Most vital segment barriers

We study continuous analogues of "vitality" for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion of "most vital arcs" for flows/paths to geometric environments. We give hardness results and efficient algorithms for various versions of the problem, (almost) completely separating hard and polynomially-solvable cases

Algorithms for distance problems in planar complexes of global nonpositive curvature

CAT(0) metric spaces and hyperbolic spaces play an important role in combinatorial and geometric group theory. In this paper, we present efficient algorithms for distance problems in CAT(0) planar complexes. First of all, we present an algorithm for answering single-point distance queries in a CAT(0) planar complex. Namely, we show that for a CAT(0) planar complex K with n vertices, one can construct in O(n^2 log n) time a data structure D of size O(n^2) so that, given a point x in K, the shortest path gamma(x,y) between x and the query point y can be computed in linear time. Our second algorithm computes the convex hull of a finite set of points in a CAT(0) planar complex. This algorithm is based on Toussaint's algorithm for computing the convex hull of a finite set of points in a simple polygon and it constructs the convex hull of a set of k points in O(n^2 log n + nk log k) time, using a data structure of size O(n^2 + k)

Securing Pathways with Orthogonal Robots

The protection of pathways holds immense significance across various domains, including urban planning, transportation, surveillance, and security. This article introduces a groundbreaking approach to safeguarding pathways by employing orthogonal robots. The study specifically addresses the challenge of efficiently guarding orthogonal areas with the minimum number of orthogonal robots. The primary focus is on orthogonal pathways, characterized by a path-like dual graph of vertical decomposition. It is demonstrated that determining the minimum number of orthogonal robots for pathways can be achieved in linear time. However, it is essential to note that the general problem of finding the minimum number of robots for simple polygons with general visibility, even in the orthogonal case, is known to be NP-hard. Emphasis is placed on the flexibility of placing robots anywhere within the polygon, whether on the boundary or in the interior.Comment: 8 pages, 5 figure

Playing Billiard in Version Space

A ray-tracing method inspired by ergodic billiards is used to estimate the theoretically best decision rule for a set of linear separable examples. While the Bayes-optimum requires a majority decision over all Perceptrons separating the example set, the problem considered here corresponds to finding the single Perceptron with best average generalization probability. For randomly distributed examples the billiard estimate agrees with known analytic results. In real-life classification problems the generalization error is consistently reduced compared to the maximal stability Perceptron.Comment: uuencoded, gzipped PostScript file, 127576 bytes To recover 1) save file as bayes.uue. Then 2) uudecode bayes.uue and 3) gunzip bayes.ps.g

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

A linear-time algorithm for constructing a circular visibility diagram

To computer circular visibility inside a simple polygon, circular arcs that emanate from a given interior point are classified with respect to the edges of the polygon they first intersect. Representing these sets of circular arcs by their centers results in a planar partition called the circular visibility diagram. An O(n) algorithm is given for constructing the circular visibility diagram for a simple polygon with n vertices.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41346/1/453_2005_Article_BF01206329.pd