23 research outputs found
Fault-tolerant routing in peer-to-peer systems
We consider the problem of designing an overlay network and routing mechanism
that permits finding resources efficiently in a peer-to-peer system. We argue
that many existing approaches to this problem can be modeled as the
construction of a random graph embedded in a metric space whose points
represent resource identifiers, where the probability of a connection between
two nodes depends only on the distance between them in the metric space. We
study the performance of a peer-to-peer system where nodes are embedded at grid
points in a simple metric space: a one-dimensional real line. We prove upper
and lower bounds on the message complexity of locating particular resources in
such a system, under a variety of assumptions about failures of either nodes or
the connections between them. Our lower bounds in particular show that the use
of inverse power-law distributions in routing, as suggested by Kleinberg
(1999), is close to optimal. We also give efficient heuristics to dynamically
maintain such a system as new nodes arrive and old nodes depart. Finally, we
give experimental results that suggest promising directions for future work.Comment: Full version of PODC 2002 paper. New version corrects missing
conditioning in Lemma 9 and some related details in the proof of Theorem 10,
with no changes to main result
An analytical framework for the performance evaluation of proximity-aware structured overlays
In this paper, we present an analytical study of proximity-aware structured peer-to-peer networks under churn. We use a master-equation-based approach, which is used traditionally in non-equilibrium statistical mechanics to describe steady-state or transient phenomena. In earlier work we have demonstrated that this methodology is in fact also well suited to describing structured overlay networks under churn, by showing how we can accurately predict the average number of hops taken by a lookup, for any value of churn, for the Chord system. In this paper, we extend the analysis so as to also be able to predict lookup latency, given an average latency for the links in the network. Our results show that there exists a region in the parameter space of the model, depending on churn, the number of nodes, the maintenance rates and the delays in the network, when the network cannot function as a small world graph anymore, due to the farthest connections of a node always being wrong or dead. We also demonstrate how it is possible to analyse proximity neighbour selection or proximity route selection within this formalism
Comparing Maintenance Strategies for Overlays
In this paper, we present an analytical tool for understanding the
performance of structured overlay networks under churn based on the
master-equation approach of physics. We motivate and derive an equation for the
average number of hops taken by lookups during churn, for the Chord network. We
analyse this equation in detail to understand the behaviour with and without
churn. We then use this understanding to predict how lookups will scale for
varying peer population as well as varying the sizes of the routing tables. We
then consider a change in the maintenance algorithm of the overlay, from
periodic stabilisation to a reactive one which corrects fingers only when a
change is detected. We generalise our earlier analysis to underdstand how the
reactive strategy compares with the periodic one.Comment: 10 pages, 8 figure
Tight Lower Bounds for Greedy Routing in Higher-Dimensional Small-World Grids
We consider Kleinberg's celebrated small world graph model (Kleinberg, 2000),
in which a D-dimensional grid {0,...,n-1}^D is augmented with a constant number
of additional unidirectional edges leaving each node. These long range edges
are determined at random according to a probability distribution (the
augmenting distribution), which is the same for each node. Kleinberg suggested
using the inverse D-th power distribution, in which node v is the long range
contact of node u with a probability proportional to ||u-v||^(-D). He showed
that such an augmenting distribution allows to route a message efficiently in
the resulting random graph: The greedy algorithm, where in each intermediate
node the message travels over a link that brings the message closest to the
target w.r.t. the Manhattan distance, finds a path of expected length O(log^2
n) between any two nodes. In this paper we prove that greedy routing does not
perform asymptotically better for any uniform and isotropic augmenting
distribution, i.e., the probability that node u has a particular long range
contact v is independent of the labels of u and v and only a function of
||u-v||.
In order to obtain the result, we introduce a novel proof technique: We
define a budget game, in which a token travels over a game board, while the
player manages a "probability budget". In each round, the player bets part of
her remaining probability budget on step sizes. A step size is chosen at random
according to a probability distribution of the player's bet. The token then
makes progress as determined by the chosen step size, while some of the
player's bet is removed from her probability budget. We prove a tight lower
bound for such a budget game, and then obtain a lower bound for greedy routing
in the D-dimensional grid by a reduction
Navigability is a Robust Property
The Small World phenomenon has inspired researchers across a number of
fields. A breakthrough in its understanding was made by Kleinberg who
introduced Rank Based Augmentation (RBA): add to each vertex independently an
arc to a random destination selected from a carefully crafted probability
distribution. Kleinberg proved that RBA makes many networks navigable, i.e., it
allows greedy routing to successfully deliver messages between any two vertices
in a polylogarithmic number of steps. We prove that navigability is an inherent
property of many random networks, arising without coordination, or even
independence assumptions