Robust geometric computing and optimal visibility coverage

• Authors: Eli Packer
• Venue: Robust geometric computing and optimal visibility coverage
• Year: 2008

This thesis is divided into two parts: Robust Geometric Computing and Optimal Visibility Coverage. I next introduce both. Robust Geometric Computing. Implementation of geometric algorithms is generally difficult because one must deal with both precision problems and degenerate input. While these issues are usually ignored when describing geometric algorithms in theory, overlooking them in practice may result in program crashes and incorrect results. I present techniques I developed to approximate arrangements of line segments to make them more robust for further manipulation. The first technique, Iterated Snap Rounding with Bounded Drift improves previous algorithms that approximate each original segment by a polygonal chain. It is the first algorithm of this kind to maintain sufficient separation between vertices and non-incident edges while assuring a close approximation of the input. The second technique belongs to the family of Controlled Perturbation algorithms, in which the input is perturbed to overcome potential degeneracies that are risky for finite-precision manipulation. In this work I focus on a novel method for significantly decreasing the perturbation magnitude. Optimal Visibility Coverage. I present techniques for optimizing visibility coverage. Given a polygonal domain, we optimize two different coverage problems whose goal is to cover the interior of the domain with point guards. The first, known as the Art Gallery problem, deals with stationary guards; the goal is to minimize the number of guards that cover the domain. The second, known as the Mobile Watchmen problem, allows guards to move along predefined routes. In this case, the number of guards is an input parameter of the problem and the goal is to minimize the length of the routes. Both optimization problems are NP-hard, and both are challenging problems with only limited results known about approximation algorithms. Motivated by these difficulties, we develop heuristics for both problems. We also develop heuristics for finding lower bounds based on the idea of visibility-independent “witness points”. We present these heuristics and show experimental results that show their practical effectiveness, based on a comparison of the upper and lower bounds we compute. As far as we know, this is the first comprehensive experimental analysis on these classic problems.