Using geometric rewrite rules for solving geometric problems symbolically

Solving Geometric Optimization Problems

We show how to use graphics hardware for tackling optimization problems arising in the field of computational geometry. We exemplarily discuss three problems, where combinatorial algorithms are inefficient or hard to implement. Given a set S of n point in the plane, the first two problems are to determine the smallest homothetic scaling of a star shaped polygon P enclosing S and to find the lar...

Solving Geometric Problems in Space-Conscious Models

When dealing with massive data sets, standard algorithms may easily “run out of memory”. In this thesis, we design efficient algorithms in space-conscious models. In particular, in-place algorithms, multi-pass algorithms, read-only algorithms, and stream-sort algorithms are studied, and the focus is on fundamental geometric problems, such as 2D convex hulls, 3D convex hulls, Voronoi diagrams an...

Solving Geometric Problems with Real Quantifier Elimination

Solving Geometric Problems with the Rotating Calipers *

Shamos [1] recently showed that the diameter of a convex n-sided polygon could be computed in O(n) time using a very elegant and simple procedure which resembles rotating a set of calipers around the polygon once. In this paper we show that this simple idea can be generalized in two ways: several sets of calipers can be used simultaneously on one convex polygon, or one set of calipers can be us...

Solving Geometric Covering Problems by Data Reduction

We consider a scenario where stops are to be placed along an already existing public transportation network in order to improve its attractiveness for the customers. The core problem is a geometric set covering problem which is NP-hard in general. However, if the corresponding covering matrix has the consecutive ones property, it is solvable in polynomial time. In this paper, we present data re...