Online Maximum Independent Set of Hyperrectangles

The maximum independent set problem is a classical NP-hard problem in theoretical computer science. In this work, we study a special case where the family of graphs considered is restricted to intersection graphs of sets of axis-aligned hyperrectangles and the input is provided in an online fashion. We prove bounds on the competitive ratio of an optimal online algorithm under the adaptive offline, adaptive online, and oblivious adversary models, for several classes of hyperrectangles and restrictions on the order of the input. We are the first to present results on this problem under the oblivious adversary model. We prove bounds on the competitive ratio for unit hypercubes, $\sigma$-bounded hypercubes, unit-volume hypercubes, arbitrary hypercubes, and arbitrary hyperrectangles, in both arbitrary and non-dominated order. We are also the first to present results under the adaptive offline and adaptive online adversary models with input in non-dominated order, proving bounds on the competitive ratio for the same classes of hyperrectangles; for input in arbitrary order, we present the first results on $\sigma$-bounded hypercubes, unit-volume hyperrectangles, arbitrary hypercubes, and arbitrary hyperrectangles. For input in dominating order, we show that the performance of the naive greedy algorithm matches the performance of an optimal offline algorithm in all cases. We also give lower bounds on the competitive ratio of a probabilistic greedy algorithm under the oblivious adversary model. We conclude by discussing several promising directions for future work.Comment: 27 pages, 12 figure

Maximum Clique in Geometric Intersection Graphs

An intersection graph is a graph that represents some geometric objects as vertices, and joins edges between the nodes corresponding to the items that intersect. The maximum clique in a geometric intersec- tion graph is the largest mutually intersecting set of objects. In this thesis, the primary focus is to study the maximum clique in various geometric intersection graphs. We develop three results motivated by the maximum clique problem in the intersection graph of disks in the Euclidean plane. First, we improve the time complexity of calculating the maximum clique in unit disk graphs from O(n3 log n) to O(n2.5 log n). Second, we introduce a new technique called pair-oriented labelling. This method is used to show the NP- hardness of finding a maximum clique in various geometric intersection graphs, acting as a way to augment the commonly used co-2-subdivision approach. Finally, finding maximum clique in two classes of geometric intersection graphs are proven to be NP-hard. These are the intersection graph of disks and axis-aligned rectangles, and the outer triangle graph. The former is previously known to be NP-hard, and so this proof represents the use of pair-oriented labelling in a problem that was otherwise considered difficult to prove NP-hard using a co-2-subdivision approach. The outer triangle graph is a novel intersection graph, which therefore provides new NP-hardness results for finding a maximum clique in geometric intersection graphs

Numerical modelling of the fluid-structure interaction in complex vascular geometries

A complex network of vessels is responsible for the transportation of blood throughout the body and back to the heart. Fluid mechanics and solid mechanics play a fundamental role in this transport phenomenon and are particularly suited for computer simulations. The latter may contribute to a better comprehension of the physiological processes and mechanisms leading to cardiovascular diseases, which are currently the leading cause of death in the western world. In case these computational models include patient-specific geometries and/or the interaction between the blood flow and the arterial wall, they become challenging to develop and to solve, increasing both the operator time and the computational time. This is especially true when the domain of interest involves vascular pathologies such as a local narrowing (stenosis) or a local dilatation (aneurysm) of the arterial wall. To overcome these issues of high operator times and high computational times when addressing the bio(fluid)mechanics of complex geometries, this PhD thesis focuses on the development of computational strategies which improve the generation and the accuracy of image-based, fluid-structure interaction (FSI) models. First, a robust procedure is introduced for the generation of hexahedral grids, which allows for local grid refinements and automation. Secondly, a straightforward algorithm is developed to obtain the prestress which is implicitly present in the arterial wall of a – by the blood pressure – loaded geometry at the moment of medical image acquisition. Both techniques are validated, applied to relevant cases, and finally integrated into a fluid-structure interaction model of an abdominal mouse aorta, based on in vivo measurements

Codesign of edge intelligence and automated guided vehicle control

Abstract. In recent years, edge Artificial Intelligence (AI) coupled with other technologies such as autonomous systems have gained a lot of attention. This work presents a harmonic design of Autonomous Guided Vehicles (AGV) control, edge intelligence, and human input to enable autonomous transportation in industrial environments. The AGV has the capability to navigate between a source and destinations and pick/place objects. The human input implicitly provides the preferences of the destination and exact drop point, which are derived from the AI at the network edge and shared with the AGV over a wireless network. Design and integration of autonomous control of AGV, edge intelligence, and communication therein are carried out in this work and presented as a unified demonstration. The demonstration indicates that the proposed design of hardware, software, and intelligence design achieves the Technology Readiness Level (TRL) of range 4–5

Characterizing the Realistic-ness of Word Problems in Secondary Mathematics Textbooks

Word problems are an integral part of any secondary mathematics curriculum and one purpose has been to prepare students for the real-world – for everyday events as well as workplace problem-solving. Prior literature suggests that word problems have not met this objective, in part, because the textbook problems do not mirror the kinds of problems commonly found in real life situations. In this dissertation, I investigate a sample of word problems from two contemporary non-traditional textbooks to uncover the aspects that may influence if and how the problems might be used in the classroom. I utilize a qualitative content analysis with a directed approach, using the literature to guide my initial codes and categories, and allowing other categories and subcategories to emerge during the analysis. I also conduct a numerical analysis of the data to reveal aspects which may be a common thread between the two books. These analyses allow me to answer the research question: Given that the two books chosen for this study have different approaches, what aspects of realistic-ness exist in the textbooks’ word problems that encourage students to use their real-world knowledge of the context of the problems? This study suggests that changes to the manner in which problems are presented can be beneficial to re-negotiating the didactical contract. Textbook word problems should be posed in a variety of ways, breaking from the tradition of the three-component structure. Additionally, secondary mathematics textbooks should use scaffolding throughout the curricula to afford students the opportunities to grapple with problems as they would in the real world. This study recommends a digital database to organize and update problems with a real-world context

Octahedron-based Projections as Intermediate Representations for Computer Imaging: TOAST, TEA, and More

This paper defines and discusses a set of rectangular all-sky projections that have no singular points, notably the Tesselated Octahedral Adaptive Spherical Transformation (or TOAST) developed initially for the WorldWide Telescope. These have proven to be useful as intermediate representations for imaging data where the application transforms dynamically from a standardized internal format to a specific format (projection, scaling, orientation, etc.) requested by the user. TOAST is strongly related to the Hierarchical Triangular Mesh pixelization and is particularly well adapted to situations where one wishes to traverse a hierarchy of images increasing in resolution. Because it can be recursively computed using a very simple algorithm it is particularly adaptable to use with graphical processing units

Analysis and Development of a Computer Science Program for Use in Secondary School Mathematics

Higher Educatio

Visualizing Set Relations and Cardinalities Using Venn and Euler Diagrams

In medicine, genetics, criminology and various other areas, Venn and Euler diagrams are used to visualize data set relations and their cardinalities. The data sets are represented by closed curves and the data set relationships are depicted by the overlaps between these curves. Both the sets and their intersections are easily visible as the closed curves are preattentively processed and form common regions that have a strong perceptual grouping effect. Besides set relations such as intersection, containment and disjointness, the cardinality of the sets and their intersections can also be depicted in the same diagram (referred to as area-proportional) through the size of the curves and their overlaps. Size is a preattentive feature and so similarities, differences and trends are easily identified. Thus, such diagrams facilitate data analysis and reasoning about the sets. However, drawing these diagrams manually is difficult, often impossible, and current automatic drawing methods do not always produce appropriate diagrams. This dissertation presents novel automatic drawing methods for different types of Euler diagrams and a user study of how such diagrams can help probabilistic judgement. The main drawing algorithms are: eulerForce, which uses a force-directed approach to lay out Euler diagrams; eulerAPE, which draws area-proportional Venn diagrams with ellipses. The user study evaluated the effectiveness of area- proportional Euler diagrams, glyph representations, Euler diagrams with glyphs and text+visualization formats for Bayesian reasoning, and a method eulerGlyphs was devised to automatically and accurately draw the assessed visualizations for any Bayesian problem. Additionally, analytic algorithms that instantaneously compute the overlapping areas of three general intersecting ellipses are provided, together with an evaluation of the effectiveness of ellipses in drawing accurate area-proportional Venn diagrams for 3-set data and the characteristics of the data that can be depicted accurately with ellipses