Many systems in the world \u2014 such as cellular networks, the post service, or transportation pathways \u2014 can be modeled as networks or graphs. The practical applications of graph algorithms generally seek to achieve some goal while minimizing some cost such as money or distance. While the minimum linear arrangement (MLA) problem has been widely-studied amongst graph ordering and embedding problems, there have been no developments into versions of the problem involving degree higher than 2. An application of our problem can be seen in overlay networks in telecommunications. An overlay network is a virtual network that is built on top of another network. It is a logical network where the links between nodes represent the physical paths connecting the nodes in the underlying infrastructure. The underlying physical network may be incomplete, but as long as it is connected, we can build a complete overlay network on top of it. Since some nodes may be overloaded by traffic, we can reduce the strain on the overlay network by limiting the communication between nodes. Some edges, however, may have more importance than others so we must be careful about our selection of which nodes are allowed to communicate with each other. The balance of reducing the degree of the network while maximizing communication forms the basis of our d-degree minimum arrangement problem. In this thesis we will look at several approaches to solving the generalized d-degree minimum arrangement d-MA problem where we embed a graph onto a subgraph of a given degree. We first look into the requirements and challenges of solving the d-MA problem. We will then present a polynomial-time heuristic and compare its performance with the optimal solution derived from integer linear programming. We will show that a simple (d-1)-ary tree construction provides the optimal structure for uniform graphs with large requests sets. Finally, we will present experimental data gathered from running simulations on a variety of graphs to evaluate the efficiency of our heuristic and tree construction.

In today's data-driven world, every datum is connected to a large amount of data. Relational databases have been proving itself a pioneer in the field of data storage and manipulation since 1970s. But more recently they have been challenged by NoSQL graph databases in handling data models which have an inherent graphical representation. Graph databases with the ability to store physical relationships between two nodes and native graph processing technique have been doing exceptionally well in graph data storage and management for applications like recommendation engines, biological modeling, network modeling, social media applications, etc.

Instructional Module Development System (IMODS) is a web-based software system that guides STEM instructors through the complex task of curriculum design, ensures tight alignment between various components of a course (i.e., learning objectives, content, assessments), and provides relevant information about research-based pedagogical and assessment strategies. The data model of IMODS is highly connected and has an inherent graphical representation between all its entities with numerous relationships between them. This thesis focuses on developing an algorithm to determine completeness of course design developed using IMODS. As part of this research objective, the study also analyzes the data model for best fit database to run these algorithms. As part of this thesis, two separate applications abstracting the data model of IMODS have been developed - one with Neo4j (graph database) and another with PostgreSQL (relational database). The research objectives of the thesis are as follows: (i) evaluate the performance of Neo4j and PostgreSQL in handling complex queries that will be fired throughout the life cycle of the course design process; (ii) devise an algorithm to determine the completeness of a course design developed using IMODS. This thesis presents the process of creating data model for PostgreSQL and converting it into a graph data model to be abstracted by Neo4j, creating SQL and CYPHER scripts for undertaking experiments on both platforms, testing and elaborate analysis of the results and evaluation of the databases in the context of IMODS.