To make computations on large data sets more efficient, algorithms will frequently divide information into smaller, more manageable, pieces. This idea, for example, forms the basis of the common algorithmic approach known as Divide and Conquer.
If we wish to use this principle in planar geometric computations, however, we may require specialized techniques for decomposing our data. This is due to the fact that the data sets are typically points, lines, regions, or polygons. This motivates algorithms that can break-up polygons into simpler pieces. Algorithms that perform such computations are said to compute polygon decompositions. There are many ways
that we can decompose a polygon, and there are also many types of polygons that we could decompose. Both applications and theoretical interest demand algorithms for a wide variety of decomposition problems.
In this thesis we study two different polygon decomposition problems. The first
problem that we study is a polygon decomposition problem that is equivalent to the Rectilinear Art Gallery problem. In this problem we seek a decomposition of a polygon into so-called r-stars. These r-stars model visibility in an orthogonal setting.
We show that we can compute a certain type of decomposition, known as a Steinercover, of a simple orthogonal polygon into r-stars in polynomial time. In the second problem, we explore the complexity of decomposing polygons into components that have an upper bound on their size. In this problem, the size of a polygon refers
to the size of its bounding-box. This problem is motivated by a polygon collision detection heuristic that approximates a polygon by its bounding-box to determine whether an exact collision detection computation should take place. We show that it is NP-complete to decide whether a polygon that contains holes can be decomposed
into a specified number of size-constrained components.
| Identifer | oai:union.ndltd.org:USASK/oai:usask.ca:etd-08032004-194707 |
| Date | 09 August 2004 |
| Creators | Worman, Chris |
| Contributors | Mould, David, Keil, J. Mark, Cheston, Grant A., Neufeld, Eric |
| Publisher | University of Saskatchewan |
| Source Sets | University of Saskatchewan Library |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://library.usask.ca/theses/available/etd-08032004-194707/ |
| Rights | unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to University of Saskatchewan or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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