A distributed polylogarithmic time algorithm for self-stabilizing skip graphs

Abstract

Peer-to-peer systems rely on scalable overlay networks that enable efficient routing between its members. Hypercubic topologies facilitate such operations while each node only needs to connect to a small number of other nodes. In contrast to static communication networks, peer-to-peer networks allow nodes to adapt their neighbor set over time in order to react to join and leave events and failures. This paper shows how to maintain such networks in a robust manner. Concretely, we present a distributed and self-stabilizing algorithm that constructs a (variant of the) skip graph in polylogarithmic time from any initial state in which the overlay network is still weakly connected. This is an exponential improvement compared to previously known self-stabilizing algorithms for overlay networks. In addition, individual joins and leaves are handled locally and require little work.

Enabling Robust and Efficient Distributed Computation in Dynamic Peer-to-Peer Networks

“…On the other hand, an important aspect of our protocol is that it works even when the network is continually changing in an highly dynamic and adversarial manner. Most prior algorithms (e.g., [24], [35], [36], [23], [37], [38]) will only work under the assumption that the network will eventually stabilize and stop changing or there is a "repair" time for maintenance when there are no further changes (till the repair/maintenance is finished); these algorithms do not work under high continuous adversarial churn. Law and Siu [24] provide a distributed algorithm for maintaining an expander in the presence of limited number of insertions/deletions; their algorithm does not work for high continuous adversarial churn.…”

“…In the same model, [23] present a protocol that maintains constant node degrees and constant expansion (both with probability 1) against an adaptive adversary, while requiring only logarithmic (in the network size) messages, time, and topology changes per deletion/insertion. [35] presented a self-stabilizing algorithm that converges from any weakly connected graph to a SKIP graph (which is an expander with high probability) in time polylogarithmic in the network size. In [37] the authors introduce the hyperring, which is a search data structure supporting insertions and deletions, while being able to handle concurrent requests with low congestion and dilation, while guaranteeing O(1/ log n) expansion and O(log n) node degree.…”

Locally Self-Adjusting Tree Networks

“…In this sense, a peer-to-peer system can never be fully "repaired" but must always be fully functional. Today, peer-to-peer networking is a relatively mature field of research, and there are many solutions to maintain desirable network properties under both randomized [22] as well as worst case [14] membership changes, and some peer-to-peer networks are even self-stabilizing [12] in the sense that they quickly converge to a desirable topology (e.g., a hypercube) from an arbitrary connected structure. However, none of these systems are self-adjusting to the demand.…”

A Self-Stabilization Process for Small-World Networks

“…Jacob et al [13] generalize insights gained from graph linearization to two dimensions and present a self-stabilizing construction for Delaunay graphs. In another paper, Jacob et al [12] present a self-stabilizing variant of the skip graph and show that it can recover its network topology from any weakly connected state in O(log 2 n) communication rounds with high probability. Recently, also a self-stabilizing variant of the popular Chord network was presented [15].…”