Discrete Mathematics and Combinatorics Commons^™

Combinatorial Algorithms For Graph Discovery And Experimental Design, Raghavendra K. Addanki

Doctoral Dissertations

In this thesis, we study the design and analysis of algorithms for discovering the structure and properties of an unknown graph, with applications in two different domains: causal inference and sublinear graph algorithms. In both these domains, graph discovery is possible using restricted forms of experiments, and our objective is to design low-cost experiments. First, we describe efficient experimental approaches to the causal discovery problem, which in its simplest form, asks us to identify the causal relations (edges of the unknown graph) between variables (vertices of the unknown graph) of a given system. For causal discovery, we study algorithms …

On The Total Set Chromatic Number Of Graphs, Mark Anthony C. Tolentino, Gerone Russel J. Eugenio, Mari-Jo P. Ruiz

Theory and Applications of Graphs

Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χ_s(G) of a graph G is the minimum number of colors required in a set coloring of G. In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = (V, E) …

Using Magic To Teach Computer Programming, Dale F. Reed, Ronald I. Greenberg

Computer Science: Faculty Publications and Other Works

Magic can be used in project-based instruction to motivate students and provide a meaningful context for learning computer programming. This work describes several magic programs of the “Choose a Number” and “Pick a Card” varieties, making connections to underlying computing concepts.

Magic tricks presented as demonstrations and programming assignments elicit wonder and captivate students’ attention, so that students want to understand and replicate the work to show it to friends and family members. Capturing student interest and curiosity motivates them to learn the underlying programming concepts.

Two “Choose a Number” programs are shown where the computer is able to identify …

Moving Polygon Methods For Incompressible Fluid Dynamics, Chris Chartrand

Doctoral Dissertations

Hybrid particle-mesh numerical approaches are proposed to solve incompressible fluid flows. The methods discussed in this work consist of a collection of particles each wrapped in their own polygon mesh cell, which then move through the domain as the flow evolves. Variables such as pressure, velocity, mass, and momentum are located either on the mesh or on the particles themselves, depending on the specific algorithm described, and each will be shown to have its own advantages and disadvantages. This work explores what is required to obtain local conservation of mass, momentum, and convergence for the velocity and pressure in a …

Games For One, Games For Two: Computationally Complex Fun For Polynomial-Hierarchical Families, Kye Shi

HMC Senior Theses

In the first half of this thesis, we explore the polynomial-time hierarchy, emphasizing an intuitive perspective that associates decision problems in the polynomial hierarchy to combinatorial games with fixed numbers of turns. Specifically, problems in 𝐏 are thought of as 0-turn games, 𝐍𝐏 as 1-turn “puzzle” games, and in general 𝚺ₖ𝐏 as 𝑘-turn games, in which decision problems answer the binary question, “can the starting player guarantee a win?” We introduce the formalisms of the polynomial hierarchy through this perspective, alongside definitions of 𝑘-turn CIRCUIT SATISFIABILITY games, whose 𝚺ₖ𝐏-completeness is assumed from prior work (we briefly justify this assumption …

Stroke Clustering And Fitting In Vector Art, Khandokar Shakib

Senior Independent Study Theses

Vectorization of art involves turning free-hand drawings into vector graphics that can be further scaled and manipulated. In this paper, we explore the concept of vectorization of line drawings and study multiple approaches that attempt to achieve this in the most accurate way possible. We utilize a software called StrokeStrip to discuss the different mathematics behind the parameterization and fitting involved in the drawings.

Matrix Interpretations And Tools For Investigating Even Functionals, Benjamin Stringer

Theses and Dissertations--Computer Science

Even functionals are a set of polynomials evaluated on the terms of hollow symmetric matrices. Their properties lend themselves to applications such as counting subgraph embeddings in generic (weighted or unweighted) host graphs and computing moments of binary quadratic forms, which occur in combinatorial optimization. This research focuses primarily on counting subgraph embeddings, which is traditionally accomplished with brute-force algorithms or algorithms curated for special types of graphs. Even functionals provide a method for counting subgraphs algebraically in time proportional to matrix multiplication and is not restricted to particular graph types. Counting subgraph embeddings can be accomplished by evaluating a …

Lasso: Listing All Subset Sums Obediently For Evaluating Unbounded Subset Sums, Christopher N. Burgoyne, Travis J. Wheeler

Graduate Student Theses, Dissertations, & Professional Papers

In this study we present a novel algorithm, LASSO, for solving the unbounded and bounded subset sum problem. The LASSO algorithm was designed to solve the unbounded SSP quickly and to return all subsets summing to a target sum. As speed was the highest priority, we benchmarked the run time performance of LASSO against implementations of some common approaches to the bounded SSP, as well as the only comparable implementation for solving the unbounded SSP that we could find. In solving the bounded SSP, our algorithm had a significantly faster run time than the competing algorithms when the target sum …