Self-healing Deterministic Expanders

We present fully-distributed algorithms that construct and maintain deterministic expander networks (i.e., the expansion properties are deterministically guaranteed) in the presence of an adaptive adversary. To the best of our knowledge, these algorithms are the first distributed implementations of deterministic expanders that work even under an all-powerful adaptive adversary (that has unlimited computational power and knowledge of the entire network state, can decide which nodes join and leave and at what time, and knows the past random choices made by the algorithm). Previous distributed expander constructions typically provide only probabilistic guarantees and these rapidly degrade over a series of network changes, so here we provide a much needed solution. Our algorithms are in a self-healing model where at each step, the omniscient adversary either inserts or deletes a node and the algorithm responds by locally adding or dropping edges (connections) in the network. Our algorithms provide fast healing with high probability in O(log n) rounds on average, where n is the number of nodes currently in the network, while deterministically maintaining a constant degree expander. Moreover, our algorithms guarantee that the number of added/dropped edges per round is constant on average. Our communication model follows the CONGEST model, allowing messages of size only O(log n). This ensures that our algorithms are highly scalable. We also present a lower bound of Theta(log n) rounds on average, for any distributed expander self-healing algorithm. This shows that our algorithms are asymptotically the best possible (up to constant factors) in this model. Our work also yields an improved version of the self-healing algorithm Xheal [PODC 2011], which previously relied on expander constructions with only probabilistic guarantees.