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Time-Optimal and Energy-Efficient Deterministic Consensus

Shachar Meir, Hugo Mirault, David Peleg, Peter Robinson

TL;DR

This work analyzes deterministic crash-tolerant consensus in the sleeping model, where nodes can be awake or asleep each round and energy is measured by the maximum awake rounds. It proves that optimal time $f+1$ rounds is achievable under crash faults, and provides two energy-efficient deterministic algorithms: (i) multi-value consensus with energy $O\big(\lceil f^2/n\rceil\big)$ and time $f+1$ (plus $O\big(f^3+nf\big)$ messages), and (ii) binary consensus with energy $O\big(\lceil f/\sqrt{n}\rceil\big)$ and time $f+1$ (plus $O(nf)$ messages). The multi-value approach uses fixed committees and phased value propagation to converge to the maximum observed input, while the binary approach reduces committee sizes to balance information spreading and energy, employing a four-phase protocol with 1-bit messages. Together, these results quantify the energy-time tradeoffs in sleeping-model consensus under crash faults and extend the sleeping-model literature by achieving fault tolerance with provably optimal time complexity. The findings have implications for energy-aware distributed systems and blockchain protocols where participant activity is sporadic or adversarially constrained.

Abstract

We study fault-tolerant consensus in a variant of the synchronous message passing model, where, in each round, every node can choose to be awake or asleep. This is known as the sleeping model (Chatterjee, Gmyr, Pandurangan PODC 2020) and defines the awake complexity (also called \emph{energy complexity}), which measures the maximum number of rounds that any node is awake throughout the execution. Only awake nodes can send and receive messages in a given round and all messages sent to sleeping nodes are lost. We present new deterministic consensus algorithms that tolerate up to $f<n$ crash failures, where $n$ is the number of nodes. Our algorithms match the optimal time complexity lower bound of $f+1$ rounds. For multi-value consensus, where the input values are chosen from some possibly large set, we achieve an energy complexity of ${O}(\lceil f^2 / n \rceil)$ rounds, whereas for binary consensus, we show that ${O}(\lceil f / \sqrt{n} \rceil)$ rounds are possible.

Time-Optimal and Energy-Efficient Deterministic Consensus

TL;DR

This work analyzes deterministic crash-tolerant consensus in the sleeping model, where nodes can be awake or asleep each round and energy is measured by the maximum awake rounds. It proves that optimal time rounds is achievable under crash faults, and provides two energy-efficient deterministic algorithms: (i) multi-value consensus with energy and time (plus messages), and (ii) binary consensus with energy and time (plus messages). The multi-value approach uses fixed committees and phased value propagation to converge to the maximum observed input, while the binary approach reduces committee sizes to balance information spreading and energy, employing a four-phase protocol with 1-bit messages. Together, these results quantify the energy-time tradeoffs in sleeping-model consensus under crash faults and extend the sleeping-model literature by achieving fault tolerance with provably optimal time complexity. The findings have implications for energy-aware distributed systems and blockchain protocols where participant activity is sporadic or adversarially constrained.

Abstract

We study fault-tolerant consensus in a variant of the synchronous message passing model, where, in each round, every node can choose to be awake or asleep. This is known as the sleeping model (Chatterjee, Gmyr, Pandurangan PODC 2020) and defines the awake complexity (also called \emph{energy complexity}), which measures the maximum number of rounds that any node is awake throughout the execution. Only awake nodes can send and receive messages in a given round and all messages sent to sleeping nodes are lost. We present new deterministic consensus algorithms that tolerate up to crash failures, where is the number of nodes. Our algorithms match the optimal time complexity lower bound of rounds. For multi-value consensus, where the input values are chosen from some possibly large set, we achieve an energy complexity of rounds, whereas for binary consensus, we show that rounds are possible.

Paper Structure

This paper contains 10 sections, 10 theorems, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Model and Preliminaries
  4. Multi-value Consensus
  5. Description of the Algorithm
  6. Proof of Theorem \ref{['thm:multi']}
  7. Binary Consensus
  8. Description of the Algorithm
  9. Notation and Terminology
  10. Proof of Theorem \ref{['thm:binary']}

Key Result

Theorem 1

Suppose that each player starts with an input value from a set of integers $D$. Algorithm alg:multi solves consensus in $f+1$ rounds, sends $\mathcal{O}(f^3+nf)$ messages of size $\mathcal{O}(\log |D|)$ bits, and achieves an energy complexity of $\mathcal{O}(\lceil{f^2}/{n}\rceil)$ rounds, if at mos

Figures (2)

  • Figure 1: High-level Overview of Algorithm \ref{['alg:multi']}
  • Figure 2: An Example of the Execution of Binary Consensus (Algorithm \ref{['alg:binary']}): Phase 1: Every player with input value $X=1$ sends a message to $p_i$, then crashes. Phase 2: $p_i$ is only able to send a message to $p_j \in C_2$ as it crashes. The same happens in each of the rounds of Phase 2, i.e., the player sending $1$ can only send its message to one member of the next committee. Phase 3: Assume that $f > n' - \sqrt{n'}$. All committees are now of size $f+1$. Player $p_m$, who received the message in the last round of Phase 2, tries to send a message to everyone in $C_h$ but crashes and the message is only received by $p_n$; in the remaining rounds of Phase 3, this process continues analogously. Phase 4: Every alive player with $Y=1$ try to send messages in $C_f$ in round $f$, they may crash before fully propagating 1 to $C_f$. $p_a\in C_f$ receives a message. In round $f+1$, $p_a$ try to broadcast $1$, and the adversary here can not crash $p_a$ without exceeding the threshold $f$. Consensus is reached.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Claim 10
  • Corollary 11
  • Lemma 12
  • Claim 15
  • ...and 5 more