Persistent Link:
http://hdl.handle.net/10150/338714
Title:
How To Color A Map
Author:
Veeramoni Mythili, Sankaranarayanan
Issue Date:
2014
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 or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
We study the maximum differential coloring problem, where an n-vertex graph must be colored with colors numbered 1, 2...n such that the minimal difference between the two colors of any edge is maximized. This problem is motivated by coloring maps in which not all countries are contiguous. Since it is known that this problem is NP-hard for general graphs; we consider planar graphs and subclasses thereof. In Chapter 1 we introduce the topic of this thesis and in Chapter 2 we review relevant definitions and basic results. In Chapter 3 we prove that the maximum differential coloring problem remains NP-hard even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs and close-to-optimal differential coloring algorithms for general caterpillars and biconnected triangle-free outer-planar graphs. In Chapter 4 we introduce the (d, kn)-differential coloring problem. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, in this chapter we provide a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the ([2n/3], 2n)-differential coloring problem, even when input graph is planar. In Chapter 5 we experimentally evaluate and compare several algorithms for coloring a map. Motivated by different application scenarios, we classify our approaches into two categories, depending on the dimensionality of the underlying color space. To cope with the one dimensional color space (e.g., gray-scale colors), we employ the (d, kn)-differential coloring. In Chapter 6 we describe a practical approach for visualizing multiple relationships defined on the same dataset using a geographic map metaphor, where clusters of nodes form countries and neighboring countries correspond to nearby clusters. The aim is to provide a visualization that allows us to compare two or more such maps. In the case where we are considering multiple relationships we also provide an interactive tool to visually explore the effect of combining two or more such relationships. Our method ensures good readability and mental map preservation, based on dynamic node placement with node stability, dynamic clustering with cluster stability, and dynamic coloring with color stability. Finally in Chapter 7 we discuss future work and open problems.
Type:
text; Electronic Dissertation
Keywords:
coloring maps; differential coloring; dual bandwidth; planar graphs; separation number; anti bandwidth; Computer Science
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Computer Science
Degree Grantor:
University of Arizona
Advisor:
Kubourov, Stephen

Full metadata record

DC FieldValue Language
dc.language.isoen_USen
dc.titleHow To Color A Mapen_US
dc.creatorVeeramoni Mythili, Sankaranarayananen_US
dc.contributor.authorVeeramoni Mythili, Sankaranarayananen_US
dc.date.issued2014-
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractWe study the maximum differential coloring problem, where an n-vertex graph must be colored with colors numbered 1, 2...n such that the minimal difference between the two colors of any edge is maximized. This problem is motivated by coloring maps in which not all countries are contiguous. Since it is known that this problem is NP-hard for general graphs; we consider planar graphs and subclasses thereof. In Chapter 1 we introduce the topic of this thesis and in Chapter 2 we review relevant definitions and basic results. In Chapter 3 we prove that the maximum differential coloring problem remains NP-hard even for planar graphs. Then, we present tight bounds for regular caterpillars and spider graphs and close-to-optimal differential coloring algorithms for general caterpillars and biconnected triangle-free outer-planar graphs. In Chapter 4 we introduce the (d, kn)-differential coloring problem. While it was known that the problem of determining whether a general graph is (2, n)-differential colorable is NP-complete, in this chapter we provide a complete characterization of bipartite, planar and outerplanar graphs that admit (2, n)-differential colorings. We show that it is NP-complete to determine whether a graph admits a (3, 2n)-differential coloring. The same negative result holds for the ([2n/3], 2n)-differential coloring problem, even when input graph is planar. In Chapter 5 we experimentally evaluate and compare several algorithms for coloring a map. Motivated by different application scenarios, we classify our approaches into two categories, depending on the dimensionality of the underlying color space. To cope with the one dimensional color space (e.g., gray-scale colors), we employ the (d, kn)-differential coloring. In Chapter 6 we describe a practical approach for visualizing multiple relationships defined on the same dataset using a geographic map metaphor, where clusters of nodes form countries and neighboring countries correspond to nearby clusters. The aim is to provide a visualization that allows us to compare two or more such maps. In the case where we are considering multiple relationships we also provide an interactive tool to visually explore the effect of combining two or more such relationships. Our method ensures good readability and mental map preservation, based on dynamic node placement with node stability, dynamic clustering with cluster stability, and dynamic coloring with color stability. Finally in Chapter 7 we discuss future work and open problems.en_US
dc.typetexten
dc.typeElectronic Dissertationen
dc.subjectcoloring mapsen_US
dc.subjectdifferential coloringen_US
dc.subjectdual bandwidthen_US
dc.subjectplanar graphsen_US
dc.subjectseparation numberen_US
dc.subjectanti bandwidthen_US
dc.subjectComputer Scienceen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineComputer Scienceen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorKubourov, Stephenen_US
dc.contributor.committeememberKubourov, Stephenen_US
dc.contributor.committeememberDowney, Peteren_US
dc.contributor.committeememberEfrat, Alonen_US
dc.contributor.committeememberGlickenstein, Daviden_US
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