New results on stabbing segments with a polygon

We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Loftier and van Kreveld [Algorithmica 56(2), 236-269 (2010)] [16] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard. (C) 2014 Elsevier B.V. All rights reserved.Peer ReviewedPostprint (author's final draft

Locating a service facility and a rapid transit line

In this paper we study a facility location problem in the plane in which a single point (facility) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time of the clients to the facility, using the L 1 or Manhattan metric. The rapid transit line is represented by a line segment with fixed length and arbitrary orientation. The highway is an alternative transportation system that can be used by the clients to reduce their travel time to the facility. This problem was introduced by Espejo and Rodríguez-Chía in [8]. They gave both a characterization of the optimal solutions and an algorithm running in O(n 3logn) time, where n represents the number of clients. In this paper we show that the Espejo and Rodríguez-Chía’s algorithm does not always work correctly. At the same time, we provide a proper characterization of the solutions with a simpler proof and give an algorithm solving the problem in O(n 3) time.Peer ReviewedPostprint (published version

The 1-Center and 1-Highway Problem

In this paper we extend the Rectilinear 1-center as follows: Given a set S of n points in the plane, we are interested in locating a facility point f and a rapid transit line (highway) H that together minimize the expression max p ∈ S d H (p,f), where d H (p,f) is the travel time between p and f. A point p ∈ S uses H to reach f if H saves time for p. We solve the problem in O(n 2) or O(nlogn) time, depending on whether or not the highway’s length is fixed.Peer ReviewedPostprint (published version

Geometric biplane graphs I: maximal graphs

We study biplane graphs drawn on a finite planar point set in general position. This is the family of geometric graphs whose vertex set is and can be decomposed into two plane graphs. We show that two maximal biplane graphs-in the sense that no edge can be added while staying biplane-may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over -element point sets.Peer ReviewedPostprint (author's final draft

Geometric biplane graphs II: graph augmentation

We study biplane graphs drawn on a finite point set in the plane in general position. This is the family of geometric graphs whose vertex set is and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.Peer ReviewedPostprint (author's final draft

Geometric biplane graphs I: maximal graphs

Postprint (author’s final draft

Geometric Biplane Graphs II: Graph Augmentation

We study biplane graphs drawn on a nite point set S in the plane in general position. This is the family of geometric graphs whose vertex set is S and which can be decomposed into two plane graphs. We show that every su ciently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6- connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.Peer ReviewedPostprint (author’s final draft

Economic comparison of the processes of 'coal steelmaking' and 'steelmaking on remelted iron'

30.00; Translated from Chinese (Iron Steel (China) 1986 v. 21(7) p. 57-62)SIGLEAvailable from British Library Document Supply Centre- DSC:9022.06(BISI-Trans--26061)T / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Geodesic order types

The geodesic between two points a and b in the interior of a simple polygon P is the shortest polygonal path inside P that connects a to b. It is thus the natural generalization of straight line segments on unconstrained point sets to polygonal environments. In this paper we use this extension to generalize the concept of the order type of a set of points in the Euclidean plane to geodesic order types. In particular, we show that, for any set S of points and an ordered subset of at least four points, one can always construct a polygon P such that the points of define the geodesic hull of S w.r.t. P, in the specified order. Moreover, we show that an abstract order type derived from the dual of the Pappus arrangement can be realized as a geodesic order type.Peer ReviewedPostprint (published version

Computing a visibility polygon using few variables

We present several algorithms for computing the visibility polygon of a simple polygon P of n vertices (out of which r are reflex) from a viewpoint inside P, when P resides in read-only memory and only few working variables can be used. The first algorithm uses a constant number of variables, and outputs the vertices of the visibility polygon in O (n (r) over bar) time, where (r) over bar denotes the number of reflex vertices of P that are part of the output. Whenever we are allowed to use O(s) variables, the running time decreases to O (nr/2(s) + n log(2) r) (or O (nr/2(s) + n log r) randomized expected time), where s is an element of O (log r). This is the first algorithm in which an exponential space-time trade-off for a geometric problem is obtained. (C) 2014 Elsevier B.V. All rights reserved.Peer Reviewe