A Peer-to-Peer (P2P) network is a dynamic collection of nodes that connect with each other via virtual overlay links built upon an underlying network (usually, the Internet). Typical P2P networks, however, are highly dynamic and can experience very heavy churn, i.e., a large number of nodes join/leave the network every time step. Building and maintaining a stable overlay network despite such heavy churn is therefore an important problem that has been studied extensively for nearly two decades.We present an overlay design called Sparse Robust Addressable Network (Spartan) that can tolerate heavy adversarial churn. We show that Spartan can be built efficiently in a fully distributed manner within O(log n) rounds. Furthermore, the Spartan overlay structure can be maintained, again, in a fully distributed manner despite adversarially controlled churn (i.e., nodes joining and leaving) and significant variation in the number of nodes. When the number of nodes in the network lies in [n, f n] for any fixed f ≥ 1 the adversary can remove up to n nodes and add up to n nodes (for some small but fixed > 0) within any period of P rounds for some P ∈ O(log log n). Moreover, the adversary can add or remove nodes from the network at will and without any forewarning.Despite such uncertainty in the network, Spartan maintains Θ(n/ log n) committees that are stable and addressable collections of Θ(log n) nodes each. Any node that enters the network will be able to gain membership in one of these committees within O(1) rounds. The committees are also capable of performing sustained computation and passing messages between each other. Thus, any protocol designed for static networks can be simulated on Spartan with minimal overhead. This makes Spartan an ideal platform for developing applications. All our results hold with high probability. *
We study graph realization problems from a distributed perspective. The problem is naturally applicable to the distributed construction of overlay networks that must satisfy certain degree or connectivity properties, and we study it in the node capacitated clique (NCC) model of distributed computing, recently introduced for representing peer-to-peer networks.We focus on two central variants, degree-sequence realization and minimum threshold-connectivity realization. In the degree sequence problem, each node v is associated with a degree d(v), and the resulting degree sequence is realizable if it is possible to construct an overlay network in which the degree of each node v is d(v). The minimum threshold-connectivity problem requires us to construct an overlay network that satisfies connectivity constraints specified between every pair of nodes. Overlay network realizations can be either explicit or implicit. Explicit realizations require both endpoints of any edge in the realized graph to be aware of the edge. In implicit realizations, on the other hand, at least one endpoint of each edge of the realized graph needs to be aware of the edge. The main realization algorithms we present are the following.• An Õ(min{ √ m, ∆}) time algorithm for implicit realization of a degree sequence. Here, ∆ = max v d(v) is the maximum degree and m = (1/2) v d(v) is the number of edges in the final realization.• Õ(∆) time algorithm for an explicit realization of a degree sequence. We first compute an implicit realization and then transform it into an explicit one in Õ(∆) additional rounds.• An Õ(∆) time algorithm for the threshold connectivity problem that obtains an explicit solution and an improved Õ(1) algorithm for implicit realization when all nodes know each other's IDs. These algorithms are 2-approximations w.r.t. the number of edges.We complement our upper bounds with lower bounds to show that the above algorithms are tight up to factors of log n. Additionally, we provide algorithms for realizing trees and an Õ(1) round algorithm for approximate degree sequence realization.
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