65,923 research outputs found
Quasirandom Rumor Spreading: An Experimental Analysis
We empirically analyze two versions of the well-known "randomized rumor
spreading" protocol to disseminate a piece of information in networks. In the
classical model, in each round each informed node informs a random neighbor. In
the recently proposed quasirandom variant, each node has a (cyclic) list of its
neighbors. Once informed, it starts at a random position of the list, but from
then on informs its neighbors in the order of the list. While for sparse random
graphs a better performance of the quasirandom model could be proven, all other
results show that, independent of the structure of the lists, the same
asymptotic performance guarantees hold as for the classical model. In this
work, we compare the two models experimentally. This not only shows that the
quasirandom model generally is faster, but also that the runtime is more
concentrated around the mean. This is surprising given that much fewer random
bits are used in the quasirandom process. These advantages are also observed in
a lossy communication model, where each transmission does not reach its target
with a certain probability, and in an asynchronous model, where nodes send at
random times drawn from an exponential distribution. We also show that
typically the particular structure of the lists has little influence on the
efficiency.Comment: 14 pages, appeared in ALENEX'0
Perfect Matchings as IID Factors on Non-Amenable Groups
We prove that in every bipartite Cayley graph of every non-amenable group,
there is a perfect matching that is obtained as a factor of independent uniform
random variables. We also discuss expansion properties of factors and improve
the Hoffman spectral bound on independence number of finite graphs.Comment: 16 pages; corrected missing reference in v
Geographic Gossip: Efficient Averaging for Sensor Networks
Gossip algorithms for distributed computation are attractive due to their
simplicity, distributed nature, and robustness in noisy and uncertain
environments. However, using standard gossip algorithms can lead to a
significant waste in energy by repeatedly recirculating redundant information.
For realistic sensor network model topologies like grids and random geometric
graphs, the inefficiency of gossip schemes is related to the slow mixing times
of random walks on the communication graph. We propose and analyze an
alternative gossiping scheme that exploits geographic information. By utilizing
geographic routing combined with a simple resampling method, we demonstrate
substantial gains over previously proposed gossip protocols. For regular graphs
such as the ring or grid, our algorithm improves standard gossip by factors of
and respectively. For the more challenging case of random
geometric graphs, our algorithm computes the true average to accuracy
using radio
transmissions, which yields a factor improvement over
standard gossip algorithms. We illustrate these theoretical results with
experimental comparisons between our algorithm and standard methods as applied
to various classes of random fields.Comment: To appear, IEEE Transactions on Signal Processin
Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball
graphs) in -dimensional hyperbolic space, which we denote by .
Using a new separator theorem, we show that unit ball graphs in
enjoy similar properties as their Euclidean counterparts, but in one dimension
lower: many standard graph problems, such as Independent Set, Dominating Set,
Steiner Tree, and Hamiltonian Cycle can be solved in
time for any fixed , while the same problems need
time in . We also show that these algorithms in
are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we
introduce a new technique to bound the treewidth of noisy uniform disk graphs.
The bounds yield quasi-polynomial () algorithms for all of the
studied problems, while in the case of Hamiltonian Cycle and -Coloring we
even get polynomial time algorithms. Furthermore, if the underlying noisy disks
in have constant maximum degree, then all studied problems can
be solved in polynomial time. This contrasts with the fact that these problems
require time under ETH in constant maximum degree
Euclidean unit disk graphs.
Finally, we complement our quasi-polynomial algorithm for Independent Set in
noisy uniform disk graphs with a matching lower bound
under ETH. This shows that the hyperbolic plane is a potential source of
NP-intermediate problems.Comment: Short version appears in SODA 202
Developments on Spectral Characterizations of Graphs
In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs
- …