The topology of randomized symmetry-breaking distributed computing

Abstract

Studying distributed computing through the lens of algebraic topology has been the source of many significant breakthroughs during the last 2 decades, especially in the design of lower bounds or impossibility results. Despite hundred of results considering deterministic algorithms, none apply to randomized 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 design a topological framework, which allows analyzing such tasks and determining their solvability. The pivotal technical observation is that, unlike in deterministic algorithm, where solvability means that the topological complex describing the protocol can be globally mapped into an output protocol, in our framework the solvability is determined “locally”, i.e., for each simplex of the protocol complex individually, without requiring any global consistency. As an interesting application, we derive necessary and sufficient conditions for solving leader election in shared-memory and message-passing models in which there might be correlations between the randomness provided to the nodes. 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.