Beyond 2-Edge-Connectivity: Algorithms and Impossibility for Content-Oblivious Leader Election

Abstract: The content-oblivious model, introduced by Censor-Hillel, Cohen, Gelles, and Sel (PODC 2022; Distributed Computing 2023), captures an extremely weak form of communication where nodes can only send asynchronous, content-less pulses. Censor-Hillel, Cohen, Gelles, and Sel showed that no non-constant function $f(x,y)$ can be computed correctly by two parties using content-oblivious communication over a single edge, where one party holds $x$ and the other holds $y$. This seemingly ruled out many natural graph problems on non-2-edge-connected graphs. In this work, we show that, with the knowledge of network topology $G$, leader election is possible in a wide range of graphs. Impossibility: Graphs symmetric about an edge admit no randomized terminating leader election algorithm, even when nodes have unique identifiers and full knowledge of $G$. Leader election algorithms: Trees that are not symmetric about any edge admit a quiescently terminating leader election algorithm with topology knowledge, even in anonymous networks, using $O(n^2)$ messages, where $n$ is the number of nodes. Moreover, even-diameter trees admit a terminating leader election given only the knowledge of the network diameter $D = 2r$, with message complexity $O(nr)$. Necessity of topology knowledge: In the family of graphs $\mathcal{G} = {P_3, P_5}$, both the 3-path $P_3$ and the 5-path $P_5$ admit a quiescently terminating leader election if nodes know the topology exactly. However, if nodes only know that the underlying topology belongs to $\mathcal{G}$, then terminating leader election is impossible.