Inmaculada Ventura

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.

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.

Bichromatic separability with two boxes: A general approach

Let S be a set of n points on the plane in general position such that its elements are colored red or blue. We study the following problem: Find a largest subset of S which can be enclosed by the union of two, not necessarily disjoint, axis-aligned rectanglesRandBsuch thatR (resp.B) contains only red (resp. blue) points. We prove that this problem can be solved in O(n^2logn) time and O(n) space. Our approach is based on solving some instances of Bentleys maximum-sum consecutive subsequence problem. We introduce the first known data structure to dynamically maintain the optimal solution of this problem. We show that our techniques can be used to efficiently solve a more general class of problems in data analysis.

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.

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 class cover problem with boxes

In this paper we study the following problem: Given sets R and B of r red and b blue points respectively in the plane, find a minimum-cardinality set H of axis-aligned rectangles (boxes) so that every point in B is covered by at least one rectangle of H, and no rectangle of H contains a point of R. We prove the NP-hardness of the stated problem, and give either exact or approximate algorithms depending on the type of rectangles considered. If the covering boxes are vertical or horizontal strips we give an efficient algorithm that runs in O(rlogr+blogb+rb) time. For covering with oriented half-strips an optimal O((r+b)log(min{r,b}))-time algorithm is shown. We prove that the problem remains NP-hard if the covering boxes are half-strips oriented in any of the four orientations, and show that there exists an O(1)-approximation algorithm. We also give an NP-hardness proof if the covering boxes are squares. In this situation, we show that there exists an O(1)-approximation algorithm.

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 from the clients to the facility, using the L1 or Manhattan metric. The rapid transit line is given by a segment with any length and orientation, and is an alternative transportation line that can be used by the clients to reduce their travel time to the facility. We study the variant of the problem in which clients can enter and exit the highway at any point. We provide an O(n 3 )-time algorithm that solves this variant, where n is the number of clients. We also present a detailed characterization of the solutions, which depends on the speed given in the highway.

Computing an obnoxious anchored segment

In this paper we address a maximin facility location problem for n points in the plane such that the facility is an anchored line segment of a fixed length. We show how to solve this problem in O(nlogn) time, which is optimal in the algebraic computation tree model. Other variations of this problem are also discussed.

The 1-median and 1-highway problem

In this paper we study a facility location problem in the plane in which a single point (median) 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 L1 or Manhattan metric. The highway is an alternative transportation system that can be used by the clients to reduce their travel time to the facility. We represent the highway by a line segment with fixed length and arbitrary orientation. This problem was introduced in [Computers & Operations Research 38(2) (2011) 525–538]. They gave both a characterization of the optimal solutions and an algorithm running in O(n3logn) time, where n represents the number of clients. In this paper we show that the previous characterization does not work in general. Moreover, we provide a complete characterization of the solutions and give an algorithm solving the problem in O(n3) time.

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 ∈ Sd 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(n2) or O(nlogn) time, depending on whether or not the highway’s length is fixed.