The Topology of Randomized Symmetry-Breaking Distributed Computing

TL;DR

This paper explores the use of algebraic topology to analyze randomized symmetry-breaking tasks in distributed computing, introducing a novel projection method to retain topological structure and deriving conditions for leader election solvability.

Contribution

It introduces a new topological framework with a projection technique to study randomized distributed algorithms, extending algebraic topology methods to randomized symmetry-breaking tasks.

Findings

Solvability of leader election depends on the correlation of randomness among nodes.

The projection method preserves topological structure in local analysis.

Necessary and sufficient conditions for leader election are derived for different models.

Abstract

Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last two decades, especially in the design of lower bounds or impossibility results for deterministic algorithms. This paper aims at studying randomized synchronous distributed computing through the lens of algebraic topology. We do so by studying the wide class of (input-free) symmetry-breaking tasks, e.g., leader election, in synchronous fault-free anonymous systems. We show that it is possible to redefine solvability of a task "locally", i.e., for each simplex of the protocol complex individually, without requiring any global consistency. However, this approach has a drawback: it eliminates the topological aspect of the computation, since a single facet has a trivial topological structure. To overcome this issue, we introduce a "projection" $\pi$ of both protocol and output complexes, where every simplex $\sigma$ is mapped to a complex $\pi(\sigma)$; the later has a rich structure that replaces the structure we lost by considering one single facet at a time. To show the significance and applicability of our topological approach, we derive necessary and sufficient conditions for solving leader election in synchronous fault-free anonymous shared-memory and message-passing models. In both models, we consider scenarios in which there might be correlations between the random values provided to the nodes. In particular, different parties might have access to the same randomness source so their randomness is not independent but equal. Interestingly, we find that solvability of leader election relates to the number of parties that possess correlated randomness, either directly or via their greatest common divisor, depending on the specific communication model.