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Content-Oblivious Leader Election on Rings

Fabian Frei, Ran Gelles, Ahmed Ghazy, Alexandre Nolin

TL;DR

This work removes the need for a pre-elected root in content-oblivious computation by presenting leader election algorithms on rings. In oriented rings, it achieves quiescent termination with complexity $O(n \cdot ID_{\max})$ and proves a matching lower bound of $\Omega(n \log(ID_{\max}/n))$; it also extends to non-oriented rings under stabilization (no termination) with orientation guarantees. The key ideas include parallel executions on circular channels (CW/CCW), use of quiescence and message-attribution to enable composition, and handling of non-unique IDs to reduce complexity. The results advance understanding of what can be computed under extreme noise and highlight fundamental limits tied to the maximal ID, with implications for composing content-oblivious primitives in robust distributed systems.

Abstract

In content-oblivious computation, n nodes wish to compute a given task over an asynchronous network that suffers from an extremely harsh type of noise, which corrupts the content of all messages across all channels. In a recent work, Censor-Hillel, Cohen, Gelles, and Sela (Distributed Computing, 2023) showed how to perform arbitrary computations in a content-oblivious way in 2-edge connected networks but only if the network has a distinguished node (called root) to initiate the computation. Our goal is to remove this assumption, which was conjectured to be necessary. Achieving this goal essentially reduces to performing a content-oblivious leader election since an elected leader can then serve as the root required to perform arbitrary content-oblivious computations. We focus on ring networks, which are the simplest 2-edge connected graphs. On oriented rings, we obtain a leader election algorithm with message complexity O(n*ID_max), where ID_max is the maximal assigned ID. As it turns out, this dependency on $ID_max$ is inherent: we show a lower bound of Omega(n*log(ID_max/n)) messages for content-oblivious leader election algorithms. We also extend our results to non-oriented rings, where nodes cannot tell which channel leads to which neighbor. In this case, however, the algorithm does not terminate but only reaches quiescence.

Content-Oblivious Leader Election on Rings

TL;DR

This work removes the need for a pre-elected root in content-oblivious computation by presenting leader election algorithms on rings. In oriented rings, it achieves quiescent termination with complexity and proves a matching lower bound of ; it also extends to non-oriented rings under stabilization (no termination) with orientation guarantees. The key ideas include parallel executions on circular channels (CW/CCW), use of quiescence and message-attribution to enable composition, and handling of non-unique IDs to reduce complexity. The results advance understanding of what can be computed under extreme noise and highlight fundamental limits tied to the maximal ID, with implications for composing content-oblivious primitives in robust distributed systems.

Abstract

In content-oblivious computation, n nodes wish to compute a given task over an asynchronous network that suffers from an extremely harsh type of noise, which corrupts the content of all messages across all channels. In a recent work, Censor-Hillel, Cohen, Gelles, and Sela (Distributed Computing, 2023) showed how to perform arbitrary computations in a content-oblivious way in 2-edge connected networks but only if the network has a distinguished node (called root) to initiate the computation. Our goal is to remove this assumption, which was conjectured to be necessary. Achieving this goal essentially reduces to performing a content-oblivious leader election since an elected leader can then serve as the root required to perform arbitrary content-oblivious computations. We focus on ring networks, which are the simplest 2-edge connected graphs. On oriented rings, we obtain a leader election algorithm with message complexity O(n*ID_max), where ID_max is the maximal assigned ID. As it turns out, this dependency on is inherent: we show a lower bound of Omega(n*log(ID_max/n)) messages for content-oblivious leader election algorithms. We also extend our results to non-oriented rings, where nodes cannot tell which channel leads to which neighbor. In this case, however, the algorithm does not terminate but only reaches quiescence.
Paper Structure (13 sections, 31 theorems, 1 figure, 4 algorithms)

This paper contains 13 sections, 31 theorems, 1 figure, 4 algorithms.

Table of Contents

  1. Introduction
  2. Quiescence and Composability
  3. Related Work
  4. Organization
  5. Preliminaries
  6. Leader Election in an Oriented Ring
  7. Warm-up: Leader Election Without Termination
  8. Leader Election With Quiescent Termination
  9. Leader Election in Non-Oriented Rings
  10. Lower Bound On Message Complexity in Content-Oblivious Rings
  11. Conclusion and Open Questions
  12. Deferred Proofs Regarding Achieving Quiescent Termination
  13. Anonymous Rings

Key Result

Theorem 1

There is a quiescently terminating content-oblivious algorithm of message complexity $n(2\cdot \mathsf{ID}_{\max}\xspace + 1)$ that elects a leader in oriented rings of $n$ nodes with unique IDs.

Figures (1)

  • Figure 1: An oriented ring (left) and a non-oriented one (right).

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 3
  • Theorem 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Corollary 8
  • ...and 22 more