Amnesiac Flooding: Easy to break, hard to escape

TL;DR

This paper investigates Amnesiac Flooding, a stateless broadcast protocol in synchronous networks, addressing both its uniqueness under four design constraints and its robustness to faults. The authors prove that any terminating broadcast protocol satisfying Strict Statelessness, Obliviousness, Determinism, and Unit Bandwidth must behave identically to AF on all graphs with adversarial labeling, and they construct four relaxed protocols that differ when any constraint is dropped, complemented by a precise fault-sensitivity dichotomy. They introduce a termination dichotomy based on balance and faux-even cycles, providing invariant-based characterizations and a bound of $k\le 2|\!E\!|$ for balanced configurations, and they analyze fault scenarios—single message drops, unidirectional link failures, and weak Byzantine faults—that can force non-termination or non-broadcast. The results position AF as a prototypical stateless broadcasting primitive and illuminate fundamental limits of stateless algorithms for other canonical network problems.

Abstract

Broadcast is a central problem in distributed computing. Recently, Hussak and Trehan [PODC'19/DC'23] proposed a stateless broadcasting protocol (Amnesiac Flooding), which was surprisingly proven to terminate in asymptotically optimal time (linear in the diameter of the network). However, it remains unclear: (i) Are there other stateless terminating broadcast algorithms with the desirable properties of Amnesiac Flooding, (ii) How robust is Amnesiac Flooding with respect to \emph{faults}? In this paper we make progress on both of these fronts. Under a reasonable restriction (obliviousness to message content) additional to the fault-free synchronous model, we prove that Amnesiac Flooding is the \emph{only} strictly stateless deterministic protocol that can achieve terminating broadcast. We achieve this by identifying four natural properties of a terminating broadcast protocol that Amnesiac Flooding uniquely satisfies. In contrast, we prove that even minor relaxations of \textit{any} of these four criteria allow the construction of other terminating broadcast protocols. On the other hand, we prove that Amnesiac Flooding can become non-terminating or non-broadcasting, even if we allow just one node to drop a single message on a single edge in a single round. As a tool for proving this, we focus on the set of all \textit{configurations} of transmissions between nodes in the network, and obtain a \textit{dichotomy} characterizing the configurations, starting from which, Amnesiac Flooding terminates. Additionally, we characterise the structure of sets of Byzantine agents capable of forcing non-termination or non-broadcast of the protocol on arbitrary networks.

Figures (3)

• Figure 1: Left: An $FEC_{x,y,z}$. Right: The corresponding even cycle representation. Please note that this depiction only holds for $y\geq 2$. For $y=1$: $a_0$ and $c_0$ are connected directly by an edge in both sub figures (as are $a_{-1}$ and $c_{-1}$). For $y=0$: $a_0=c_0$ and $a_{-1}=c_{-1}$.
• Figure 2: An illustration of the walk described in the interleaved case in the proof of \ref{['lemma: Odd Cycle Lemma']}
• Figure 3: The forbidden sub-digraph used in the argument of lemma \ref{['lemma: ramsey']}.

Theorems & Definitions (68)

• Definition 1.1
• Theorem 1.1: Uniqueness of Amnesiac Flooding
• Theorem 1.2: Existence of relaxed Algorithms
• Theorem 1.3: Single Message Failure
• Theorem 1.4: Uni-directional link failure
• Theorem 1.5: Byzantine Failure
• Theorem 2.1: Termination of Amnesiac Flooding (adapted from HT-AFCases-Arxiv20hussak2023terminationturau2021amnesiac)
• Definition 3.1
• Definition 3.2
• Lemma 3.1: Multicast Amnesiac Flooding