On the Oß-hull of a planar point set

© 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We study the Oß-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle ß. Given a set P of n points in the plane, we show how to maintain the Oß-hull of P while ß runs from 0 to p in T(n log n) time and O(n) space. With the same complexity, we also find the values of ß that maximize the area and the perimeter of the Oß-hull and, furthermore, we find the value of ß achieving the best fitting of the point set P with a two-joint chain of alternate interior angle ß.Peer ReviewedPostprint (author's final draft

Production matrices for geometric graphs

We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot numbers, and Fibonacci numbers arise in a natural way, and also new formulas are obtained, such as a formula for the number of non-crossing geometric graphs with root vertex of given degree. The characteristic polynomials of some of these production matrices are also presented. The proofs make use of generating trees and Riordan arrays.Postprint (updated version

On finding widest empty curved corridors

Open archive-ElsevierAn α-siphon of width w is the locus of points in the plane that are at the same distance w from a 1-corner polygonal chain C such that α is the interior angle of C. Given a set P of n points in the plane and a fixed angle α, we want to compute the widest empty α-siphon that splits P into two non-empty sets.We present an efficient O(n log3 n)-time algorithm for computing the widest oriented α-siphon through P such that the orientation of a half-line of C is known.We also propose an O(n3 log2 n)-time algorithm for the widest arbitrarily-oriented version and an (nlog n)-time algorithm for the widest arbitrarily-oriented α-siphon anchored at a given point

The siphon problem

An α-siphon is the locus of points in the plane that are at the same distance ǫ from a polygonal chain consisting of two half-lines emanating from a common point such that α is the interior angle of the half-lines. Given a set S of n points in the plane and a fixed angle α, we want to compute an α-siphon of largest width ǫ such that no points of S lies in its interior. We present an efficient O(n2)-time algorithm for computing an orthogonal siphon. The approach can be handled to solve the problem of the oriented α-siphon for which the orientation of a half-line is known. We also propose an O(n3 log n)-time algorithm for the arbitrarily oriented version.Ministerio de Ciencia y TecnologiaFondo Europeo de Desarrollo RegionalGeneralitat de Cataluny

Characteristic polynomials of production matrices for geometric graphs

An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production matrices for non-crossing geometric graphs on point sets in convex position [Huemer, C., A. Pilz, C. Seara, and R.I. Silveira, Production matrices for geometric graphs, Electronic Notes in Discrete Mathematics 54 (2016) 301–306]. In this note, we determine the characteristic polynomials of these matrices. Then, the Cayley-Hamilton theorem implies relations among the numbers of geometric graphs with different numbers of vertices. Further, relations between characteristic polynomials of production matrices for geometric graphs and Fibonacci numbers are revealed.This project has received funding from the European Union’s Horizon 89 2020 research and innovation programme under the Marie Sk lodowska- 90 Curie grant agreement No 734922. 91 C. H., C. S., and R. I. S. were partially supported by projects MINECO MTM2015- 92 63791-R and Gen. Cat. DGR2014SGR46. R. I. S. was also supported by MINECO 93 through the Ramon y Cajal programPostprint (published version

Separating bichromatic point sets in the plane by restricted orientation convex hulls

The version of record is available online at: http://dx.doi.org/10.1007/s10898-022-01238-9We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and O be a set of k=2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of O for which the O-convex hull of R contains no points of B. For k=2 orthogonal lines we have the rectilinear convex hull. In optimal O(nlogn) time and O(n) space, n=|R|+|B|, we compute the set of rotation angles such that, after simultaneously rotating the lines of O around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where O is formed by k=2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of O, let ai be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in O(1/T·NlogN) time and O(1/T·N) space, where T=min{a1,…,ak} and N=max{k,|R|+|B|}. We finally consider the case in which O is formed by k=2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to p. We show that this last case can also be solved in optimal O(nlogn) time and O(n) space, where n=|R|+|B|.Carlos Alegría: Research supported by MIUR Proj. “AHeAD” no 20174LF3T8. David Orden: Research supported by Project PID2019-104129GB-I00 / AEI / 10.13039/501100011033 of the Spanish Ministry of Science and Innovation. Carlos Seara: Research supported by Project PID2019-104129GB-I00 / AEI / 10.13039/501100011033 of the Spanish Ministry of Science and Innovation. Jorge Urrutia: Research supported in part by SEP-CONACYThis project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska–Curie Grant Agreement No 734922.Peer ReviewedPostprint (published version

Capturing points with a rotating polygon (and a 3D extension)

This is a post-peer-review, pre-copyedit version of an article published in Theory of computing systems: an international journal. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00224-018-9885-yWe study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a segment or a line. We also solve an extension to 3D where we rotate a polyhedron around a given point to contain the maximum number of elements from a set of points in the space.Peer ReviewedPostprint (author's final draft

Maximum box problem on stochastic points

This is a post-peer-review, pre-copyedit version of an article published in Algorithmica. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00453-021-00882-z.Given a finite set of weighted points in Rd (where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that the sum of the weights of the points that it contains is maximized. We consider that each point of the input has a probability of being present in the final random point set, and these events are mutually independent; then, the total weight of a maximum box is a random variable. We aim to compute both the probability that this variable is at least a given parameter, and its expectation. We show that even in d=1 these computations are #P-hard, and give pseudo-polynomial time algorithms in the case where the weights are integers in a bounded interval. For d=2, we consider that each point is colored red or blue, where red points have weight +1 and blue points weight -8. The random variable is the maximum number of red points that can be covered with a box not containing any blue point. We prove that the above two computations are also #P-hard, and give a polynomial-time algorithm for computing the probability that there is a box containing exactly two red points, no blue point, and a given point of the plane.This work has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922.Peer ReviewedPostprint (published version

On the determining number and the metric dimension of graphs

This paper initiates a study on the problem of computing the difference between the metric dimension and the determining number of graphs. We provide new proofs and results on the determining number of trees and Cartesian products of graphs, and establish some lower bounds on the difference between the two parameters.Postprint (published version

Covering point sets with two disjoint disks or squares

Open archive-ElsevierWe study the following problem: Given a set of red points and a set of blue points on the plane, find two unit disks CR and CB with disjoint interiors such that the number of red points covered by CR plus the number of blue points covered by CB is maximized. We give an algorithm to solve this problem in O(n8/3 log2 n) time, where n denotes the total number of points. We also show that the analogous problem of finding two axis-aligned unit squares SR and SB instead of unit disks can be solved in O(nlog n) time, which is optimal. If we do not restrict ourselves to axis-aligned squares, but require that both squares have a common orientation, we give a solution using O(n3 log n) time