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Partitioning problems in discrete and computational geometry
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Partitioning problems in discrete and computational geometry
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ETD_2315
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http://hdl.rutgers.edu/1782.2/rucore10001600001.ETD.000052168
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eng
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theses
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Computer Science
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Topic
Discrete geometry
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Topic
Geometry--Data processing
Abstract
Many interesting problems in Discrete and Computational Geometry involve partitioning. A main question is whether a given set, or sets, may be separated into parts satisfying certain properties. Sometime we also need to find an efficient way to do it - in other words an algorithm. In this thesis, we discuss several of problems and results of this kind.
First we give a combinatorial proof of the existence and the uniqueness of the generalized ham-sandwich cut for well separated point sets in R^d, that have the weak general position property. The combinatorial proof allows us to derive an O(n(log n)^d−3) running time algorithm to find a generalized cut for d given well separated point sets in R^d.
A second problem concerns the 6-way partition of a given convex set in R^2 using 3 lines. We reopen an old question and show that when the direction of one line is fixed, there is unique partition such that 6 regions have the same area.
In the Voronoi game two players A and B each play n points in a compact set S in R^d. They obtain scores equal to the total volume of the Voronoi cells of their points. There are one round and alternating versions of the game. We give the player A's best strategy to minimize the area of player B's one(first) Voronoi cell.
First we give a combinatorial proof of the existence and the uniqueness of the generalized ham-sandwich cut for well separated point sets in R^d, that have the weak general position property. The combinatorial proof allows us to derive an O(n(log n)^d−3) running time algorithm to find a generalized cut for d given well separated point sets in R^d.
A second problem concerns the 6-way partition of a given convex set in R^2 using 3 lines. We reopen an old question and show that when the direction of one line is fixed, there is unique partition such that 6 regions have the same area.
In the Voronoi game two players A and B each play n points in a compact set S in R^d. They obtain scores equal to the total volume of the Voronoi cells of their points. There are one round and alternating versions of the game. We give the player A's best strategy to minimize the area of player B's one(first) Voronoi cell.
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electronic resource
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viii, 65 p. : ill.
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Ph.D.
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Includes bibliographical references (p. 62-64)
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by Jihui Zhao
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Zhao
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Jihui
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1971-
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Jihui Zhao
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Steiger
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William
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chair
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William Steiger
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Szegedy
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Mario
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Mario Szegedy
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Kalantari
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Bahman
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Bahman Kalantari
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Imre
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Imre Bárány
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Rutgers University
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Graduate School - New Brunswick
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2010
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2010-01
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xx
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Rutgers University Electronic Theses and Dissertations
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ETD
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Graduate School - New Brunswick Electronic Theses and Dissertations
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rucore19991600001
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doi:10.7282/T3X34XM8
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ETD doctoral
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Rights
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The author owns the copyright to this work.
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Copyright protected
Notice
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Open
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Permission or license
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Zhao
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Jihui
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2009-12-17 16:57:16
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Jihui Zhao
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Rutgers University. Graduate School - New Brunswick
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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