David Flores-Peñaloza

Modem illumination of monotone polygons

Abstract We study a generalization of the classical problem of the illumination of polygons. Instead of modeling a light source we model a wireless device whose radio signal can penetrate a given number k of walls. We call these objects k-modems and study the minimum number of k-modems sufficient and sometimes necessary to illuminate monotone and monotone orthogonal polygons. We show that every monotone polygon with n vertices can be illuminated with ⌈ n − 2 2 k + 3 ⌉ k-modems. In addition, we exhibit examples of monotone polygons requiring at least ⌈ n − 2 2 k + 3 ⌉ k-modems to be illuminated. For monotone orthogonal polygons with n vertices we show that for k = 1 and for even k, every such polygon can be illuminated with ⌈ n − 2 2 k + 4 ⌉ k-modems, while for odd k ≥ 3 , ⌈ n − 2 2 k + 6 ⌉ k-modems are always sufficient. Further, by presenting according examples of monotone orthogonal polygons, we show that both bounds are tight.

On crossing numbers of geometric proximity graphs

Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straight-line segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the k-nearest neighbor graph, the k-relative neighborhood graph, the k-Gabriel graph and the k-Delaunay graph. For k=0 (k=1 in the case of the k-nearest neighbor graph) these graphs are plane, but for higher values of k in general they contain crossings. In this paper, we provide lower and upper bounds on their minimum and maximum number of crossings. We give general bounds and we also study particular cases that are especially interesting from the viewpoint of applications. These cases include the 1-Delaunay graph and the k-nearest neighbor graph for small values of k.

Embedding the double circle in a square grid of minimum size

In 1926, Jarnik investigated the drawing of a curve that visits a large number of lattice points relative to its curvature. To this end, he constructed a convex n-gon with vertices on a “small” integer grid [0, c.n3/2]2, where c > 0 is a constant, and proved that this grid size is optimal up to a constant factor. We consider a similar construction for the double circle of 2n points and prove that it can be embedded in a grid of the same asymptotic size. Moreover, we give an O(n)-time algorithm to generate the corresponding point set.

Proximity graphs inside large weighted graphs

Given a large weighted graph G = (V;E) and a subset U of V , we de¯ne several graphs with vertex set U in which two vertices are adjacent if they satisfy some prescribed proximity rule. These rules use the shortest path distance in G and generalize the proximity rules that generate some of the most common proximity graphs in Euclidean spaces. We prove basic properties of the de¯ned graphs and provide algorithms for their computation.

Min-energy broadcast in mobile ad hoc networks with restricted motion

This paper concerns about energy-efficient broadcasts in mobile ad hoc networks, yet in a model where each station moves on the plane with uniform rectilinear motion. Such restriction is imposed to discern which issues arise from the introduction of movement in the wireless ad hoc networks.Given a transmission range assignment for a set of n stations S, we provide an polynomial O(n2)-time algorithm to decide whether a broadcast operation from a source station could be performed or not. Additionally, we study the problem of computing a transmission range assignment for S that minimizes the energy required in a broadcast operation. An O(n3log n)-time algorithm for this problem is presented, under the assumption that all stations have equally sized transmission ranges. However, we prove that the general version of such problem is NP-hard and not approximable within a (1−o(1))ln n factor (unless NP⊂DTIME(nO(log log n))). We then propose a polynomial time approximation algorithm for a restricted version of that problem.

Covering moving points with anchored disks

Consider a set of mobile clients represented by n points in the plane moving at constant speed along n different straight lines. We study the problem of covering all mobile clients using a set of k disks centered at k fixed centers. Each disk exists only at one instant and while it does, covers any client within its coverage radius. The task is to select an activation time and a radius for each disk such that every mobile client is covered by at least one disk. In particular, we study the optimization problem of minimizing the maximum coverage radius. First we prove that, although the static version of the problem is polynomial, the kinetic version is NP-hard. Moreover, we show that the problem is not approximable by a constant factor (unless P=NP). We then present a generic framework to solve it for fixed values of k, which in turn allows us to solve more general optimization problems. Our algorithms are efficient for constant values of k.

Computing balanced islands in two colored point sets in the plane

Abstract Let S be a set of n points in general position in the plane, r of which are red and b of which are blue. In this paper we present algorithms to find convex sets containing a balanced number of red and blue points. We provide an O ( n 4 ) time algorithm that for a given α ∈ [ 0 , 1 2 ] finds a convex set containing exactly ⌈ α r ⌉ red points and exactly ⌈ α b ⌉ blue points of S. If ⌈ α r ⌉ + ⌈ α b ⌉ is not much larger than 1 3 n , we improve the running time to O ( n log ⁡ n ) . We also provide an O ( n 2 log ⁡ n ) time algorithm to find a convex set containing exactly ⌈ r + 1 2 ⌉ red points and exactly ⌈ b + 1 2 ⌉ blue points of S, and show that balanced islands with more points do not always exist.

Optimizing some constructions with bars: new geometric knapsack problems

A set of vertical bars planted on given points of a horizontal line defines a fence composed of the quadrilaterals bounded by successive bars. A set of bars in the plane, each having one endpoint at the origin, defines an umbrella composed of the triangles bounded by successive bars. Given a collection of bars, we study how to use them to build the fence or the umbrella of maximum total area. We present optimal algorithms for these constructions. The problems introduced in this paper are related to the Geometric Knapsack problems (Arkin et al. in Algorithmica 10:399–427, 1993) and the Rearrangement Inequality (Wayne in Scripta Math 12(2):164–169, 1946).