965 research outputs found
The Maximum Degree-and-Diameter-Bounded Subgraph in the Mesh
The problem of finding the largest connected subgraph of a given undirected
host graph, subject to constraints on the maximum degree and the
diameter , was introduced in \cite{maxddbs}, as a generalization of the
Degree-Diameter Problem. A case of special interest is when the host graph is a
common parallel architecture. Here we discuss the case when the host graph is a
-dimensional mesh. We provide some general bounds for the order of the
largest subgraph in arbitrary dimension , and for the particular cases of
and , we give constructions that result in
sharper lower bounds.Comment: accepted, 18 pages, 7 figures; Discrete Applied Mathematics, 201
A transfer principle and applications to eigenvalue estimates for graphs
In this paper, we prove a variant of the Burger-Brooks transfer principle
which, combined with recent eigenvalue bounds for surfaces, allows to obtain
upper bounds on the eigenvalues of graphs as a function of their genus. More
precisely, we show the existence of a universal constants such that the
-th eigenvalue of the normalized Laplacian of a graph
of (geometric) genus on vertices satisfies where denotes the maximum valence of
vertices of the graph. This result is tight up to a change in the value of the
constant , and improves recent results of Kelner, Lee, Price and Teng on
bounded genus graphs.
To show that the transfer theorem might be of independent interest, we relate
eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple
graph models, and discuss an application to the mesh partitioning problem,
extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang
to arbitrary meshes.Comment: Major revision, 16 page
Scaling limits of loop-erased random walks and uniform spanning trees
The uniform spanning tree (UST) and the loop-erased random walk (LERW) are
related probabilistic processes. We consider the limits of these models on a
fine grid in the plane, as the mesh goes to zero. Although the existence of
scaling limits is still unproven, subsequential scaling limits can be defined
in various ways, and do exist. We establish some basic a.s. properties of the
subsequential scaling limits in the plane. It is proved that any LERW
subsequential scaling limit is a simple path, and that the trunk of any UST
subsequential scaling limit is a topological tree, which is dense in the plane.
The scaling limits of these processes are conjectured to be conformally
invariant in 2 dimensions. We make a precise statement of the conformal
invariance conjecture for the LERW, and show that this conjecture implies an
explicit construction of the scaling limit, as follows. Consider the Loewner
differential equation
with boundary values , in the range z\in\U=\{w\in\C\st |w|<1\},
. We choose \zeta(t):= \B(-2t), where \B(t) is Brownian motion on
\partial \U starting at a random-uniform point in \partial \U. Assuming the
conformal invariance of the LERW scaling limit in the plane, we prove that the
scaling limit of LERW from 0 to \partial\U has the same law as that of the
path . We believe that a variation of this process gives the
scaling limit of the boundary of macroscopic critical percolation clusters.Comment: (for V2) inserted another figure and two more reference
Distributed match-making
In many distributed computing environments, processes are concurrently executed by nodes in a store- and-forward communication network. Distributed control issues as diverse as name server, mutual exclusion, and replicated data management involve making matches between such processes. We propose a formal problem called distributed match-making as the generic paradigm. Algorithms for distributed match-making are developed and the complexity is investigated in terms of messages and in terms of storage needed. Lower bounds on the complexity of distributed match-making are established. Optimal algorithms, or nearly optimal algorithms, are given for particular network topologies
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