From Art Galleries to Terrain Modelling  A Meandering Path through Computational Geometry
(2001) Abstract
 We give approximation and online algorithms as well as data structures for some well studied problems in computational geometry. The thesis is divided into three parts.
In part one, we study problems related to guarding, exploring and searching geometric environments. We show inapproximability results for guarding lines and <i>2</i>link polygons, question stated time bounds for computing shortest watchman routes and give a competitive strategy for exploring rectilinear polygons. We also give matching upper and lower bounds for two large classes of strategies for searching concurrent rays in parallel.
The second part considers generalisations of the travelling salesman problem. We give... (More)  We give approximation and online algorithms as well as data structures for some well studied problems in computational geometry. The thesis is divided into three parts.
In part one, we study problems related to guarding, exploring and searching geometric environments. We show inapproximability results for guarding lines and <i>2</i>link polygons, question stated time bounds for computing shortest watchman routes and give a competitive strategy for exploring rectilinear polygons. We also give matching upper and lower bounds for two large classes of strategies for searching concurrent rays in parallel.
The second part considers generalisations of the travelling salesman problem. We give online strategies for the time dependent travelling salesman problem and approximation algorithms and inapproximability results for versions of the kinetic travelling salesman problem. A highlight of the thesis is the exponential lower bound on the approximation ratio for the kinetic travelling salesman problem restricted to expanding point sets.
The last part is devoted to data structures in geographic information systems. We give a pioneer algorithm for constructing Rtrees optimised for point location queries. This data structure is used in databases for geometrical objects containing an exceptional amount of data. Finally, bringing the thesis to a close, we suggest a generalisation of the Delaunay triangulation that we call the <i>k</i>order Delaunay triangulation. This geometric structure corresponds to a similar generalisation of the Voronoi diagram, and is predicted to be of value in automating the removal of artifacts in terrain modelling. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/42106
 author
 Hammar, Mikael ^{LU}
 supervisor
 opponent

 Mitchell, J. S. B., Professor at the Department of Applied Mathematics and Statistics, SUNYSB, New York
 organization
 publishing date
 2001
 type
 Thesis
 publication status
 published
 subject
 keywords
 system, numerisk analys, Datalogi, systems, control, numerical analysis, Approximation Algorithms, Computational Geometry, Online Algorithms, Art Gallery Problem, Linear Search, Traveling Salesman Problem, RTree, Delaunay Triangulation, Polygon Exploration, Computer science, Shortest Watchman Routes, kontroll, Mathematics, Matematik
 pages
 176 pages
 publisher
 Department of Computer Science, Lund University
 defense location
 EBuilding room E:1406, Ole Römers väg 3
 defense date
 20011130 10:15:00
 ISBN
 9162850105
 language
 English
 LU publication?
 yes
 id
 52266fcd41ca495cb50dc666c429f964 (old id 42106)
 date added to LUP
 20160401 15:46:31
 date last changed
 20181121 20:36:15
@phdthesis{52266fcd41ca495cb50dc666c429f964, abstract = {We give approximation and online algorithms as well as data structures for some well studied problems in computational geometry. The thesis is divided into three parts.<br/><br> <br/><br> In part one, we study problems related to guarding, exploring and searching geometric environments. We show inapproximability results for guarding lines and <i>2</i>link polygons, question stated time bounds for computing shortest watchman routes and give a competitive strategy for exploring rectilinear polygons. We also give matching upper and lower bounds for two large classes of strategies for searching concurrent rays in parallel.<br/><br> <br/><br> The second part considers generalisations of the travelling salesman problem. We give online strategies for the time dependent travelling salesman problem and approximation algorithms and inapproximability results for versions of the kinetic travelling salesman problem. A highlight of the thesis is the exponential lower bound on the approximation ratio for the kinetic travelling salesman problem restricted to expanding point sets.<br/><br> <br/><br> The last part is devoted to data structures in geographic information systems. We give a pioneer algorithm for constructing Rtrees optimised for point location queries. This data structure is used in databases for geometrical objects containing an exceptional amount of data. Finally, bringing the thesis to a close, we suggest a generalisation of the Delaunay triangulation that we call the <i>k</i>order Delaunay triangulation. This geometric structure corresponds to a similar generalisation of the Voronoi diagram, and is predicted to be of value in automating the removal of artifacts in terrain modelling.}, author = {Hammar, Mikael}, isbn = {9162850105}, language = {eng}, publisher = {Department of Computer Science, Lund University}, school = {Lund University}, title = {From Art Galleries to Terrain Modelling  A Meandering Path through Computational Geometry}, year = {2001}, }