Theory and network applications of balanced kautz tree structures

In order to improve scalability and to reduce the maintenance overhead for structured peer-to-peer (P2P) networks, researchers have proposed architectures based on several interconnection networks with a fixed-degree and a logarithmical diameter. Among existing fixed-degree interconnection networks, the Kautz digraph has many distinctive topological properties compared to others. It, however, requires that the number of peers have the some given values, determined by peer degree and network diameter. In practice, we cannot guarantee how many peers will join a P2P network at a given time, since a P2P network is typically dynamic with peers frequently entering and leaving. To address such an issue, we propose the balanced Kautz tree and Kautz ring structures. We further design a novel structured P2P system, called BAKE, based on the two structures that has the logarithmical diameter and constant degree, even the number of peers is an arbitrary value. By keeping a total ordering of peers and employing a robust locality-preserved resource placement strategy, resources that are similar in a single or multidimensional attributes space are stored on the same peer or neighboring peers. Through analysis and simulation, we show that BAKE achieves the optimal diameter and as good a connectivity as the Kautz digraph does (almost achieves the Moore bound), and supports the exact as well as the range queries efficiently. Indeed, the structures of balanced Kautz tree and Kautz ring we propose can also be applied to other interconnection networks after minimal modifications, for example, the de Bruijn digraph.