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. T
his 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.
In recent years, protocols that are based on the properties of random walks on graphs have found many applications in communication and information networks, such as wireless networks, peer-to-peer networks and the Web. For wireless networks (and oth
er networks), graphs are actually not the correct model of the communication; instead hyper-graphs better capture the communication over a wireless shared channel. Motivated by this example, we study in this paper random walks on hyper-graphs. First, we formalize the random walk process on hyper-graphs and generalize key notions from random walks on graphs. We then give the novel definition of radio cover time, namely, the expected time of a random walk to be heard (as opposed to visit) by all nodes. We then provide some basic bounds on the radio cover, in particular, we show that while on graphs the radio cover time is O(mn), in hyper-graphs it is O(mnr) where n, m and r are the number of nodes, the number of edges and the rank of the hyper-graph, respectively. In addition, we define radio hitting times and give a polynomial algorithm to compute them. We conclude the paper with results on specific hyper-graphs that model wireless networks in one and two dimensions.
