Stochastic Analysis of a Churn-Tolerant Structured Peer-to-Peer Scheme

We present and analyze a simple and general scheme to build a churn (fault)-tolerant structured Peer-to-Peer (P2P) network. Our scheme shows how to "convert" a static network into a dynamic distributed hash table(DHT)-based P2P network such that all the good properties of the static network are guaranteed with high probability (w.h.p). Applying our scheme to a cube-connected cycles network, for example, yields a $O(\log N)$ degree connected network, in which every search succeeds in $O(\log N)$ hops w.h.p., using $O(\log N)$ messages, where $N$ is the expected stable network size. Our scheme has an constant storage overhead (the number of nodes responsible for servicing a data item) and an $O(\log N)$ overhead (messages and time) per insertion and essentially no overhead for deletions. All these bounds are essentially optimal. While DHT schemes with similar guarantees are already known in the literature, this work is new in the following aspects: (1) It presents a rigorous mathematical analysis of the scheme under a general stochastic model of churn and shows the above guarantees; (2) The theoretical analysis is complemented by a simulation-based analysis that validates the asymptotic bounds even in moderately sized networks and also studies performance under changing stable network size; (3) The presented scheme seems especially suitable for maintaining dynamic structures under churn efficiently. In particular, we show that a spanning tree of low diameter can be efficiently maintained in constant time and logarithmic number of messages per insertion or deletion w.h.p. Keywords: P2P Network, DHT Scheme, Churn, Dynamic Spanning Tree, Stochastic Analysis

Quantitative Coding and Complexity Theory of Compact Metric Spaces

Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is usually straightforward and/or complexity-theoretically inessential (up to polynomial time, say); but concerning continuous data, already real numbers naturally suggest various encodings with very different computational properties. With respect to qualitative computability, Kreitz and Weihrauch (1985) had identified ADMISSIBILITY as crucial property for 'reasonable' encodings over the Cantor space of infinite binary sequences, so-called representations [doi:10.1007/11780342_48]: For (precisely) these does the sometimes so-called MAIN THEOREM apply, characterizing continuity of functions in terms of continuous realizers. We rephrase qualitative admissibility as continuity of both the representation and its multivalued inverse, adopting from [doi:10.4115/jla.2013.5.7] a notion of sequential continuity for multifunctions. This suggests its quantitative refinement as criterion for representations suitable for complexity investigations. Higher-type complexity is captured by replacing Cantor's as ground space with Baire or any other (compact) ULTRAmetric space: a quantitative counterpart to equilogical spaces in computability [doi:10.1016/j.tcs.2003.11.012]

Second-Order Linear-Time Computability with Applications to Computable Analysis

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