113,275 research outputs found
Constructions of Large Graphs on Surfaces
We consider the degree/diameter problem for graphs embedded in a surface,
namely, given a surface and integers and , determine the
maximum order of a graph embeddable in with
maximum degree and diameter . We introduce a number of
constructions which produce many new largest known planar and toroidal graphs.
We record all these graphs in the available tables of largest known graphs.
Given a surface of Euler genus and an odd diameter , the
current best asymptotic lower bound for is given by
Our constructions produce
new graphs of order \begin{cases}6\Delta^{\lfloor k/2\rfloor}& \text{if
$\Sigma$ is the Klein bottle}\\
\(\frac{7}{2}+\sqrt{6g+\frac{1}{4}}\)\Delta^{\lfloor k/2\rfloor}&
\text{otherwise,}\end{cases} thus improving the former value by a factor of
4.Comment: 15 pages, 7 figure
Space-Efficient Routing Tables for Almost All Networks and the Incompressibility Method
We use the incompressibility method based on Kolmogorov complexity to
determine the total number of bits of routing information for almost all
network topologies. In most models for routing, for almost all labeled graphs
bits are necessary and sufficient for shortest path routing. By
`almost all graphs' we mean the Kolmogorov random graphs which constitute a
fraction of of all graphs on nodes, where is an arbitrary
fixed constant. There is a model for which the average case lower bound rises
to and another model where the average case upper bound
drops to . This clearly exposes the sensitivity of such bounds
to the model under consideration. If paths have to be short, but need not be
shortest (if the stretch factor may be larger than 1), then much less space is
needed on average, even in the more demanding models. Full-information routing
requires bits on average. For worst-case static networks we
prove a lower bound for shortest path routing and all
stretch factors in some networks where free relabeling is not allowed.Comment: 19 pages, Latex, 1 table, 1 figure; SIAM J. Comput., To appea
Theoretically Efficient Parallel Graph Algorithms Can Be Fast and Scalable
There has been significant recent interest in parallel graph processing due
to the need to quickly analyze the large graphs available today. Many graph
codes have been designed for distributed memory or external memory. However,
today even the largest publicly-available real-world graph (the Hyperlink Web
graph with over 3.5 billion vertices and 128 billion edges) can fit in the
memory of a single commodity multicore server. Nevertheless, most experimental
work in the literature report results on much smaller graphs, and the ones for
the Hyperlink graph use distributed or external memory. Therefore, it is
natural to ask whether we can efficiently solve a broad class of graph problems
on this graph in memory.
This paper shows that theoretically-efficient parallel graph algorithms can
scale to the largest publicly-available graphs using a single machine with a
terabyte of RAM, processing them in minutes. We give implementations of
theoretically-efficient parallel algorithms for 20 important graph problems. We
also present the optimizations and techniques that we used in our
implementations, which were crucial in enabling us to process these large
graphs quickly. We show that the running times of our implementations
outperform existing state-of-the-art implementations on the largest real-world
graphs. For many of the problems that we consider, this is the first time they
have been solved on graphs at this scale. We have made the implementations
developed in this work publicly-available as the Graph-Based Benchmark Suite
(GBBS).Comment: This is the full version of the paper appearing in the ACM Symposium
on Parallelism in Algorithms and Architectures (SPAA), 201
Highly intensive data dissemination in complex networks
This paper presents a study on data dissemination in unstructured
Peer-to-Peer (P2P) network overlays. The absence of a structure in unstructured
overlays eases the network management, at the cost of non-optimal mechanisms to
spread messages in the network. Thus, dissemination schemes must be employed
that allow covering a large portion of the network with a high probability
(e.g.~gossip based approaches). We identify principal metrics, provide a
theoretical model and perform the assessment evaluation using a high
performance simulator that is based on a parallel and distributed architecture.
A main point of this study is that our simulation model considers
implementation technical details, such as the use of caching and Time To Live
(TTL) in message dissemination, that are usually neglected in simulations, due
to the additional overhead they cause. Outcomes confirm that these technical
details have an important influence on the performance of dissemination schemes
and that the studied schemes are quite effective to spread information in P2P
overlay networks, whatever their topology. Moreover, the practical usage of
such dissemination mechanisms requires a fine tuning of many parameters, the
choice between different network topologies and the assessment of behaviors
such as free riding. All this can be done only using efficient simulation tools
to support both the network design phase and, in some cases, at runtime
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