Distributed Algorithms for Maintaining Dynamic Expander Graphs

We consider the problem of maintaining expansion in an overlay network with dynamic node insertions. We study this problem in two models: one, where insertions are chosen benevolently, and the other, where insertions are chosen by an adversary. For the model of benevolent insertions, the problem has been previously studied empirically by Gkantsidis, Mihail, and Saberi [6]. In this paper, we present the first provable upper bound that has constant cost per insertion; this algorithm is based on the zig-zag construction of expander graphs. For the case of adversarial insertions, we provide an improved analysis of a natural algorithm proposed by Law and Siu [10], which is based on approximate sampling via random walks. This algorithm was previously known to work with constant probability; we show that it actually works with high probability. Several open problems are discussed. ∗Department of Computer Science, Yale University. †Supported in part by NSF grant CNS-0435201. Email: [email protected]. ‡Supported by a Kempner Foundation Fellowship and NSF grant CNS-0435201. Email: [email protected].