# Communication cost of consensus for nodes with limited memory

## TL;DR

This paper investigates the communication complexity of reaching consensus among nodes with limited memory, establishing thresholds for memory size that enable or prevent linear communication costs in both asynchronous and synchronous models.

## Contribution

It introduces a novel analysis of how memory constraints affect the communication cost of consensus, providing tight bounds for when linear communication is achievable.

## Key findings

- Linear communication cost is achievable if memory exceeds 3 log log log n in the asynchronous model.
- Achieving linear communication cost is impossible if memory is less than log log log n in the asynchronous model.
- In the synchronous model, the thresholds are 2 log log log n for upper bounds and log log log n for lower bounds on memory.

## Abstract

Motivated by applications in blockchains and sensor networks, we consider a model of $n$ nodes trying to reach consensus on their majority bit. Each node $i$ is assigned a bit at time zero, and is a finite automaton with $m$ bits of memory (i.e., $2^m$ states) and a Poisson clock. When the clock of $i$ rings, $i$ can choose to communicate, and is then matched to a uniformly chosen node $j$. The nodes $j$ and $i$ may update their states based on the state of the other node. Previous work has focused on minimizing the time to consensus and the probability of error, while our goal is minimizing the number of communications. We show that when $m>3 \log\log\log(n)$, consensus can be reached at linear communication cost, but this is impossible if $m<\log\log\log(n)$. We also study a synchronous variant of the model, where our upper and lower bounds on $m$ for achieving linear communication cost are $2\log\log\log(n)$ and $\log\log\log(n)$, respectively. A key step is to distinguish when nodes can become aware of knowing the majority bit and stop communicating. We show that this is impossible if their memory is too low.