Theory and Algorithms Commons^™

Partitions Of R^N With Maximal Seclusion And Their Applications To Reproducible Computation, Jason Vander Woude

Department of Mathematics: Dissertations, Theses, and Student Research

We introduce and investigate a natural problem regarding unit cube tilings/partitions of Euclidean space and also consider broad generalizations of this problem. The problem fits well within a historical context of similar problems and also has applications to the study of reproducibility in randomized computation.

Given $k\in\mathbb{N}$ and $\epsilon\in(0,\infty)$, we define a $(k,\epsilon)$-secluded unit cube partition of $\mathbb{R}^{d}$ to be a unit cube partition of $\mathbb{R}^{d}$ such that for every point $\vec{p}\in\R^d$, the closed $\ell_{\infty}$ $\epsilon$-ball around $\vec{p}$ intersects at most $k$ cubes. The problem is to construct such partitions for each dimension $d$ with the primary goal of minimizing …

The Dope Distance Is Sic: A Stable, Informative, And Computable Metric On Ordered Merge Trees, Jose Arbelo, Antonio Delgado, Charley Kirk, Zach Schlamowitz

Mathematics Summer Fellows

When analyzing time series data, it is often of interest to categorize them based on how different they are. We define a new dissimilarity measure between time series: Dynamic Ordered Persistence Editing (DOPE). DOPE satisfies metric properties, is stable to noise, is as informative as alternative approaches, and efficiently computable. Satisfying these properties simultaneously makes DOPE of interest to both theoreticians and data scientists alike.

Finding Optimal Cayley Map Embeddings Using Genetic Algorithms, Jacob Buckelew

Honors Program Theses

Genetic algorithms are a commonly used metaheuristic search method aimed at solving complex optimization problems in a variety of fields. These types of algorithms lend themselves to problems that can incorporate stochastic elements, which allows for a wider search across a search space. However, the nature of the genetic algorithm can often cause challenges regarding time-consumption. Although the genetic algorithm may be widely applicable to various domains, it is not guaranteed that the algorithm will outperform other traditional search methods in solving problems specific to particular domains. In this paper, we test the feasibility of genetic algorithms in solving a …

Multilateration Index., Chip Lynch

Electronic Theses and Dissertations

We present an alternative method for pre-processing and storing point data, particularly for Geospatial points, by storing multilateration distances to fixed points rather than coordinates such as Latitude and Longitude. We explore the use of this data to improve query performance for some distance related queries such as nearest neighbor and query-within-radius (i.e. “find all points in a set P within distance d of query point q”). Further, we discuss the problem of “Network Adequacy” common to medical and communications businesses, to analyze questions such as “are at least 90% of patients living within 50 miles of a covered emergency …

Bayesian Topological Machine Learning, Christopher A. Oballe

Doctoral Dissertations

Topological data analysis encompasses a broad set of ideas and techniques that address 1) how to rigorously define and summarize the shape of data, and 2) use these constructs for inference. This dissertation addresses the second problem by developing new inferential tools for topological data analysis and applying them to solve real-world data problems. First, a Bayesian framework to approximate probability distributions of persistence diagrams is established. The key insight underpinning this framework is that persistence diagrams may be viewed as Poisson point processes with prior intensities. With this assumption in hand, one may compute posterior intensities by adopting techniques …

A 3d Image-Guided System To Improve Myocardial Revascularization Decision-Making For Patients With Coronary Artery Disease, Haipeng Tang

Dissertations

OBJECTIVES. Coronary artery disease (CAD) is the most common type of heart disease and kills over 360,000 people a year in the United States. Myocardial revascularization (MR) is a standard interventional treatment for patients with stable CAD. Fluoroscopy angiography is real-time anatomical imaging and routinely used to guide MR by visually estimating the percent stenosis of coronary arteries. However, a lot of patients do not benefit from the anatomical information-guided MR without functional testing. Single-photon emission computed tomography (SPECT) myocardial perfusion imaging (MPI) is a widely used functional testing for CAD evaluation but limits to the absence of anatomical information. …

Invariance And Invertibility In Deep Neural Networks, Han Zhang

Theses and Dissertations

Machine learning is concerned with computer systems that learn from data instead of being explicitly programmed to solve a particular task. One of the main approaches behind recent advances in machine learning involves neural networks with a large number of layers, often referred to as deep learning. In this dissertation, we study how to equip deep neural networks with two useful properties: invariance and invertibility. The first part of our work is focused on constructing neural networks that are invariant to certain transformations in the input, that is, some outputs of the network stay the same even if the input …

Extensions Of The Morse-Hedlund Theorem, Eben Blaisdell

Honors Theses

Bi-infinite words are sequences of characters that are infinite forwards and backwards; for example "...ababababab...". The Morse-Hedlund theorem says that a bi-infinite word f repeats itself, in at most n letters, if and only if the number of distinct subwords of length n is at most n. Using the example, "...ababababab...", there are 2 subwords of length 3, namely "aba" and "bab". Since 2 is less than 3, we must have that "...ababababab..." repeats itself after at most 3 letters. In fact it does repeat itself every two letters. …

Normal Surfaces And 3-Manifold Algorithms, Josh D. Hews

Honors Theses

This survey will develop the theory of normal surfaces as they apply to the S3 recognition algorithm. Sections 2 and 3 provide necessary background on manifold theory. Section 4 presents the theory of normal surfaces in triangulations of 3-manifolds. Section 6 discusses issues related to implementing algorithms based on normal surfaces, as well as an overview of the Regina, a program that implements many 3-manifold algorithms. Finally section 7 presents the proof of the 3-sphere recognition algorithm and discusses how Regina implements the algorithm.

Ε-Kernel Coresets For Stochastic Points, Haitao Wang, Lingxiao Huang, Jian Li, Jeff Mark Phillips

Computer Science Faculty and Staff Publications

With the dramatic growth in the number of application domains that generate probabilistic, noisy and uncertain data, there has been an increasing interest in designing algorithms for geometric or combinatorial optimization problems over such data. In this paper, we initiate the study of constructing epsilon-kernel coresets for uncertain points. We consider uncertainty in the existential model where each point's location is fixed but only occurs with a certain probability, and the locational model where each point has a probability distribution describing its location. An epsilon-kernel coreset approximates the width of a point set in any direction. We consider approximating the …

Algorithmic Foundations Of Heuristic Search Using Higher-Order Polygon Inequalities, Newton Henry Campbell Jr.

CCE Theses and Dissertations

The shortest path problem in graphs is both a classic combinatorial optimization problem and a practical problem that admits many applications. Techniques for preprocessing a graph are useful for reducing shortest path query times. This dissertation studies the foundations of a class of algorithms that use preprocessed landmark information and the triangle inequality to guide A* search in graphs. A new heuristic is presented for solving shortest path queries that enables the use of higher order polygon inequalities. We demonstrate this capability by leveraging distance information from two landmarks when visiting a vertex as opposed to the common single landmark …

Skeleton Structures And Origami Design, John C. Bowers

Doctoral Dissertations

In this dissertation we study problems related to polygonal skeleton structures that have applications to computational origami. The two main structures studied are the straight skeleton of a simple polygon (and its generalizations to planar straight line graphs) and the universal molecule of a Lang polygon. This work builds on results completed jointly with my advisor Ileana Streinu. Skeleton structures are used in many computational geometry algorithms. Examples include the medial axis, which has applications including shape analysis, optical character recognition, and surface reconstruction; and the Voronoi diagram, which has a wide array of applications including geographic information systems …

Algorithms To Compute Characteristic Classes, Martin Helmer

Electronic Thesis and Dissertation Repository

In this thesis we develop several new algorithms to compute characteristics classes in a variety of settings. In addition to algorithms for the computation of the Euler characteristic, a classical topological invariant, we also give algorithms to compute the Segre class and Chern-Schwartz-MacPherson (CSM) class. These invariants can in turn be used to compute other common invariants such as the Chern-Fulton class (or the Chern class in smooth cases).

We begin with subschemes of a projective space over an algebraically closed field of characteristic zero. In this setting we give effective algorithms to compute the CSM class, Segre class and …

Efficient Estimation Of Cluster Population, Sanjeev K C

UNLV Theses, Dissertations, Professional Papers, and Capstones

Partitioning a given set of points into clusters is a well known problem in pattern recognition, data mining, and knowledge discovery. One of the well known methods for identifying clusters in Euclidean space is the K-mean algorithm. In using the K-mean clustering algorithm it is necessary to know the value of k (the number of clusters) in advance. We propose to develop algorithms for good estimation of k for points distributed in two dimensions. The techniques we pursue include a bucketing method, g-hop neighbors, and Voronoi diagrams. We also present experimental results for examining the performances of the bucketing method …

Approaches For Generating 2d Shapes, Pratik Shankar Hada

UNLV Theses, Dissertations, Professional Papers, and Capstones

Constructing a two dimensional shape from given a set of point sites is a well known problem in computation geometry. We present a critical review of the existing algorithms for constructing polygonal shapes. We present a new approach calledinward dentingfor constructing simple polygons. We then extend the proposed approach for modeling polygons with holes. This is the

first known algorithm for modeling holes in the interior of 2d shapes. We also present experimental investigations of the quality of the solutions generated by the proposed algorithms.

For this we implemented the proposed algorithms in Java programming language. The prototype program can …

Degree Constrained Triangulation, Roshan Gyawali

UNLV Theses, Dissertations, Professional Papers, and Capstones

Triangulation of simple polygons or sets of points in two dimensions is a widely investigated problem in computational geometry. Some researchers have considered variations of triangulation problems that include minimum weight triangulation, de-launay triangulation and triangulation refinement. In this thesis we consider a constrained version of the triangulation problem that asks for triangulating a given domain (polygon or point sites) so that the resulting triangulation has an increased number of even degree vertices. This problem is called Degree Constrained Triangulation (DCT). We propose four algorithms to solve DCT problems. We also present experimental results based on the implementation of the …

Sharp Feature Identification In A Polygon, Joseph P. Scanlan

UNLV Theses, Dissertations, Professional Papers, and Capstones

This thesis presents an efficient algorithm for recognizing and extracting sharp-features from polygonal shapes. As used here, a sharp-feature is a distinct portion of a polygon that is long and skinny. The algorithm executes in O(n^2) time, where n is the number of vertices in the polygon. Experimental results from a Java implementation of the algorithm are also presented.