Beyond 2-Edge-Connectivity: Algorithms and Impossibility for Content-Oblivious Leader Election
Yi-Jun Chang, Lyuting Chen, Haoran Zhou
TL;DR
This work characterizes the feasibility of terminating leader election in the content-oblivious model on trees with topology knowledge, showing that termination is possible if and only if the tree is not symmetric about any edge. It provides two main algorithms: a diameter-aware approach for even-diameter trees achieving O(nr) pulses and a topology-aware approach for general asymmetric trees achieving O(n^2) pulses, while proving impossibility for edge-symmetric graphs even with topology knowledge. The results establish that topology knowledge is essential in non-2-edge-connected graphs and offer a stabilizing, uniform leader election method that does not rely on prior network-size knowledge. Overall, the paper advances understanding of content-oblivious computation beyond 2-edge-connectivity and lays out open questions about extending these techniques to broader graph classes and improving efficiency.
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.