Spartan: A Framework For Sparse Robust Addressable Networks

Abstract

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. *

Time-Optimal Construction of Overlay Networks

“…In this category, we have algorithms that maintain an overlay topology under randomized or adversarial errors. These works focus on quickly reconfiguring the network to distribute the load evenly (under churn) or to reach an unpredictable topology (in the presence of an adversary) [8,9,16,29]. However, a common assumption is that the overlay starts in some well-defined initial state.…”

“…Furthermore, the degree of is (log ). To obtain a well-formed tree , we first perform a BFS on starting from the node with lowest identifier 9 . This requires time (log ) and gives us a rooted tree with degree and diameter (log ).…”

“…8 We will see that this case does not occur w.h.p. 9 Since a node cannot locally check whether it has the lowest identifier, the implementation of this step is slightly more complex: Every node simultaneously floods the graph with a token message that contains its identifier. Every node that receives one or more tokens only forwards the token with the lowest identifier.…”

“…These problems include monitoring the graph's node and edge count, its bipartiteness, and the approximate and exact weight of an MST. Second, for the churn resistant overlays in [5,8,9,16,29], the assumption that the graph starts in a well-initialized overlay can be dropped. Instead, the desired topology can be constructed in the first (log( )) rounds.…”

Distributed Graph Realizations

“…For example, Fiat and Saia [12] introduced an overlay using butterfly networks that is tolerant of faults. This was used in [3,4] to provide a low diameter structure that is also easily addressable. Detailed surveys of various overlays and their properties can be found in [25,26].…”