[Show abstract][Hide abstract] ABSTRACT: We study a problem about shortest paths in Delaunay triangulations. Given two nodes s, t in the Delaunay triangulation of a point set S, we look for a new point p ∉ S that can be added, such that the shortest path from s to t, in the Delaunay triangulation of S∪{p}, improves as much as possible. We study several properties of the problem, and give efficient algorithms to find such a point when the graph-distance used is Euclidean and for the link-distance. Several other variations of the problem are also discussed.
International Journal of Computational Geometry & Applications 04/2013; 22(06). DOI:10.1142/S0218195912500161 · 0.08 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Consider a set of receptors belonging to two competitive telecommunication firms, the blue firm and the red firm. The receptors are represented as points in the plane, b are blue and belong to the blue firm and r are red and belong to the red firm. The blue firm has an emitting device represented as a point that moves along a path sending information to blue receptors as follows: At any time, the device sends information to all blue receptors covered by the largest disk centered at it and containing no red receptor. In this scenario, we study two optimization problems. The first problem is to compute a path P, such that the number of blue receptors served by a moving device is maximized. In particular, we give efficient algorithms when P is a straight line, an anchored half-line, and an axis-parallel double ray. As a second task, we study the problem of removing the minimum number of red receptors in such a way there exists a straight line path P so that if the device moves along P all blue receptors are served. We prove geometrical properties of an optimal straight line and propose efficient algorithms depending on the degrees of freedom of the line.
The Computer Journal 06/2012; 56(7). DOI:10.1093/comjnl/bxs030 · 0.79 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let S be a set of n + m sites, of which n are red and have weight w
R
, and m are blue and weigh w
B
. The objective of this paper is to calculate the minimum value of the red sites’ weight such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected region. This problem is solved for the multiplicatively-weighted Voronoi diagram in \(\mathcal{O}((n+m)^2 \log (nm))\) time and for both the additively-weighted and power Voronoi diagram in \(\mathcal{O}(nm \log (nm))\) time.
[Show abstract][Hide abstract] ABSTRACT: Coverage problems are a flourishing topic in optimization, thanks to the recent advances in the field of wireless sensor networks. The main coverage issue centres around critical conditions that require reliable monitoring and prohibit failures. This issue can be addressed by maximal-exposure paths, regarding which this article presents new results. Namely, it shows how to minimize the sensing range of a set of sensors in order to ensure the existence of a k-covered path between two points on a given region. Such a path's coverage depends on k≥2, which is fixed. The three types of regions studied are: a planar graph, the whole plane and a polygonal region.
The Computer Journal 01/2012; 55(1):69-81. DOI:10.1093/comjnl/bxr076 · 0.79 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Spatial models of two-player competition in spaces with more than one dimension almost never have pure-strategy Nash equilibria,
and the study of the equilibrium positions, if they exist, yields a disappointing result: the two players must choose the
same position to achieve equilibrium. In this work, a discrete game is proposed in which the existence of Nash equilibria
is studied using a geometric argument. This includes a definition of equilibrium which is weaker than the classical one to
avoid the uniqueness of the equilibrium position. As a result, a “region of equilibrium” appears, which can be located by
geometric methods. In this area, the players can move around in an “almost-equilibrium” situation and do not necessarily have
to adopt the same position.
KeywordsNash equilibrium–Computational geometry–Game theory–Location
International Journal of Games Theory 08/2011; 40(3):449-459. DOI:10.1007/s00182-010-0241-y · 0.58 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Given a set S of n antennas and point q on the plane, q is α-covered by S if ∃si,sj∈S such that the angle ∡(si,q,sj)≥α. It is shown how to minimise the transmission range of S to α-cover a point in O(n) time and space, and how to construct the π2-covered region in O(nlogn) time and O(n) space. Finally, this paper introduces the Coverage Voronoi diagram and an algorithm to construct it in O(n4logn) time.
Operations Research Letters 07/2011; 39(4):241-245. DOI:10.1016/j.orl.2011.04.011 · 0.62 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Un camino que conecta dos puntos s y t en el plano es de desviación mínima respecto de un conjunto de puntos S si la mayor de las distancias entre un punto del camino y S es la menor posible entre todos los caminos que conectan s y t. En este trabajo estudiamos los caminos de desviación mínima que satisfacen además que todo subcamino es también de desviación mínima, a los que llamamos caminos de desviación mínima local. Postprint (published version)
[Show abstract][Hide abstract] ABSTRACT: This paper considers a problem of political economy in which a Nash equilibrium study is performed in a proposed game with restrictions where the two major parties in a country vary their position within a politically flexible framework to increase their number of voters. The model as presented fits the reality of many countries. Moreover, it avoids the uniqueness of equilibrium positions. The problem is stated and solved from a geometric point of view.
European Journal of Operational Research 03/2010; 201(3):892-896. DOI:10.1016/j.ejor.2009.04.002 · 2.36 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k ≥ 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.
International Journal of Computational Geometry & Applications 12/2009; 19(06). DOI:10.1142/S0218195909003143 · 0.08 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper one gives a description of the forest modeler VOREST. It is a single tree type forest modeler based on generalized Voronoi diagrams. In the paper one describes why Voronoi diagrams appears in a natural way in forest growth modeling and how they are used in the design of VOREST.
[Show abstract][Hide abstract] ABSTRACT: A point q is embraced by a set of points S if q is interior to the convex hull of S [8]. In some illumination applications where points of S are lights and q is a point to be illuminated, the embracing concept is related to a good illumination [4, 6], also known as the ∆-guarding [12] and the well-covering [10]. In this paper, we are not only interested in convex dependency (which is actually the embracing notion) but also in proximity.
Suppose that the sites of S are lights or antennas with limited range; due to their limited power, they cover a disk of a given radius r centered at the sites of S. Only the points lying in such disks are illuminated. If we want to embrace the point q with the minimum range r, we need to know which is the closest light s
q
to q such that q lies in the convex hull formed by s
q
and the lights of S closer to q than s
q
. This subset of S related to point q is called the closest embracing set for q in relation to S and its cardinality is the closest embracing number of q. By assigning every point q in the convex hull of S to its closest embracing site s
q
, we obtain a partition of the convex hull that we call the embracing Voronoi diagram of S. This paper proves some properties of the embracing Voronoi diagrams and describes algorithms to compute such diagrams, as
well as the levels in which the convex hull is decomposed regarding the closest embracing number.
Journal of Mathematical Sciences 09/2009; 161(6):909-918. DOI:10.1007/s10958-009-9610-0
[Show abstract][Hide abstract] ABSTRACT: Every orthogonal polygon can be illuminated by ⌊n/4⌋ lights situated in the vertices of the polygon. In this paper we improve this bound for pyramids, showing that ⌈n/6⌉ guards situated in vertices are always sufficient and sometimes necessary for watching any pyramid of n vertices. Our proof leads to a linear time algorithm for placing those guards.
Information Processing Letters 06/2009; 109(13):719-721. DOI:10.1016/j.ipl.2009.03.014 · 0.55 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: In this paper we describe and solve the following geometric optimisation problem: given a set S of n points on the plane (antennas) and two points A and B, find the smallest radial range r∈ℜ+ (power transmission range of the antennas) so that a path with endpoints A and B exists in which all points are within the range of at least two antennas. The solution to the problem has several applications (e.g., in the planning of safe routes). We present an O(nlogn) time solution, which is based on the second order Voronoi diagram. We also show how to obtain a path with such characteristics.
Information Processing Letters 06/2009; 109:768-773. DOI:10.1016/j.ipl.2009.03.016 · 0.55 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Let S be a set of n points in the plane (antennas). An object is said to be 2-covered with range r if every point of such object is interior to at least two discs (not necessarily the same) centered at S of radius r. The following problem is considered in this pa- per: given a set S of n antennas and a planar geomet- ric graph G = (N,E), calculate the minimum power transmission range of S so that a 2-covered path be- tween two given nodes of G exists. Is is described an algorithm to solve this problem in two phases. In the first phase (preprocessing phase), graph G is trans- formed into a weighted graph Gw (using the second order Voronoi diagram of S) and then a minimum spanning tree of Gw, Tw, is found. This phase takes O(|E|◊n),|E| > logn, time. In the second phase (so- lution phase), the minimum power transmission range of S and a 2-covered path are calculated using Tw. Re- garding time complexity, this second phase is linear on the number of edges of Tw.
[Show abstract][Hide abstract] ABSTRACT: In this paper we consider polygonizations that are robust when faced with changes in the vertices that are present or in their position. We analyze the dynamic maintenance of different types of polygonizations (monotone, star-shaped…) and we introduce monotone half-convex polygonizations that are specially interesting because they provide minimum cost per insertion or deletion. If we had to delete not only one point but several external layers of the set, then the onion polygonizations would be suited, because they can be updated in constant time. We also consider the case of points that can be moved to contiguous positions and we show how to polygonize the set for updating in linear time. We deal too with security problems for a polygon: What is the maximum distance the vertices of a polygon could be moved away of their position in such a way that the topology on the boundary of the polygon (or its convexity) remains the same?.
[Show abstract][Hide abstract] ABSTRACT: Let G be a connected plane geometric graph with n vertices. In this paper, we study bounds on the number of edges required to be added to G to obtain 2-vertex or 2-edge connected plane geometric graphs. In particular, we show that for G to become 2-edge connected, additional edges are required in some cases and that additional edges are always sufficient. For the special case of plane geometric trees, these bounds decrease to and , respectively.
[Show abstract][Hide abstract] ABSTRACT: In this paper we consider a new time metric called the Heavy Luggage Metric. This metric models the behavior of a traveller in a city wishing to walk as little as possible, maybe carrying some very heavy luggage. A transportation network allows the traveller to move between any pair of stations at cost zero, while walking has a cost proportional to the length of the walked path. We propose efficient algorithms for computing the closest and farthest Voronoi Diagrams for a set of points with respect to this metric.
International Journal of Computational Geometry & Applications 08/2008; 18(04):295-306. DOI:10.1142/S0218195908002635 · 0.08 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Filtering and clustering of the data are very important aspects in data visualization. We will concentrate on these two topics
and study how can we combine them to simulate a multiresolution scheme. We will focus on the properties of Voronoi Diagrams
in order to avoid the need to compute any other time- or space-consuming data structure. Voronoi Diagrams capture deeply the
notion of proximity between elements in an environment and allow queries to be efficiently performed. In this paper we present
an application of Voronoi Diagrams and their use in visualization of georreferenced data. The input is a 2.5 data-set, and
the output is a colored map where proximity to the given locations is used in order to compute the region contours. We have
implemented the proposed techniques in C++. Examples of the results obtained with our application GeoVyS are given in this paper.
Computational Science and Its Applications - ICCSA 2008, International Conference, Perugia, Italy, June 30 - July 3, 2008, Proceedings, Part I; 01/2008
[Show abstract][Hide abstract] ABSTRACT: Every notion of depth induces a stratification of the plane in regions of points with the same depth with respect to a given set of points. The boundaries of these regions, also known as depth-contours, are an appropriate tool for data visualization and have already been studied for some depths like Turkey depth [5, 9, 10, 11] and Delaunay depth [3, 8]. The contours also have applications in quality illumination as is the case of good alpha-illumination [2]. The first alpha-depth contour is also known as the alpha-embracing contour. We prove that the first alpha-depth contour has linear size and we give an algorithm to compute it that runs in O(n<sup>2</sup> log n) time and O(n<sup>2</sup>)space.
Computational Sciences and Its Applications, 2008. ICCSA '08. International Conference on; 01/2008