The Topology of Randomized Symmetry-Breaking Distributed Computing

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. In a nutshell, this approach consists of capturing all the possible states of a distributed system at a certain time as a simplicial complex called protocol complex, and viewing computation as a simplicial map from that complex to the so-called output complex, that captures all possible legal output states of the system. 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'' of both protocol and output complexes, where every simplex σ is mapped to a complex (σ); 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.