## On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures

dc.contributor.author | Lee, Patrick | |

dc.date.accessioned | 2014-09-11 12:44:31 (GMT) | |

dc.date.available | 2014-09-11 12:44:31 (GMT) | |

dc.date.issued | 2014-09-11 | |

dc.date.submitted | 2014 | |

dc.identifier.uri | http://hdl.handle.net/10012/8783 | |

dc.description.abstract | Many standard problems in computational geometry have been solved asymptotically optimally as far as comparison-based algorithms are concerned, but there has been little work focusing on improving the constant factors hidden in big-Oh bounds on the number of comparisons needed. In this thesis, we consider orthogonal-type problems and present a number of results that achieve optimality in the constant factors of the leading terms, including: - An output-sensitive algorithm that computes the maxima for a set of n points in two dimensions using 1n log(h) + O(n sqrt(log(h))) comparisons, where h is the size of the output. - A randomized algorithm that computes the maxima in three dimensions that uses 1n log(n) + O(n sqrt(log(n))) expected number of comparisons. - A randomized output-sensitive algorithm that computes the maxima in three dimensions that uses 1n log(h) + O(n log^(2/3)(h)) expected number of comparisons, where h is the size of the output. - An output-sensitive algorithm that computes the convex hull for a set of n points in two dimensions using 1n log(h) + O(n sqrt(log(h))) comparisons and O(n sqrt(log(h))) sidedness tests, where h is the size of the output. - A randomized algorithm for detecting whether of a set of n horizontal and vertical line segments in the plane intersect that uses 1n log(n) +O(n sqrt(log(n))) expected number of comparisons. - A data structure for point location among n axis-aligned disjoint boxes in three dimensions that answers queries using at most (3/2)log(n)+ O(log(log(n))) comparisons. The data structure can be extended to higher dimensions and uses at most (d/2)log(n)+ O(log(log(n))) comparisons. - A data structure for point location among n axis-aligned disjoint boxes that form a space-filling subdivision in three dimensions that answers queries using at most (4/3)log(n)+ O(sqrt(log(n))) comparisons. The data structure can be extended to higher dimensions and uses at most ((d+1)/3)log(n)+ O(sqrt(log(n))) comparisons. Our algorithms and data structures use a variety of techniques, including Seidel and Adamy's planar point location method, weighted binary search, and height-optimal BSP trees. | en |

dc.language.iso | en | en |

dc.publisher | University of Waterloo | en |

dc.subject | Algorithms | en |

dc.subject | data structures | en |

dc.subject | computational geometry | en |

dc.title | On Constant Factors in Comparison-Based Geometric Algorithms and Data Structures | en |

dc.type | Master Thesis | en |

dc.pending | false | |

dc.subject.program | Computer Science | en |

uws-etd.degree.department | School of Computer Science | en |

uws-etd.degree | Master of Mathematics | en |

uws.typeOfResource | Text | en |

uws.peerReviewStatus | Unreviewed | en |

uws.scholarLevel | Graduate | en |