#### Date of Award

1-1-2011

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Computer Science

#### First Advisor

Mario A. Lopez

#### 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, i.e., by an x-monotone orthogonal polyline R with k 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-*epsilon*, 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 *epsilon*. After

O(*n*)preprocessing time, we solve instances of the latter in O(min{*klog n, n*}) time per instance.

We can then solve the former problem in O(min{*n*^{2}, *nklog n*}) time.

Both algorithms require O(*n*) space.

The second contribution is a heuristic for the min-*epsilon* 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(*nlogn*) 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 *logn* factor. At this point, a randomized algorithm that runs in O(*nlog*^{2}*n*)

expected time for the unweighted version is presented. It easily generalizes to the weighted case, though at the expense of an additional logn factor. Finally, we treat the maximin

problem and present an O(*n*^{3}*logn*) 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}*logn*) 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.

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*#### 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

#### Provenance

Recieved from ProQuest

#### Rights holder

Yan B. Mayster

#### File size

98 p.

#### Copyright date

2011

#### File format

application/pdf

#### Language

en

#### Discipline

Computer science, Mathematics

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