317,115 research outputs found
Counting spanning trees in a small-world Farey graph
The problem of spanning trees is closely related to various interesting
problems in the area of statistical physics, but determining the number of
spanning trees in general networks is computationally intractable. In this
paper, we perform a study on the enumeration of spanning trees in a specific
small-world network with an exponential distribution of vertex degrees, which
is called a Farey graph since it is associated with the famous Farey sequence.
According to the particular network structure, we provide some recursive
relations governing the Laplacian characteristic polynomials of a Farey graph
and its subgraphs. Then, making use of these relations obtained here, we derive
the exact number of spanning trees in the Farey graph, as well as an
approximate numerical solution for the asymptotic growth constant
characterizing the network. Finally, we compare our results with those of
different types of networks previously investigated.Comment: Definitive version accepted for publication in Physica
Parallel Graph Connectivity in Log Diameter Rounds
We study graph connectivity problem in MPC model. On an undirected graph with
nodes and edges, round connectivity algorithms have been
known for over 35 years. However, no algorithms with better complexity bounds
were known. In this work, we give fully scalable, faster algorithms for the
connectivity problem, by parameterizing the time complexity as a function of
the diameter of the graph. Our main result is a
time connectivity algorithm for diameter- graphs, using total
memory. If our algorithm can use more memory, it can terminate in fewer rounds,
and there is no lower bound on the memory per processor.
We extend our results to related graph problems such as spanning forest,
finding a DFS sequence, exact/approximate minimum spanning forest, and
bottleneck spanning forest. We also show that achieving similar bounds for
reachability in directed graphs would imply faster boolean matrix
multiplication algorithms.
We introduce several new algorithmic ideas. We describe a general technique
called double exponential speed problem size reduction which roughly means that
if we can use total memory to reduce a problem from size to , for
in one phase, then we can solve the problem in
phases. In order to achieve this fast reduction for graph
connectivity, we use a multistep algorithm. One key step is a carefully
constructed truncated broadcasting scheme where each node broadcasts neighbor
sets to its neighbors in a way that limits the size of the resulting neighbor
sets. Another key step is random leader contraction, where we choose a smaller
set of leaders than many previous works do
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Aspects of graph colouring
The four-colour conjecture of 1852, and the total colouring conjecture of 1965, have sparked off many new concepts and conjectures. In this thesis we investigate many of the outstanding conjectures, establishing various related results, and present many conjectures of our own. We give a brief historical introduction (Chapter 1) and establish some notation, terminology and techniques (Chapter 2). Next, in Chapter 3, we examine the use of latin squares to represent edge and total colourings. In Chapters 4 - 6 we deal with vertex, edge and total colourings respectively. Various ways of measuring different aspects of graphs are presented, in particular, the ‘colouring difference’ between two edge-colourings of a graph (Chapter 5) and the ‘beta parameter’ (defined in Chapter 2 and used in Chapters 3 and 6); this is a measure of how far from a type 1 graph a type 2 graph can be. In Chapter 6 we derive an upper bound for the beta value of any near type 1 graph and give the exact results for all Kn. The number of ways of colouring Kn and Kn,,n are also quantified. Chapter 6 also examines Hilton’s concept of conformability. It is shown that every graph with at least A spines is conformable, and an extension to the concept, which we call G*-conformability, is introduced. We then give new necessary conditions for a cubic graph to be type 1 in relation to G*-conformability. Various methods of manipulating graphs are considered and we present: a method to compatibly triangulate a graph G-e; a method of introducing a fourth colour thus allowing a sequence of Kempe interchanges from any edge 3-colouring of a cubic graph to any other; and a method to re-colour a near type 1 graph within a certain bound on beta. We end this thesis with a brief discussion on possible practical uses for colouring graphs. A list of the main results and conjectures is given at the end of each chapter, but a short list of the principle theorems proven is given below
Finding conserved patterns in biological sequences, networks and genomes
Biological patterns are widely used for identifying biologically interesting regions
within macromolecules, classifying biological objects, predicting functions and studying
evolution. Good pattern finding algorithms will help biologists to formulate and
validate hypotheses in an attempt to obtain important insights into the complex
mechanisms of living things.
In this dissertation, we aim to improve and develop algorithms for five biological
pattern finding problems. For the multiple sequence alignment problem, we propose
an alternative formulation in which a final alignment is obtained by preserving pairwise
alignments specified by edges of a given tree. In contrast with traditional NPhard
formulations, our preserving alignment formulation can be solved in polynomial
time without using a heuristic, while having very good accuracy.
For the path matching problem, we take advantage of the linearity of the query
path to reduce the problem to finding a longest weighted path in a directed acyclic
graph. We can find k paths with top scores in a network from the query path in
polynomial time. As many biological pathways are not linear, our graph matching
approach allows a non-linear graph query to be given. Our graph matching formulation
overcomes the common weakness of previous approaches that there is no
guarantee on the quality of the results.
For the gene cluster finding problem, we investigate a formulation based on constraining the overall size of a cluster and develop statistical significance estimates that
allow direct comparisons of clusters of different sizes. We explore both a restricted
version which requires that orthologous genes are strictly ordered within each cluster,
and the unrestricted problem that allows paralogous genes within a genome and clusters
that may not appear in every genome. We solve the first problem in polynomial
time and develop practical exact algorithms for the second one.
In the gene cluster querying problem, based on a querying strategy, we propose
an efficient approach for investigating clustering of related genes across multiple
genomes for a given gene cluster. By analyzing gene clustering in 400 bacterial
genomes, we show that our algorithm is efficient enough to study gene clusters across
hundreds of genomes
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