Robust and Scalable Renaming with Subquadratic Bits

TL;DR

This paper tackles strong renaming in synchronous distributed systems under crash and Byzantine faults, addressing the long-standing issue of prohibitive communication costs. It introduces two randomized, fault-tolerant renaming algorithms that adapt their resource usage to the actual number of failures: a crash-resilient version with subquadratic communication enabled by a committee and interval-halving, and a Byzantine-resilient version that leverages shared randomness, message authentication, fingerprinting, and divide-and-conquer consensus to achieve near-linear communication while tolerating up to $(1/3-\varepsilon_0)n$ Byzantine nodes. A lower bound shows that the proposed costs are near-optimal in many regimes, and the work situates itself within a broader resource-competitive analysis of distributed renaming. Together, the results offer scalable, failure-adaptive techniques for strong renaming with practical implications for large networks and systems requiring efficient symmetry-breaking. The methods blend committee-based communication reduction, interval-halving, fingerprinting, and Validator/Consensus primitives to achieve correctness, order-preservation (for the Byzantine scheme), and robust performance across failure scenarios.

Abstract

In the renaming problem, a set of $n$ nodes, each with a unique identity from a large namespace $[N]$, needs to obtain new unique identities in a smaller namespace $[M]$. A renaming algorithm is strong if $M=n$. Renaming is a classical problem in distributed computing with a range of applications, and there exist many time-efficient solutions for fault-tolerant renaming in synchronous message-passing systems. However, all previous algorithms send $Ω(n^2)$ messages, and many of them also send large messages each containing $Ω(n)$ bits. Moreover, most algorithms' performance do not scale with the actual number of failures. These limitations restrict their practical performance. We develop two new strong renaming algorithms, one tolerates up to $n-1$ crash failures, and the other tolerates up to $(1/3-ε_0)n$ Byzantine failures for an arbitrarily small constant $ε_0>0$. The crash-resilient algorithm is always correct and always finishes within $O(\log{n})$ rounds. It sends $\tilde{O}((f+1)\cdot n)$ messages with high probability, where $f$ is the actual number of crashes. This implies that it sends subquadratic messages as long as $f=o(n/\log{n})$. The Byzantine-resilient algorithm trades time for communication: it finishes within $\tilde{O}(\max\{f,1\})$ rounds and sends only $\tilde{O}(f+n)$ messages, with high probability. Here, $f$ is the actual number of Byzantine nodes. To obtain such strong guarantees, the Byzantine-resilient algorithm leverages shared randomness and message authentication. Both algorithms only send messages of size $O(\log{N})$ bits. Therefore, our crash-resilient algorithm incurs $o(n^2)$ communication cost as long as $f=o(n/(\log{n}\log{N}))$; and our Byzantine resilient algorithm incurs almost-linear communication cost. By deriving a lower bound, we conclude that our algorithms achieve near-optimal communication cost in many cases.

Figures (5)

• Figure 1: The crash-resilient renaming algorithm executed at each node.
• Figure 2: Procedure CommitteeAction executed at each committee member.
• Figure 3: Procedure NodeAction executed at each node.
• Figure 4: The Byzantine-resilient renaming algorithm executed at each node.
• Figure 5: Procedure Validator executed at node $v$.

Theorems & Definitions (51)

• Definition 1.1: Renaming
• Theorem 1.2: Crash-resilient algorithm
• Theorem 1.3: Byzantine-resilient algorithm
• Theorem 1.4
• Definition 2.1
• Lemma 2.1
• Lemma 2.1
• Lemma 2.1
• Lemma 2.1
• Lemma 2.1