414 research outputs found
A study of multiloop networks
Multiloop networks is a family of network topologies which is an extension of the ring topology. In this thesis we study the structural properties of bipartite double loop networks using the plane tessellation technique. We also study the problem of broadcasting in the bipartite double loop networks and in triple loop networks. For the first kind of graphs we find that the broadcast time is d + 2 where d is the diameter of the graph. For the triple loop graphs, we give a d + 5 upper bound on the broadcast time by providing an algorithm that completes broadcasting in at most d + 5 time units. We also find a d + 2 lower bound for the optimal triple loop graphs, these are the graphs with maximum number of nodes given a diameter d . Finally we give an upper bound for the broadcast time of undirected Circulant (also called multiloop) graphs of degree 2 k which is d + 2 k -
Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results
Recent results establish for 2-spin antiferromagnetic systems that the
computational complexity of approximating the partition function on graphs of
maximum degree D undergoes a phase transition that coincides with the
uniqueness phase transition on the infinite D-regular tree. For the
ferromagnetic Potts model we investigate whether analogous hardness results
hold. Goldberg and Jerrum showed that approximating the partition function of
the ferromagnetic Potts model is at least as hard as approximating the number
of independent sets in bipartite graphs (#BIS-hardness). We improve this
hardness result by establishing it for bipartite graphs of maximum degree D. We
first present a detailed picture for the phase diagram for the infinite
D-regular tree, giving a refined picture of its first-order phase transition
and establishing the critical temperature for the coexistence of the disordered
and ordered phases. We then prove for all temperatures below this critical
temperature that it is #BIS-hard to approximate the partition function on
bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to
approximate the number of k-colorings on bipartite graphs of maximum degree D
when k <= D/(2 ln D).
The #BIS-hardness result for the ferromagnetic Potts model uses random
bipartite regular graphs as a gadget in the reduction. The analysis of these
random graphs relies on recent connections between the maxima of the
expectation of their partition function, attractive fixpoints of the associated
tree recursions, and induced matrix norms. We extend these connections to
random regular graphs for all ferromagnetic models and establish the Bethe
prediction for every ferromagnetic spin system on random regular graphs. We
also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm
is torpidly mixing on random D-regular graphs at the critical temperature for
large q.Comment: To appear in SIAM J. Computin
Properties and Algorithms of the KCube Graphs
The KCube interconnection topology was rst introduced in 2010. The KCube graph
is a compound graph of a Kautz digraph and hypercubes. Compared with the at-
tractive Kautz digraph and well known hypercube graph, the KCube graph could
accommodate as many nodes as possible for a given indegree (and outdegree) and the diameter of interconnection networks. However, there are few algorithms designed for the KCube graph. In this thesis, we will concentrate on nding graph theoretical properties of the KCube graph and designing parallel algorithms that run on this network. We will explore several topological properties, such as bipartiteness, Hamiltonianicity, and symmetry property. These properties for the KCube graph are very useful to develop efficient algorithms on this network. We will then study the KCube network from the algorithmic point of view, and will give an improved routing algorithm. In addition, we will present two optimal broadcasting algorithms. They are fundamental algorithms to many applications. A literature review of the state of the art network designs in relation to the KCube network as well as some open problems in this field will also be given
Gossiping in chordal rings under the line model
The line model assumes long distance
calls between non neighboring processors. In this sense, the line
model is strongly related to circuit-switched networks, wormhole
routing, optical networks supporting wavelength division
multiplexing, ATM switching, and networks supporting connected mode
routing protocols.
Since the chordal rings are competitors of networks as meshes or
tori because of theirs short diameter and bounded degree, it is of
interest to ask whether they can support intensive communications
(typically all-to-all) as efficiently as these networks. We
propose polynomial algorithms to derive optimal or near optimal
gossip protocols in the chordal ring
Walks and games on graphs
Herrman, Rebekah Ph.D. The University of Memphis, May 2020. Walks and Games on Graphs. Major Professor: B\\u27ela Bollob\\u27as, Ph.D.Chapter 1 is joint work with Dr. Travis Humble and appears in the journal Physical Review A. In this work, we consider continuous-time quantum walks on dynamic graphs. Continuous-time quantum walks have been well studied on graphs that do not change as a function of time. We offer a mathematical formulation for how to express continuous-time quantum walks on graphs that can change in time, find a universal set of walks that can perform any operation, and use them to simulate basic quantum circuits. This work was supported in part by the Department of Energy Student Undergraduate Laboratory Internship and the National Science Foundation Mathematical Sciences Graduate Internship programs.The broadcast domination number of a graph , , is a generalization of the domination number of a graph. In Chapter 2, we consider the broadcast domination number on graphs, specifically powers of cycles, powers of paths, and infinite grids. This work is joint with Peter van Hintum and has been submitted to the journal Discrete Applied Mathematics.Bridge-burning cops and robbers is a variant of the cops and robbers game on graphs in which the robber removes an edge from the graph once it is traversed. In Chapter 3, we study the maximum time it takes the cops to capture the robber in this variant. This is joint with Peter van Hintum and Dr. Stephen Smith.In Chapter 4, we study a variant of the chip-firing game called the \emph{diffusion game}. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. Long and Narayanan asked whether there exists an for each , such that whenever we have a graph on vertices and an initial allocation with at least chips on each vertex, then the number of chips on each vertex will remain non-negative. We answer their question in the affirmative, showing further that is the best possible bound. We also consider the existence of a similar bound for each , where is the maximum degree of the graph. This work is joint with Andrew Carlotti and has been submitted to the journal Discrete Mathematics.In Chapter 5, we consider the eternal game chromatic number of random graphs. The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza \cite{klostermeyer}, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this chapter, we show that with high probability for odd , and also for even when for some . This work is joint with Vojt\u{e}ch Dvo\u{r}\\u27ak and Peter van Hintum, and has been submitted to the European Journal of Combinatorics
Distributed Symmetry Breaking in Hypergraphs
Fundamental local symmetry breaking problems such as Maximal Independent Set
(MIS) and coloring have been recognized as important by the community, and
studied extensively in (standard) graphs. In particular, fast (i.e.,
logarithmic run time) randomized algorithms are well-established for MIS and
-coloring in both the LOCAL and CONGEST distributed computing
models. On the other hand, comparatively much less is known on the complexity
of distributed symmetry breaking in {\em hypergraphs}. In particular, a key
question is whether a fast (randomized) algorithm for MIS exists for
hypergraphs.
In this paper, we study the distributed complexity of symmetry breaking in
hypergraphs by presenting distributed randomized algorithms for a variety of
fundamental problems under a natural distributed computing model for
hypergraphs. We first show that MIS in hypergraphs (of arbitrary dimension) can
be solved in rounds ( is the number of nodes of the
hypergraph) in the LOCAL model. We then present a key result of this paper ---
an -round hypergraph MIS algorithm in
the CONGEST model where is the maximum node degree of the hypergraph
and is any arbitrarily small constant.
To demonstrate the usefulness of hypergraph MIS, we present applications of
our hypergraph algorithm to solving problems in (standard) graphs. In
particular, the hypergraph MIS yields fast distributed algorithms for the {\em
balanced minimal dominating set} problem (left open in Harris et al. [ICALP
2013]) and the {\em minimal connected dominating set problem}. We also present
distributed algorithms for coloring, maximal matching, and maximal clique in
hypergraphs.Comment: Changes from the previous version: More references adde
Heuristic Algorithms For Broadcasting In Cactus Graphs
Broadcasting is an information dissemination problem in a connected network, in which one node, called the originator, disseminates a message to all other nodes by placing a series of calls along the communication lines of the network. Once informed, the nodes aid the originator in distributing the message. Finding the broadcast time of a vertex in an arbitrary graph is NP-complete. The problem is solved polynomially only for a few classes of graphs. In this thesis, we study the broadcast problem in a class of graph called a Cactus Graph. A cactus graph is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle. We review broadcasting on subclasses of cactus graphs such as, the unicyclic graphs, necklace graphs, k-cycle graphs, 2-restricted cactus graphs and k-restricted cactus graphs. We then provide four heuristic algorithms that solves broadcasting on a k-cycle graph. A k-cycle graph is a collection of k cycles of arbitrary lengths all connected to a central vertex. Finally, we run simulations of these heuristic algorithms on different sized k-cycle graphs to compare and discuss the results
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