\ldots,v_n\}$.
• Figure 2: The information flow graph $\mathsf{IF}(G,r,\{\varphi\})$ as defined in CastanedaFPRRT23 for $r=1$ and $r=2$, where $\varphi$ is the failure pattern in which $v$ crashes cleanly at the second round. No node dominates $\mathsf{IF}(G,2,\{\varphi\})$ (right), even though consensus is solvable in $G$ under $\varphi$ in $2$ rounds.
• Figure 3: A successor $\varphi'$ of a failure pattern $\varphi$ with respect to node $u$. Red nodes are faulty in $\varphi$ and white nodes are correct in it.
• Figure 6: A graph $G$ for which the adaptive algorithm is faster than the greedy algorithm 2
• Figure 7: The information flow graphs of a $3$-path, after one round and $r\geq 2$ rounds, where $t=1$ node may fail, potentially disconnecting the graph since $\kappa(G)=1$.

Theorems & Definitions (26)

• Proposition 1: Lemma 1 in CastanedaFPRRT23
• Definition 2
• Proposition 3: Theorem 2 in CastanedaFPRRT23
• Proposition 4: Theorem 3 in CastanedaFPRRT23
• Theorem 5
• Definition 6
• Definition 7
• Theorem 8
• Definition 10
• Lemma 11