José Miguel Díaz-Báñez

Continuous location of dimensional structures

Abstract A natural extension of point facility location problems are those problems in which facilities are extensive, i.e. those that cannot be represented by isolated points but as some dimensional structures, such as straight lines, line-segments, polygonal curves, or circles. In this paper a review of the existing work on the location of extensive facilities in continuous spaces is given. Gaps in the knowledge are identified and suggestions for further research are made.

Facility location problems in the plane based on reverse nearest neighbor queries

For a finite set of points S, the (monochromatic) reverse nearest neighbor (RNN) rule associates with any query point q the subset of points in S that have q as its nearest neighbor. In the bichromatic reverse nearest neighbor (BRNN) rule, sets of red and blue points are given and any blue query is associated with the subset of red points that have it as its nearest blue neighbor. In this paper we introduce and study new optimization problems in the plane based on the bichromatic reverse nearest neighbor (BRNN) rule. We provide efficient algorithms to compute a new blue point under criteria such as: (1) the number of associated red points is maximum (MAXCOV criterion); (2) the maximum distance to the associated red points is minimum (MINMAX criterion); (3) the minimum distance to the associated red points is maximum (MAXMIN criterion). These problems arise in the competitive location area where competing facilities are established. Our solutions use techniques from computational geometry, such as the concept of depth of an arrangement of disks or upper envelope of surface patches in three dimensions.

Fitting rectilinear polygonal curves to a set of points in the plane

Abstract In this paper two problems of fitting rectilinear polygonal curves to a set of points in the plane according to the minimax approximation are considered. The constraints are, respectively, on the number of vertices and length of the polygonal curve. In both cases efficient algorithms are developed.

One-to-One Coordination Algorithm for Decentralized Area Partition in Surveillance Missions with a Team of Aerial Robots

This paper presents a decentralized algorithm for area partition in surveillance missions that ensures information propagation among all the robots in the team. The robots have short communication ranges compared to the size of the area to be covered, so a distributed one-to-one coordination schema has been adopted. The goal of the team is to minimize the elapsed time between two consecutive observations of any point in the area. A grid-shape area partition strategy has been designed to guarantee that the information gathered by any robot is shared among all the members of the team. The whole proposed decentralized strategy has been simulated in an urban scenario to confirm that fulfils all the goals and requirements and has been also compared to other strategies.

Decentralized strategy to ensure information propagation in area monitoring missions with a team of UAVs under limited communications

This paper presents the decentralized strategy followed to ensure information propagation in area monitoring missions with a fleet of heterogeneous UAVs with limited communication range. The goal of the team is to detect pollution sources over a large area as soon as possible. Hence the elapsed time between two consecutive visits should be minimized. On the other hand, in order to exploit the capabilities derived from having a fleet of UAVs, an efficient area partition is performed in a distributed manner using a one-to-one coordination schema according to the limited communication ranges. Another requirement is to have the whole team informed about the location of the new pollution sources detected. This requirement is challenging because the communication range of the vehicles is small compared to the area covered in the mission. Sufficient and necessary conditions are provided to guarantee one-to-one UAV communication in grid-shape area partitions, allowing to share any new information among all the members of the team, even under strong communication constraints. The proposed decentralized strategy has been simulated to confirm that fulfils all the goals and requirements and has been also compared to other patrolling strategies.

THE LARGEST EMPTY ANNULUS PROBLEM

Given a set of n points S in the Euclidean plane, we address the problem of computing an annulus A, (open region between two concentric circles) of largest width, that partitions S into a subset of points inside and a subset of points outside the circles, such that no point p∈S lies in the interior of A. This problem can be considered as a maximin facility location problem for n points such that the facility is a circumference. We give a characterization of the centres of annuli which are locally optimal and we show that the problem can be solved in O(n3logn) time and O(n) space. We also consider the case in which the number of points in the inner circle is a fixed value k. When k∈O(n) our algorithm runs in O(n3logn) time and O(n) space, furthermore, we can simultaneously optimize for all values of k within the same time bound. When k is small, that is a fixed constant, we can solve the problem in O(n logn) time and O(n) space.

Computing optimal islands

Let S be a bicolored set of n points in the plane. A subset I of S is an island if there is a convex set C such that I=C@?S. We give an O(n^3)-time algorithm to compute a monochromatic island of maximum cardinality. Our approach is adapted to optimize similar (decomposable) objective functions. Finally, we use our algorithm to give an O(logn)-approximation for the problem of computing the minimum number of convex polygons that cover a class region.

The Velocity Assignment Problem for Conflict Resolution with Multiple Aerial Vehicles Sharing Airspace

Efficient conflict resolution methods for multiple aerial vehicles sharing airspace are presented. The problem of assigning a velocity profile to each aerial vehicle in real time, such that the separation between them is greater than a given safety distance, is considered and the total deviation from the initial planned trajectory is minimized. The proposed methods involve the use of appropriate airspace discretization. In the paper it is demonstrated that this aerial vehicle velocity assignment problem is NP-hard. Then, the paper presents three different collision detection and resolution methods based on speed planning. The paper also presents simulations and studies for several scenarios.

The block-sharing strategy for area monitoring missions using a decentralized multi-UAV system

In monitoring missions, using a cooperative team of Unmanned Aerial Vehicles (UAVs), the goal is to minimize the elapsed time between two consecutive observations of any point in the area. Techniques based in area partitioning achieve the goal when the sub-area assigned to each UAV is according to its capabilities. In previous work of the authors [1] it was presented the one-to-one strategy to obtain in a decentralized way a near optimal partition from any initial grid partition. In this paper a generalization of that strategy called the block-sharing technique is presented. The goal in this work is to accelerate the convergence to an optimal partition with respect to previous work.

Bichromatic 2-center of pairs of points

We study a class of geometric optimization problems closely related to the 2-center problem: Given a set S of n pairs of points in the plane, for every pair, we want to assign red color to a point of the pair and blue color to the other point in order to optimize the radii of the minimum enclosing ball of the red points and the minimum enclosing ball of the blue points. In particular, we consider the problems of minimizing the maximum and minimizing the sum of the two radii of the minimum enclosing balls. For each case, minmax and minsum, we consider distances measured in the L 2 and in the L ∞ metrics.