A Simple Lower Bound for Set Agreement in Dynamic Networks

TL;DR

The paper tackles the problem of establishing a lower bound on the round complexity of $k$-set agreement in dynamic, synchronous networks within the KNOW-ALL model. It introduces a self-contained, Sperner's lemma–based proof that leverages Kuhn's triangulation to map inputs and process views, demonstrating that any algorithm must take at least $r$ rounds, where $r$ is the smallest integer with $oldsymbol{gamma(H_r)}\,oldsymbol{\le}\,k$. This approach provides a simpler, more accessible alternative to previous topological arguments and yields a pedagogical path to understanding the $k=2$ case (e.g., with $n=5$) while generalizing to arbitrary $k$. The result strengthens the understanding of fundamental limits for distributed agreement in dynamic networks and offers a potential framework for teaching topological methods in distributed computing.

Abstract

Given a positive integer $k$, $k$-set agreement is the distributed task in which each process $i\in [n]$ in a group of $n$ processing nodes starts with an input value $x_i$ in the set $\{0,\dots,k\}$, and must output a value $y_i$ such that (1) for every $i \in [n]$, $y_i$ is the input value of some process, and (2)$|\{y_i : i\in [n]\}|\leq k$. That is, at most $k$ different values in total must be outputted by the processes. The case $k=1$ correspond to (binary) consensus, arguably the most studied problem in distributed computing. While lower bounds for consensus have been obtained for most of the standard distributed computing models, the design of lower bounds for $k$-set agreement with $k>1$ is notoriously known to be much more difficult, and remains open for many models. The main techniques for designing lower bounds for k-set agreement with $k>1$ use tools from algebraic topology. The algebraic topology tools are difficult to manipulate, and require a lot of care for avoiding mistakes. This difficulty increases when the communications are mediated by a network of arbitrary structure. Recently, the KNOWALL model has been specifically designed as a first attempt to understand the LOCAL model through the lens of algebraic topology, and Castañeda et al.(2021) have designed lower bounds for $k$-set agreement in the KNOWALL model, with applications to dynamic networks. In this work, we re-prove the same lower bound for $k$-set agreement in the KNOWALL model. This new proof stands out in its simplicity, which makes it accessible to a broader audience, and increases confidence in the result.

Figures (3)

• Figure 1: Input configurations $I_0,\ldots,I_n$. In $I_0$, all nodes have input $0$. For every $i\in [n]$, configuration $I_i$ is obtained from $I_{i-1}$ by changing input of node $i$ from $0$ to $1$. For two consecutive input configurations $I_{i-1}$ and $I_i$, there is exactly one node $i\in [n]$ with different inputs, and a node $w_i$ cannot distinguish $I_{i-1}$ from $I_i$.
• Figure 2: Illustration of the lower bound proof for $k$-set agreement in the $n$-node graph $\vec{C}_n$ for $k=2$ and $n=5$.
• Figure 3: The six primitive simplices that are inside a unit cube in $\mathbb{R}^3$. $ABGF = \{A,\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\}$, $ABCF = \{A,\mathbf{e}_1,\mathbf{e}_3,\mathbf{e}_2\}$, $AHGF = \{A,\mathbf{e}_2,\mathbf{e}_1,\mathbf{e}_3\}$, $AHEF = \{A,\mathbf{e}_2,\mathbf{e}_3,\mathbf{e}_1\}$, $ADCF = \{A,\mathbf{e}_3,\mathbf{e}_1,\mathbf{e}_2\}$, and $ADEF = \{A,\mathbf{e}_3,\mathbf{e}_2,\mathbf{e}_1\}$.

Theorems & Definitions (10)

• Definition 1
• Theorem 2: Castañeda, Fraigniaud, Paz, Rajsbaum, Roy, and Travers castaneda2021topological
• Lemma 3
• proof
• Corollary 4
• Lemma 5
• proof
• Lemma 6: Sperner's lemma
• Lemma 7
• proof