Mesures de regularitat per a polígons convexos

Al llarg d'aquesta memòria, hem plantejat possibles mesures de regularitat, totes elles justificades, per a un n-gon convex qualsevol que han donat lloc a problemes de geometria discreta i computacional. A més, hem estat capaços d'oferir algorismes per al seu càlcul de complexitat baixa (i, en alguns casos, òptima) i, per tant, realitzables. Els mètodes proposats són, en algunes ocasions, l'aplicació de resultats més generals, en d'altres, algorismes ad hoc, i, en d'altres, un estudi acurat permet transformar el problema que s'ha de resoldre en un altre problema d'optimització geomètrica que té una solució eficient coneguda

Mesures de regularitat per a polígons convexos

Al llarg d'aquesta memòria, hem plantejat possibles mesures de regularitat, totes elles justificades, per a un n-gon convex qualsevol que han donat lloc a problemes de geometria discreta i computacional. A més, hem estat capaços d'oferir algorismes per al seu càlcul de complexitat baixa (i, en alguns casos, òptima) i, per tant, realitzables. Els mètodes proposats són, en algunes ocasions, l'aplicació de resultats més generals, en d'altres, algorismes ad hoc, i, en d'altres, un estudi acurat permet transformar el problema que s'ha de resoldre en un altre problema d'optimització geomètrica que té una solució eficient coneguda

Terrain prickliness: theoretical grounds for high complexity viewsheds

An important task when working with terrain models is computing viewsheds: the parts of the terrain visible from a given viewpoint. When the terrain is modeled as a polyhedral terrain, the viewshed is composed of the union of all the triangle parts that are visible from the viewpoint. The complexity of a viewshed can vary significantly, from constant to quadratic in the number of terrain vertices, depending on the terrain topography and the viewpoint position. In this work we study a new topographic attribute, the prickliness, that measures the number of local maxima in a terrain from all possible perspectives. We show that the prickliness effectively captures the potential of 2.5D terrains to have high complexity viewsheds, and we present near-optimal algorithms to compute the prickliness of 1.5D and 2.5D terrains. We also report on some experiments relating the prickliness of real word 2.5D terrains to the size of the terrains and to their viewshed complexity.Peer ReviewedPostprint (author's final draft

Improving shortest paths in the Delaunay triangulation

We study a problem about shortest paths in Delaunay triangulations. Given two nodes s; t in the Delaunay triangulation of a point set P, we look for a new point p that can be added, such that the shortest path from s to t in the Delaunay triangulation of P u{p} improves as much as possible. We study 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

Mesures de regularitat per a polígons convexos

Al llarg d'aquesta memòria, hem plantejat possibles mesures de regularitat, totes elles justificades, per a un n-gon convex qualsevol que han donat lloc a problemes de geometria discreta i computacional. A més, hem estat capaços d'oferir algorismes per al seu càlcul de complexitat baixa (i, en alguns casos, òptima) i, per tant, realitzables. Els mètodes proposats són, en algunes ocasions, l'aplicació de resultats més generals, en d'altres, algorismes ad hoc, i, en d'altres, un estudi acurat permet transformar el problema que s'ha de resoldre en un altre problema d'optimització geomètrica que té una solució eficient coneguda

On the number of higher order Delaunay triangulations

Higher order Delaunay triangulations are a generalization of the Delaunay triangulation which provides a class of well-shaped triangulations, over which extra criteria can be optimized. A triangulation is order-k Delaunay if the circumcircle of each triangle of the triangulation contains at most k points. In this paper we study lower and upper bounds on the number of higher order Delaunay triangulations, as well as their expected number for randomly distributed points. We show that arbitrarily large point sets can have a single higher order Delaunay triangulation, even for large orders, whereas for first order Delaunay triangulations, the maximum number is 2n−3. Next we show that uniformly distributed points have an expected number of at least 2ρ1n(1+o(1)) first order Delaunay triangulations, where ρ1 is an analytically defined constant (ρ1 ≈ 0.525785), and for k > 1, the expected number of order-k Delaunay triangulations (which are not order-i for any i < k) is at least 2ρkn(1+o(1)), where ρk can be calculated numerically.Peer Reviewe

Hamiltonicity for convex shape Delaunay and Gabriel graphs

© 2020. 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 Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape $\mathcal {C}$ . Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k- ${DG}_{\mathcal {C}}(S)$ , has vertex set S and edge pq provided that there exists some homothet of $\mathcal {C}$ with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph k- ${GG}_{\mathcal {C}}(S)$ is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of $\mathcal {C}$ with p and q on its boundary. We provide upper bounds on the minimum value of k for which k- ${GG}_{\mathcal {C}}(S)$ is Hamiltonian. Since k- ${GG}_{\mathcal {C}}(S)$ $\subseteq$ k- ${DG}_{\mathcal {C}}(S)$ , all results carry over to k- ${DG}_{\mathcal {C}}(S)$ . In particular, we give upper bounds of 24 for every $\mathcal {C}$ and 15 for every point-symmetric $\mathcal {C}$ . We also improve the bound to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for $t \ge 10)$ . These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs.P.B. was partially supported by NSERC. P.C. was supported by CONACyT. M.S. was supported by the Czech Science Foundation, grant number GJ19-06792Y, and by institutional support RVO:67985807. R.S. was supported by MINECO through the Ram´on y Cajal program. P.C. and R.S. were also supported by projects MINECO MTM2015-63791-R and Gen. Cat. 2017SGR1640. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922.Peer ReviewedPostprint (author's final draft

Hamiltonicity for convex shape Delaunay and Gabriel graphs

We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-DGC(S), has vertex set S and edge pq provided that there exists some homothet of C with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph k-GGC(S) is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of C with p and q on its boundary. We provide upper bounds on the minimum value of k for which k-GGC(S) is Hamiltonian. Since k-GGC(S) ¿ k-DGC(S), all results carry over to k-DGC(S). In particular, we give upper bounds of 24 for every C and 15 for every point-symmetric C. We also improve the bound to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for t = 10).Peer ReviewedPostprint (published version

The dual diameter of triangulations

Let P be a simple polygon with n vertices. The dual graph T⁎ of a triangulation T of P is the graph whose vertices correspond to the bounded faces of T and whose edges connect those faces of T that share an edge. We consider triangulations of P that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in O(n3log⁡n) time using dynamic programming. If P is convex, we show that any minimizing triangulation has dual diameter exactly 2⋅⌈log2⁡(n/3)⌉ or 2⋅⌈log2⁡(n/3)⌉−1, depending on n. Trivially, in this case any maximizing triangulation has dual diameter n−2. Furthermore, we investigate the relationship between the dual diameter and the number of ears (triangles with exactly two edges incident to the boundary of P) in a triangulation. For convex P, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of n points there are triangulations with dual diameter in O(log⁡n) and in Ω(n).SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Some properties of k-Delaunay and k-Gabriel graphs

We consider two classes of higher order proximity graphs defined on a set of points in the plane, namely, the k-Delaunay graph and the k-Gabriel graph. We give bounds on the following combinatorial and geometric properties of these graphs: spanning ratio, diameter, connectivity, chromatic number, and minimum number of layers necessary to partition the edges of the graphs so that no two edges of the same layer cross.SCOPUS: cp.jinfo:eu-repo/semantics/publishedSpecial issue of selected papers from the 22nd Canadian Conference on Computational Geometry (CCCG'10), Winnipeg, Aug. 9-11, 201