Publisher
The University of Arizona.Rights
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction, presentation (such as public display or performance) of protected items is prohibited except with permission of the author.Abstract
Given a graph $G$, a \emph{graph spanner} (or simply \emph{spanner}) is a subgraph that preserves lengths of shortest paths in $G$ up to some amount of distortion or error, typically multiplicative and/or additive. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. We first discuss different types of spanners and some classical results. We next extend constructions of spanners with purely \emph{additive} error to weighted graphs. Specifically, we show that a weighted graph $G$ contains all-pairs $+2W, +4W$ and $+6W$ weighted spanners of size $O(n^{3/2}), \Oish(n^{7/5})$ and $\Oish(n^{4/3})$ respectively. We then consider the multi-level graph spanner problem which has application in large network visualization and study the problem for additive errors. We generalize the $+2$ subsetwise spanner of [Pettie 2008, Cygan et al., 2013] to the weighted setting. We experimentally measure the performance of this and several other algorithms for weighted additive spanners, in terms of runtime and the sparsity of the computed spanners. Finally, we introduce a general framework for multi-level spanners which can be applied to both multiplicative and additive errors as well as other spanner-like objects. We show that the simple top-down and bottom-up approaches, have approximation ratios $\frac{\ell+1}{2}$ and $\ell$, which depend on the number of levels $\ell$. The composite approach provides the best approximation ratio of $2.351$ for $\ell \leq 100$, however, at the expense of higher runtime complexity. The rounding up approach is the best overall with polynomial-time algorithm and constant approximation ratio of $4$.Type
textElectronic Dissertation
Degree Name
Ph.D.Degree Level
doctoralDegree Program
Graduate CollegeComputer Science