Dual embeddings of dense near polygons

Let e: S -> Sigma be a full polarized projective embedding of a dense near polygon S, i.e., for every point p of S, the set H(p) of points at non-maximal distance from p is mapped by e into a hyperplane Pi(p) of Sigma. We show that if every line of S is incident with precisely three points or if S satisfies a certain property (P(de)) then the map p bar right arrow Pi p defines a full polarized embedding e* (the so-called dual embedding of e) of S into a subspace of the dual Sigma* of Sigma. This generalizes a result of [6] where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P(de)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding

Dilation, Transport, Visibility and Fault-Tolerant Algorithms

Connecting some points in the plane by a road network is equivalent to constructing a finite planar graph G whose vertex set contains a predefined set of vertices (i. e., the possible destinations in the road network). The dilation between two vertices p and q of graph G is defined as the Euclidean length of a shortest path in G from p to q, divided by the Euclidean distance from p to q. That is, given a point set P, the goal is to place some additional crossing vertices C such that there exists a planar graph G = (P ∪ C, E) whose dilation is small. Here, the dilation of G is defined as the maximum dilation between two vertices in G. We show that, except for some special point sets P, there is a lower bound Δ(P) > 1, depending on P, on the dilation of any finite graph containing P in its vertex set. The transportation problem is the problem of finding a transportation plan that minimizes the total transport cost. We are given a set of suppliers, and each supplier produces a fixed amount of some commodity, say, bread. Furthermore, there is a set of customers, and each customer has some demand of bread, such that the total demand equals the amount of bread the suppliers produce. The task is to assign each unit of bread produced to some customer, such that the total transportation cost becomes a minimum. A first idea is to assign each unit of bread to the client to which the transport cost of this unit is minimal. Clearly, this gives rise to a transportation plan which minimizes the total transportation cost. However, it is likely that not every customer will obtain the required amount of bread. Therefore, we need to use a different algorithm for distributing the supplier's bread. We show that if the bread produced by the suppliers is given by a continuous probability density function and the set of customers is discrete, then every optimal transport plan can be characterized by a unique additively weighted Voronoi diagram for the customers. When managing the construction process of a building by a digital model of the building, it is necessary to compute essential parts between walls of the building. Given two walls A and B, the essential part between A and B is the set of line segments s where one endpoint belongs to A, the other endpoint belongs to B, and s does not intersect A or B. We give an algorithm that computes, in linear time, the essential parts between A and B. Our algorithm is based on computing the visibility polygon of A and of B, and two shortest paths connecting points of A with points of B. We conclude the thesis by giving fault-tolerant algorithms for some fundamental geometric problems. We assume that a basic primitive operation used by an algorithm fails with some small probability p. Depending on the results of the primitive operations, it is possible that the algorithm will not work correctly. For example, one faulty comparison when executing a sorting algorithm can result in some numbers being placed far away from their true positions. An algorithm is called tolerant, if with high probability a good answer is given, if the error probability p is small. We provide tolerant algorithms that find the maximum of n numbers, search for a key in a sorted sequence of n keys, sort a set of n numbers, and solve Linear Programming in R2

The Offset Filtration of Convex Objects

We consider offsets of a union of convex objects. We aim for a filtration, a sequence of nested simplicial complexes, that captures the topological evolution of the offsets for increasing radii. We describe methods to compute a filtration based on the Voronoi partition with respect to the given convex objects. The size of the filtration and the time complexity for computing it are proportional to the size of the Voronoi diagram and its time complexity, respectively. Our approach is inspired by alpha-complexes for point sets, but requires more involved machinery and analysis primarily since Voronoi regions of general convex objects do not form a good cover. We show by experiments that our approach results in a similarly fast and topologically more stable method for computing a filtration compared to approximating the input by a point sample

Collection of abstracts of the 24th European Workshop on Computational Geometry

International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum