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.

Properties and Algorithms of the KCube Interconnection Networks

The KCube interconnection network was first introduced in 2010 in order to exploit the good characteristics of two well-known interconnection networks, the hypercube and the Kautz graph. KCube links up multiple processors in a communication network with high density for a fixed degree. Since the KCube network is newly proposed, much study is required to demonstrate its potential properties and algorithms that can be designed to solve parallel computation problems. In this thesis we introduce a new methodology to construct the KCube graph. Also, with regard to this new approach, we will prove its Hamiltonicity in the general KC(m; k). Moreover, we will find its connectivity followed by an optimal broadcasting scheme in which a source node containing a message is to communicate it with all other processors. In addition to KCube networks, we have studied a version of the routing problem in the traditional hypercube, investigating this problem: whether there exists a shortest path in a Qn between two nodes 0n and 1n, when the network is experiencing failed components. We first conditionally discuss this problem when there is a constraint on the number of faulty nodes, and subsequently introduce an algorithm to tackle the problem without restrictions on the number of nodes.

Properties and algorithms of the (n, k)-star graphs

The (n, k)-star interconnection network was proposed in 1995 as an attractive alternative to the n-star topology in parallel computation. The (n, k )-star has significant advantages over the n-star which itself was proposed as an attractive alternative to the popular hypercube. The major advantage of the (n, k )-star network is its scalability, which makes it more flexible than the n-star as an interconnection network. In this thesis, we will focus on finding graph theoretical properties of the (n, k )-star as well as developing parallel algorithms that run on this network. The basic topological properties of the (n, k )-star are first studied. These are useful since they can be used to develop efficient algorithms on this network. We then study the (n, k )-star network from algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms for basic communication, prefix computation, and sorting, etc. A literature review of the state-of-the-art in relation to the (n, k )-star network as well as some open problems in this area are also provided.

Properties and algorithms of the (n, k)-arrangement graphs

The (n, k)-arrangement interconnection topology was first introduced in 1992. The (n, k )-arrangement graph is a class of generalized star graphs. Compared with the well known n-star, the (n, k )-arrangement graph is more flexible in degree and diameter. However, there are few algorithms designed for the (n, k)-arrangement graph up to present. In this thesis, we will focus on finding graph theoretical properties of the (n, k)- arrangement graph and developing parallel algorithms that run on this network. The topological properties of the arrangement graph are first studied. They include the cyclic properties. We then study the problems of communication: broadcasting and routing. Embedding problems are also studied later on. These are very useful to develop efficient algorithms on this network. We then study the (n, k )-arrangement network from the algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms such as prefix sums computation, sorting, merging and basic geometry computation: finding convex hull on the (n, k )-arrangement graph. A literature review of the state-of-the-art in relation to the (n, k)-arrangement network is also provided, as well as some open problems in this area.

Properties and algorithms of the hyper-star graph and its related graphs

The hyper-star interconnection network was proposed in 2002 to overcome the drawbacks of the hypercube and its variations concerning the network cost, which is defined by the product of the degree and the diameter. Some properties of the graph such as connectivity, symmetry properties, embedding properties have been studied by other researchers, routing and broadcasting algorithms have also been designed. This thesis studies the hyper-star graph from both the topological and algorithmic point of view. For the topological properties, we try to establish relationships between hyper-star graphs with other known graphs. We also give a formal equation for the surface area of the graph. Another topological property we are interested in is the Hamiltonicity problem of this graph. For the algorithms, we design an all-port broadcasting algorithm and a single-port neighbourhood broadcasting algorithm for the regular form of the hyper-star graphs. These algorithms are both optimal time-wise. Furthermore, we prove that the folded hyper-star, a variation of the hyper-star, to be maixmally fault-tolerant.

On the Performance of Polynomial-time CLIQUE Approximation Algorithms on Very Large Graphs

The performance of a randomized version of the subgraph-exclusion algorithm (called Ramsey) for CLIQUE by Boppana and Halldorsson is studied on very large graphs. We compare the performance of this algorithm with the performance of two common heuristic algorithms, the greedy heuristic and a version of simulated annealing. These algorithms are tested on graphs with up to 10,000 vertices on a workstation and graphs as large as 70,000 vertices on a Connection Machine. Our implementations establish the ability to run clique approximation algorithms on very large graphs. We test our implementations on a variety of different graphs. Our conclusions indicate that on randomly generated graphs minor changes to the distribution can cause dramatic changes in the performance of the heuristic algorithms. The Ramsey algorithm, while not as good as the others for the most common distributions, seems more robust and provides a more even overall performance. In general, and especially on deterministically generated graphs, a combination of simulated annealing with either the Ramsey algorithm or the greedy heuristic seems to perform best. This combined algorithm works particularly well on large Keller and Hamming graphs and has a competitive overall performance on the DIMACS benchmark graphs.

Asymmetric Ramsey Properties of Random Graphs Involving Cliques

Consider the following problem: Forgiven graphs G and F(1),..., F(k), find a coloring of the edges of G with k colors such that G does not contain F; in color i. Rodl and Rucinski studied this problem for the random graph G,,, in the symmetric case when k is fixed and F(1) = ... = F(k) = F. They proved that such a coloring exists asymptotically almost surely (a.a.s.) provided that p <= bn(-beta) for some constants b = b(F,k) and beta = beta(F). This result is essentially best possible because for p >= Bn(-beta), where B = B(F, k) is a large constant, such an edge-coloring does not exist. Kohayakawa and Kreuter conjectured a threshold function n(-beta(F1,..., Fk)) for arbitrary F(1), ..., F(k). In this article we address the case when F(1),..., F(k) are cliques of different sizes and propose an algorithm that a.a.s. finds a valid k-edge-coloring of G(n,p) with p <= bn(-beta) for some constant b = b(F(1),..., F(k)), where beta = beta(F(1),..., F(k)) as conjectured. With a few exceptions, this algorithm also works in the general symmetric case. We also show that there exists a constant B = B(F,,..., Fk) such that for p >= Bn(-beta) the random graph G(n,p) a.a.s. does not have a valid k-edge-coloring provided the so-called KLR-conjecture holds. (C) 2008 Wiley Periodicals, Inc. Random Struct. Alg., 34, 419-453, 2009

Efficient algorithms for computing Reeb graphs

The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3-manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(nlogn+nlogg(loglogg)3) time, where n is the number of triangles in the tetrahedral mesh representing the 3-manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of d-manifolds in O(nlogn(loglogn)3) time, where n is the number of triangles in the simplicial complex that represents the d-manifold. Our result is a significant improvement over the previously known O(n2) algorithm. Finally, we present experimental results of our implementation and demonstrate that our algorithm for 3-manifolds performs efficiently in practice.

Algorithms for Constructing Hierarchical Graphs

An algorithm is described for developing a hierarchy among a set of elements having certain precedence relations. This algorithm, which is based on tracing a path through the graph, is easily implemented by a computer.

Algorithms for Constructing Hierarchical Graphs

An algorithm is described for developing a hierarchy among a set of elements having certain precedence relations. This algorithm, which is based on tracing a path through the graph, is easily implemented by a computer.

Parallel genetic algorithms for the solution of inverse heat conduction problems

A parallel genetic algorithm (PGA) is proposed for the solution of two-dimensional inverse heat conduction problems involving unknown thermophysical material properties. Experimental results show that the proposed PGA is a feasible and effective optimization tool for inverse heat conduction problems

Regional ocean-colour chlorophyll algorithms for the Red Sea

The Red Sea is a semi-enclosed tropical marine ecosystem that stretches from the Gulf of Suez and Gulf of Aqaba in the north, to the Gulf of Aden in the south. Despite its ecological and economic importance, its biological environment is relatively unexplored. Satellite ocean-colour estimates of chlorophyll concentration (an index of phytoplankton biomass) offer an observational platform to monitor the health of the Red Sea. However, little is known about the optical properties of the region. In this paper, we investigate the optical properties of the Red Sea in the context of satellite ocean-colour estimates of chlorophyll concentration. Making use of a new merged ocean-colour product, from the European Space Agency (ESA) Climate Change Initiative, and in situ data in the region, we test the performance of a series of ocean-colour chlorophyll algorithms. We find that standard algorithms systematically overestimate chlorophyll when compared with the in situ data. To investigate this bias we develop an ocean-colour model for the Red Sea, parameterised to data collected during the Tara Oceans expedition, that estimates remote-sensing reflectance as a function of chlorophyll concentration. We used the Red Sea model to tune the standard chlorophyll algorithms and the overestimation in chlorophyll originally observed was corrected. Results suggest that the overestimation was likely due to an excess of CDOM absorption per unit chlorophyll in the Red Sea when compared with average global conditions. However, we recognise that additional information is required to test the influence of other potential sources of the overestimation, such as aeolian dust, and we discuss uncertainties in the datasets used. We present a series of regional chlorophyll algorithms for the Red Sea, designed for a suite of ocean-colour sensors, that may be used for further testing.

CoastColour Round Robin datasets: a database to evaluate the performance of algorithms for the retrieval of water quality parameters in coastal waters.

The use of in situ measurements is essential in the validation and evaluation of the algorithms that provide coastal water quality data products from ocean colour satellite remote sensing. Over the past decade, various types of ocean colour algorithms have been developed to deal with the optical complexity of coastal waters. Yet there is a lack of a comprehensive intercomparison due to the availability of quality checked in situ databases. The CoastColour Round Robin (CCRR) project, funded by the European Space Agency (ESA), was designed to bring together three reference data sets using these to test algorithms and to assess their accuracy for retrieving water quality parameters. This paper provides a detailed description of these reference data sets, which include the Medium Resolution Imaging Spectrometer (MERIS) level 2 match-ups, in situ reflectance measurements, and synthetic data generated by a radiative transfer model (HydroLight). These data sets, representing mainly coastal waters, are available from doi:10.1594/PANGAEA.841950. The data sets mainly consist of 6484 marine reflectance (either multispectral or hyperspectral) associated with various geometrical (sensor viewing and solar angles) and sky conditions and water constituents: total suspended matter (TSM) and chlorophyll a (CHL) concentrations, and the absorption of coloured dissolved organic matter (CDOM). Inherent optical properties are also provided in the simulated data sets (5000 simulations) and from 3054 match-up locations. The distributions of reflectance at selected MERIS bands and band ratios, CHL and TSM as a function of reflectance, from the three data sets are compared. Match-up and in situ sites where deviations occur are identified. The distributions of the three reflectance data sets are also compared to the simulated and in situ reflectances used previously by the International Ocean Colour Coordinating Group (IOCCG, 2006) for algorithm testing, showing a clear extension of the CCRR data which covers more turbid waters.

Paving the way for the use of the SDQ in economic evaluations of school-based population health interventions: an empirical analysis of the external validity of SDQ mapping algorithms to the CHU9D in an educational setting

Purpose
The Strengths and Difficulties Questionnaire (SDQ) is a behavioural screening tool for children. The SDQ is increasingly used as the primary outcome measure in population health interventions involving children, but it is not preference based; therefore, its role in allocative economic evaluation is limited. The Child Health Utility 9D (CHU9D) is a generic preference-based health-related quality of-life measure. This study investigates the applicability of the SDQ outcome measure for use in economic evaluations and examines its relationship with the CHU9D by testing previously published mapping algorithms. The aim of the paper is to explore the feasibility of using the SDQ within economic evaluations of school-based population health interventions.
Methods
Data were available from children participating in a cluster randomised controlled trial of the school-based roots of empathy programme in Northern Ireland. Utility was calculated using the original and alternative CHU9D tariffs along with two SDQ mapping algorithms. t tests were performed for pairwise differences in utility values from the preference-based tariffs and mapping algorithms.
Results
Mean (standard deviation) SDQ total difficulties and prosocial scores were 12 (3.2) and 8.3 (2.1). Utility values obtained from the original tariff, alternative tariff, and mapping algorithms using five and three SDQ subscales were 0.84 (0.11), 0.80 (0.13), 0.84 (0.05), and 0.83 (0.04), respectively. Each method for calculating utility produced statistically significantly different values except the original tariff and five SDQ subscale algorithm.
Conclusion
Initial evidence suggests the SDQ and CHU9D are related in some of their measurement properties. The mapping algorithm using five SDQ subscales was found to be optimal in predicting mean child health utility. Future research valuing changes in the SDQ scores would contribute to this research.

The CoastColour Round Robin datasets: a database to evaluate the performance of algorithms for the retrieval of water quality parameters in coastal waters

The CoastColour project Round Robin (CCRR) project (http://www.coastcolour.org) funded by the European Space Agency (ESA) was designed to bring together a variety of reference datasets and to use these to test algorithms and assess their accuracy for retrieving water quality parameters. This information was then developed to help end-users of remote sensing products to select the most accurate algorithms for their coastal region. To facilitate this, an inter-comparison of the performance of algorithms for the retrieval of in-water properties over coastal waters was carried out. The comparison used three types of datasets on which ocean colour algorithms were tested. The description and comparison of the three datasets are the focus of this paper, and include the Medium Resolution Imaging Spectrometer (MERIS) Level 2 match-ups, in situ reflectance measurements and data generated by a radiative transfer model (HydroLight). The datasets mainly consisted of 6,484 marine reflectance associated with various geometrical (sensor viewing and solar angles) and sky conditions and water constituents: Total Suspended Matter (TSM) and Chlorophyll-a (CHL) concentrations, and the absorption of Coloured Dissolved Organic Matter (CDOM). Inherent optical properties were also provided in the simulated datasets (5,000 simulations) and from 3,054 match-up locations. The distributions of reflectance at selected MERIS bands and band ratios, CHL and TSM as a function of reflectance, from the three datasets are compared. Match-up and in situ sites where deviations occur are identified. The distribution of the three reflectance datasets are also compared to the simulated and in situ reflectances used previously by the International Ocean Colour Coordinating Group (IOCCG, 2006) for algorithm testing, showing a clear extension of the CCRR data which covers more turbid waters.