## Date of Award

1-1-2011

## Document Type

Dissertation

## Degree Name

Ph.D.

## Organizational Unit

Daniel Felix Ritchie School of Engineering and Computer Science

## First Advisor

Mario A. Lopez, Ph.D.

## Second Advisor

Ronald DeLyser

## Third Advisor

Scott Leutenegger

## Fourth Advisor

Alvaro Arias

## Fifth Advisor

Chris Gauthier-Dickey

## Keywords

Approximation algorithms, Computational geometry, Facility location, Pattern recognition, Routing

## Abstract

This thesis addresses several problems in the facility location sub-area of computational geometry. Let *S* be a set of *n* points in the plane. We derive algorithms for approximating *S* by a step function curve of size *k < n*, i.e., by an *x*-monotone orthogonal polyline ℜ with *k < n* horizontal segments. We use the vertical distance to measure the quality of the approximation, i.e., the maximum distance from a point in *S* to the horizontal segment directly above or below it. We consider two types of problems: min-*ε*, where the goal is to minimize the error for a given number of horizontal segments *k *and min-*#*, where the goal is to minimize the number of segments for a given allowed error *ε*. After *O*(*n*) preprocessing time, we solve instances of the latter in *O*(min{*k* log* n, n*}) time per instance. We can then solve the former problem in *O*(min{*n*^{2}, *nk *log* n*}) time. Both algorithms require *O*(*n*) space. The second contribution is a heuristic for the min-*ε *problem that computes a solution within a factor of 3 of the optimal error for *k* segments, or with at most the same error as the *k*-optimal but using *2k - 1* segments. Furthermore, experiments on real data show even better results than what is guaranteed by the theoretical bounds. Both approximations run in *O*(*n *log* n*) time and *O*(*n*) space. Then, we present an exact algorithm for the weighted version of this problem that runs in *O*(*n*^{2}) time and generalize the heuristic to handle weights at the expense of an additional log *n* factor. At this point, a randomized algorithm that runs in *O*(*n *log^{2 }*n*) expected time for the unweighted version is presented. It easily generalizes to the weighted case, though at the expense of an additional log *n* factor. Finally, we treat the maximin problem and present an *O*(*n*^{3 }log* n*) solution to the problem of finding the furthest separating line through a set of weighted points. We conclude with solutions to the "obnoxious" wedge problem: an *O*(*n*^{2 }log* n*) algorithm for the general case of a wedge with its apex on the boundary of the convex hull of *S* and an *O*(*n*^{2}) algorithm for the case of the apex of a wedge coming from the input set *S*.

## Publication Statement

Copyright is held by the author. User is responsible for all copyright compliance.

## Rights Holder

Yan B. Mayster

## Provenance

Received from ProQuest

## File Format

application/pdf

## Language

en

## File Size

98 p.

## Recommended Citation

Mayster, Yan B., "Minimax and Maximin Fitting of Geometric Objects to Sets of Points" (2011). *Electronic Theses and Dissertations*. 412.

https://digitalcommons.du.edu/etd/412

## Copyright date

2011

## Discipline

Computer science, Mathematics