On approximating shortest paths in weighted triangular tessellations

© 2023 Elsevier. 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 quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path , which is a shortest path from s to t in the space; a weighted shortest vertex path , which is an any-angle shortest path; and a weighted shortest grid path , which is a shortest path whose edges are edges of the tessellation. Given any arbitrary weight assignment to the faces of a triangular tessellation, thus extending recent results by Bailey et al. (2021) [6], we prove upper and lower bounds on the ratios , , , which provide estimates on the quality of the approximation. It turns out, surprisingly, that our worst-case bounds are independent of any weight assignment. Our main result is that in the worst case, and this is tight. As a corollary, for the weighted any-angle path we obtain the approximation result .P. B. is partially supported by NSERC. G. E., D. O. and R. I. S. are partially supported by H2020-MSCA-RISE project 734922 - CONNECT and project PID2019-104129GB-I00 funded by MCIN/AEI/10.13039/501100011033. G. E. and D. O. are also supported by PIUAH21/IA-062 and CM/JIN/2021-004. G. E. is also funded by an FPU of the Universidad de Alcalá.Peer ReviewedPostprint (published version

On approximating shortest paths in weighted triangular tessellations

We study the quality of weighted shortest paths when a continuous 2-dimensional space is discretized by a weighted triangular tessellation. In order to evaluate how well the tessellation approximates the 2-dimensional space, we study three types of shortest paths: a weighted shortest path SPw(s,t) , which is a shortest path from s to t in the space; a weighted shortest vertex path SVPw(s,t) , which is a shortest path where the vertices of the path are vertices of the tessellation; and a weighted shortest grid path SGPw(s,t) , which is a shortest path whose edges are edges of the tessellation. The ratios ¿SGPw(s,t)¿¿SPw(s,t)¿ , ¿SVPw(s,t)¿¿SPw(s,t)¿ , ¿SGPw(s,t)¿¿SVPw(s,t)¿ provide estimates on the quality of the approximation. Given any arbitrary weight assignment to the faces of a triangular tessellation, we prove upper and lower bounds on the estimates that are independent of the weight assignment. Our main result is that ¿SGPw(s,t)¿¿SPw(s,t)¿=23v˜1.15 in the worst case, and this is tight.Peer ReviewedPostprint (author's final draft

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

On polynomials associated to Voronoi diagrams of point sets and crossing numbers

Three polynomials are defined for sets S of n points in general position in the plane: The Voronoi polynomial with coefficients the numbers of vertices of the order-k Voronoi diagrams of S, the circle polynomial with coefficients the numbers of circles through three points of S enclosing k points, and the E=k polynomial with coefficients the numbers of (at most k)-edges of S. We present several formulas for the rectilinear crossing number of S in terms of these polynomials and their roots. We also prove that the roots of the Voronoi polynomial lie on the unit circle if and only if S is in convex position. Further, we present bounds on the location of the roots of these polynomials.Postprint (published version

Generalized kernels of polygons under rotation

Given a set $\mathcal{O}$ of $k$ orientations in the plane, two points inside a simple polygon $P$ $\mathcal{O}$-see each other if there is an $\mathcal{O}$-staircase contained in $P$ that connects them. The $\mathcal{O}$-kernel of $P$ is the subset of points which $\mathcal{O}$-see all the other points in $P$. This work initiates the study of the computation and maintenance of the $\mathcal{O}$-${\rm Kernel}$ of a polygon $P$ as we rotate the set $\mathcal{O}$ by an angle $\theta$, denoted $\mathcal{O}$-${\rm Kernel}_{\theta}(P)$. In particular, we design efficient algorithms for (i) computing and maintaining $\{0^{o}\}$-${\rm Kernel}_{\theta}(P)$ while $\theta$ varies in $[-\frac{\pi}{2},\frac{\pi}{2})$, obtaining the angular intervals where the $\{0^{o}\}$-${\rm Kernel}_{\theta}(P)$ is not empty and (ii) for orthogonal polygons $P$, computing the orientation $\theta\in[0, \frac{\pi}{2})$ such that the area and/or the perimeter of the $\{0^{o},90^{o}\}$-${\rm Kernel}_{\theta}(P)$ are maximum or minimum. These results extend previous works by Gewali, Palios, Rawlins, Schuierer, and Wood.Comment: 12 pages, 4 figures, a version omitting some proofs appeared at the 34th European Workshop on Computational Geometry (EuroCG 2018

Measuring cocircularity in a point set

In a given set S of n points in the plane, how close are four points of S to be cocircular? We define several measures to study this question, and present bounds on this almost-cocircularity in a point set. Algorithms for cocircularity are presented as well.Peer ReviewedPostprint (published version

A922 Sequential measurement of 1 hour creatinine clearance (1-CRCL) in critically ill patients at risk of acute kidney injury (AKI)

Meeting abstrac

Tratamientos psicológicos empíricamente apoyados para adultos: Una revisión selectiva

Antecedentes: los tratamientos psicológicos han mostrado su eficacia, efectividad y eficiencia para el abordaje de los trastornos mentales; no obstante, considerando el conocimiento científico generado en los últimos años, no se dispone de trabajos de actualización en español sobre cuáles son los tratamientos psicológicos con respaldo empírico. El objetivo fue realizar una revisión selectiva de los principales tratamientos psicológicos empíricamente apoyados para el abordaje de trastornos mentales en personas adultas. Método: se recogen niveles de evidencia y grados de recomendación en función de los criterios propuestos por el Sistema Nacional de Salud de España (en las Guías de Práctica Clínica) para diferentes trastornos psicológicos. Resultados: los resultados sugieren que los tratamientos psicológicos disponen de apoyo empírico para el abordaje de un amplio elenco de trastornos psicológicos. El grado de apoyo empírico oscila de bajo a alto en función del trastorno psicológico analizado. La revisión sugiere que ciertos campos de intervención necesitan una mayor investigación. Conclusiones: a partir de esta revisión selectiva, los profesionales de la psicología podrán disponer de información rigurosa y actualizada que les permita tomar decisiones informadas a la hora de implementar aquellos procedimientos psicoterapéuticos empíricamente fundamentados en función de las características de las personas que demandan ayuda. Background: Psychological treatments have shown their efficacy, effectiveness, and efficiency in dealing with mental disorders. However, considering the scientific knowledge generated in recent years, in the Spanish context, there are no updating studies about empirically supported psychological treatments. The main goal was to carry out a selective review of the main empirically supported psychological treatments for mental disorders in adults. Method: Levels of evidence and degrees of recommendation were collected based on the criteria proposed by the Spanish National Health System (Clinical Practice Guidelines) for different psychological disorders. Results: The results indicate that psychological treatments have empirical support for the approach to a wide range of psychological disorders. These levels of empirical evidence gathered range from low to high depending on the psychological disorder analysed. The review indicates the existence of certain fields of intervention that need further investigation. Conclusions: Based on this selective review, psychology professionals will be able to have rigorous, up-to-date information that allows them to make informed decisions when implementing empirically based psychotherapeutic procedures based on the characteristics of the people who require help

Spanning ratio of shortest paths in weighted square tessellations

Continuous 2-dimensional space is often discretized by considering a grid of weighted square cells. In this work we study how well a weighted tessellation approximates the space, with respect to shortest paths. In particular, we consider a shortest path SPw(s, t), which is a shortest path from s to t in the continuous weighted 2D-space, and a shortest grid path SGPw(s, t), which is a shortest path in the tessellated weighted 2D-space. Our main result is that the ratio kSGPw(s,t)k kSPw(s,t)k is at most v 2 2+v 2 ˜ 1.08, irrespective of the weight assignment.P. B. is partially supported by NSERC. G. E., D. O. and R. I. S. are partially supported by H2020-MSCA-RISE project 734922 - CONNECT and project PID2019-104129GB-I00 funded by MCIN/AEI/10.13039/501100011033. G. E. and D. O. are also supported by PIUAH21/IA-062 and CM/JIN/2021-004. G. E. is funded by an FPU of the Universidad de Alcalá.Peer ReviewedPostprint (author's final draft