Date of Award

1-1-2019

Document Type

Dissertation

Degree Name

Ph.D.

Department

Mathematics

First Advisor

Paul Horn

Second Advisor

Mario A. Lopez

Keywords

Computational geometry, Extremal problems, Sensor networks

Abstract

A sensor network is typically modeled as a collection of spatially distributed objects with the same shape, generally for the purpose of surveilling or protecting areas and locations. In this dissertation we address several questions relating to sensors with linear shapes: line, line segment, and rays in the plane, and hyperplanes in higher dimensions.

First we explore ray sensor networks in the plane, whose resilience is the number of sensors that must be crossed by an agent traveling between two known locations. The coverage of such a network is described by a particular tripartite graph, the barrier graph of the network. We show that barrier graphs are perfect (Berge) graphs and have a rigid neighborhood structure due to the rays' geometry.

We introduce two extremal problems for networks in the plane made of line sensors, line segment sensors, or ray sensors, which informally ask how well it is possible to simultaneously protect k locations with n (line/ray/segment)-shaped sensors from intruders. The first question allows any number of intruders, while the second assumes there is a lone intruder. We show these are questions to be answered separately, and provide complete answers for k = 2 in both cases. We provide asymptotically tight answers for question (1) when k = 3, 4 and the locations are in convex position. We also provide asymptotic lower bounds for question (1) for any k.

Finally, we generalize these extremal problems to d dimensions. For the d-dimensional version of question (1) we provide asymptotic lower and upper bounds for any combination of k and d, though these bounds do not meet.

Provenance

Recieved from ProQuest

Rights holder

Kirk Anthony Boyer

File size

110 p.

File format

application/pdf

Language

en

Discipline

Mathematics

Share

COinS