DSpace Collection:
http://hdl.handle.net/2117/3197
2015-04-25T06:43:17ZNote on the number of obtuse angles in point sets
http://hdl.handle.net/2117/27270
Title: Note on the number of obtuse angles in point sets
Authors: Fabila-Monroy, Ruy; Huemer, Clemens; Tramuns, Eulàlia
Abstract: In $1979$ Conway, Croft, Erd\H{o}s and Guy proved that every set $S$ of $n$ points in general position in the plane determines at least $\frac{n^3}{18}-O(n^2)$ obtuse angles and also presented a special set of $n$ points to show the upper bound $\frac{2n^3}{27}-O(n^2)$ on the minimum number of obtuse angles among all sets $S$.
We prove that every set $S$ of $n$ points in convex position determines at least $\frac{2n^3}{27}-o(n^3)$ obtuse angles, hence matching the upper bound (up to sub-cubic terms) in this case.
Also on the other side, for point sets with low rectilinear crossing number, the lower bound on the minimum number of obtuse angles is improved.2015-04-13T10:29:20ZCompatible spanning trees
http://hdl.handle.net/2117/26968
Title: Compatible spanning trees
Authors: Garcia Olaverri, Alfredo Martin; Huemer, Clemens; Hurtado Díaz, Fernando Alfredo; Tejel Altarriba, Francisco Javier
Abstract: Two plane geometric graphs are said to be compatible when their union is a plane geometric graph. Let S be a set of n points in the Euclidean plane in general position and let T be any given plane geometric spanning tree of S. In this work, we study the problem of finding a second plane geometric tree T' spanning S, such that is compatible with T and shares the minimum number of edges with T. We prove that there is always a compatible plane geometric tree T' having at most #n - 3#/4 edges in common with T, and that for some plane geometric trees T, any plane tree T' spanning S, compatible with T, has at least #n - 2#/5 edges in common with T. #C# 2013 Elsevier B.V. All rights reserved.2015-03-24T10:58:00ZEmpty monochromatic simplices
http://hdl.handle.net/2117/26662
Title: Empty monochromatic simplices
Authors: Aichholzer, Oswin; Fabila Monroy, Ruy; Hackl, Thomas; Huemer, Clemens; Urrutia Galicia, Jorge
Abstract: Let S be a k-colored (finite) set of n points in , da parts per thousand yen3, in general position, that is, no (d+1) points of S lie in a common (d-1)-dimensional hyperplane. We count the number of empty monochromatic d-simplices determined by S, that is, simplices which have only points from one color class of S as vertices and no points of S in their interior. For 3a parts per thousand currency signka parts per thousand currency signd we provide a lower bound of and strengthen this to Omega(n (d-2/3)) for k=2.; On the way we provide various results on triangulations of point sets in . In particular, for any constant dimension da parts per thousand yen3, we prove that every set of n points (n sufficiently large), in general position in , admits a triangulation with at least dn+Omega(logn) simplices.2015-03-11T11:14:31ZLower bounds for the number of small convex k-holes
http://hdl.handle.net/2117/26660
Title: Lower bounds for the number of small convex k-holes
Authors: Aichholzer, Oswin; Fabila Monroy, Ruy; Hackl, Thomas; Huemer, Clemens; Pilz, Alexander; Vogtenhuber, Birgit
Abstract: Let S be a set of n points in the plane in general position, that is, no three points of S are on a line. We consider an Erdos-type question on the least number h(k)(n) of convex k-holes in S, and give improved lower bounds on h(k)(n), for 3 <= k <= 5. Specifically, we show that h(3)(n) >= n(2) - 32n/7 + 22/7, h(4)(n) >= n(2)/2 - 9n/4 - o(n), and h(5)(n) >= 3n/4 - o(n). We further settle several questions on sets of 12 points posed by Dehnhardt in 1987. (C) 2013 Elsevier B.V. All rights reserved.2015-03-11T10:56:59Z4-Holes in point sets
http://hdl.handle.net/2117/26659
Title: 4-Holes in point sets
Authors: Aichholzer, Oswin; Fabila Monroy, Ruy; Gonzalez Aguilar, Hernan; Hackl, Thomas; Heredia, Marco A.; Huemer, Clemens; Urrutia Galicia, Jorge; Vogtenhuber, Birgit
Abstract: We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n >= 9, the maximum number of general 4-holes is ((pi)(4)); the minimum number of general 4-holes is at least 5/2 n(2) - circle minus(n); and the maximum number of non-convex 4-holes is at least 1/2 n(3) - circle minus(n(2) logn) and at most 1/2 n(3) - circle minus(n(2)). 2014 (c) Elsevier B.V. All rights reserved.2015-03-11T10:45:04ZThe number of empty four-gons in random point sets
http://hdl.handle.net/2117/26500
Title: The number of empty four-gons in random point sets
Authors: Fabila-Monroy, Ruy; Huemer, Clemens; Mitsche, Dieter
Abstract: Let S be a set of n points distributed uniformly and independently in the unit square. Then the expected number of empty four-gons with vertices from S is T(n^2 log¿ n). A four-gon is empty if it contains no points of S in its interior.2015-02-25T11:16:10ZWitness rectangle graphs
http://hdl.handle.net/2117/26485
Title: Witness rectangle graphs
Authors: Aronov, Boris; Dulieu, Muriel; Hurtado Díaz, Fernando Alfredo
Abstract: In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open isothetic rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a connected co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. In addition, we prove that a WRG with no isolated vertices has domination number at most four. Moreover, we show that any combinatorial graph can be drawn as a WRG using a combination of positive and negative witnesses. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.2015-02-24T12:19:53ZThe degree/diameter problem in maximal planar bipartite graphs
http://hdl.handle.net/2117/26448
Title: The degree/diameter problem in maximal planar bipartite graphs
Authors: Dalfó Simó, Cristina; Huemer, Clemens; Salas Piñon, Julián
Abstract: The (¿;D) (degree/diameter) problem consists of nding the largest possible number of vertices n among all the graphs with maximum degree ¿ and diameter D. We consider the (¿;D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the (¿; 2) problem, the number of vertices is n = ¿+2; and for the (¿; 3) problem, n = 3¿¿1 if ¿ is odd and n = 3¿ ¿ 2 if ¿ is even. Then, we study the general case (¿;D)
and obtain that an upper bound on n is approximately 3(2D + 1)(¿ ¿ 2)¿D=2¿ and another one is C(¿ ¿ 2)¿D=2¿ if ¿ D and C is a sufficiently large constant. Our upper bound improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately (¿ ¿ 2)k if D = 2k, and 3(¿ ¿ 3)k if D = 2k + 1, for ¿ and D sufficiently large in both cases.2015-02-20T12:42:33ZBalanced partitions of 3-colored geometric sets in the plane
http://hdl.handle.net/2117/26089
Title: Balanced partitions of 3-colored geometric sets in the plane
Authors: Bereg, Sergey; Hurtado, Ferran; Kano, Mikio; Korman, Matias; Lara, Dolores; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio; Urrutia Galicia, Jorge; Verbeek, Kevin
Abstract: Let SS be a finite set of geometric objects partitioned into classes or colors . A subset S'¿SS'¿S is said to be balanced if S'S' contains the same amount of elements of SS from each of the colors. We study several problems on partitioning 33-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m2m lines of each color, there is a segment intercepting mm lines of each color. (b) Given nn red points, nn blue points and nn green points on any closed Jordan curve ¿¿, we show that for every integer kk with 0=k=n0=k=n there is a pair of disjoint intervals on ¿¿ whose union contains exactly kk points of each color. (c) Given a set SS of nn red points, nn blue points and nn green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of SS.2015-01-27T09:56:02ZNecklaces, convolutions, and X plus Y
http://hdl.handle.net/2117/24887
Title: Necklaces, convolutions, and X plus Y
Authors: Bremner, David; Chan, Timothy M.; Demaine, Erik D.; Erickson, Jeff; Hurtado Díaz, Fernando Alfredo; Iacono, John; Langerman, Stefan; Patrascu, Mihai; Taslakian, Perouz
Abstract: We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the ℓ p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p=1, p=2, and p=∞. For p=2, we reduce the problem to standard convolution, while for p=∞ and p=1, we reduce the problem to (min,+) convolution and (median,+) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n 2) time, whereas the obvious algorithms for these problems run in Θ(n 2) time2014-11-30T18:52:19ZLD-graphs and global location-domination in bipartite graphs
http://hdl.handle.net/2117/24527
Title: LD-graphs and global location-domination in bipartite graphs
Authors: Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
Abstract: A dominating set S of a graph G is a locating-dominating-set, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S . Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number, ¿(G)¿(G). An LD-set S of a graph G is global if it is an LD-set for both G and its complement, View the MathML sourceG¯. One of the main contributions of this work is the definition of the LD-graph, an edge-labeled graph associated to an LD-set, that will be very helpful to deduce some properties of location-domination in graphs. Concretely, we use LD-graphs to study the relation between the location-domination number in a bipartite graph and its complement.2014-10-31T12:08:44ZThe graph distance game and some graph operations
http://hdl.handle.net/2117/24526
Title: The graph distance game and some graph operations
Authors: Cáceres, Jose; Puertas, M. Luz; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
Abstract: In the graph distance game, two players alternate in constructing a maximal path. The objective function is the distance between the two endpoints of the path, which one player tries to maximize and the other tries to minimize. In this paper we examine the distance game for various graph operations: the join, the corona and the lexicographic product of graphs. We provide general bounds and exact results for special graphs2014-10-31T12:00:32ZComputing a visibility polygon using few variables
http://hdl.handle.net/2117/24062
Title: Computing a visibility polygon using few variables
Authors: Barba, Luis; Korman Cozzetti, Matías; Langerman, Stefan; Silveira, Rodrigo Ignacio
Abstract: We present several algorithms for computing the visibility polygon of a simple polygon P of n vertices (out of which r are reflex) from a viewpoint inside P, when P resides in read-only memory and only few working variables can be used. The first algorithm uses a constant number of variables, and outputs the vertices of the visibility polygon in O (n (r) over bar) time, where (r) over bar denotes the number of reflex vertices of P that are part of the output. Whenever we are allowed to use O(s) variables, the running time decreases to O (nr/2(s) + n log(2) r) (or O (nr/2(s) + n log r) randomized expected time), where s is an element of O (log r). This is the first algorithm in which an exponential space-time trade-off for a geometric problem is obtained. (C) 2014 Elsevier B.V. All rights reserved.2014-09-15T19:28:08ZComputing correlation between piecewise-linear functions
http://hdl.handle.net/2117/23696
Title: Computing correlation between piecewise-linear functions
Authors: Agarwal, Pankaj; Aronov, Boris; Van Kreveld, Matias; Löffler, Maarten; Silveira, Rodrigo Ignacio
Abstract: We study the problem of computing correlation between two piecewise-linear bivariate functions defined over a common domain, where the surfaces they define in three dimensions---polyhedral terrains---can be transformed vertically by a linear transformation of the third coordinate (scaling and translation). We present a randomized algorithm that minimizes the maximum vertical distance between the graphs of the two functions, over all linear transformations of one of the terrains, in $O(n^{4/3}\operatorname{polylog}n)$ expected time, where $n$ is the total number of vertices in the graphs of the two functions. We also present approximation algorithms for minimizing the mean distance between the graphs of univariate and bivariate functions. For univariate functions we present a $(1+\varepsilon)$-approximation algorithm that runs in $O(n (1 + \log^2 (1/\varepsilon)))$ expected time for any fixed $\varepsilon >0$. The $(1+\varepsilon)$-approximation algorithm for bivariate functions runs in $O(n/\varepsilon)$ time, for any fixed $\varepsilon >0$, provided the two functions are defined over the same triangulation of their domain.2014-08-29T10:51:05ZGeneralizing the Steiner-Lehmus theorem using the Gröbner cover
http://hdl.handle.net/2117/23574
Title: Generalizing the Steiner-Lehmus theorem using the Gröbner cover
Authors: Montes Lozano, Antonio; Recio Muñiz, Tomás
Abstract: In this note we present an application of a new tool (the Gröbner cover method, to discuss parametric polynomial systems of equations) in the realm of automatic discovery of theorems in elementary geometry. Namely, we describe, through a relevant example, how the Gröbner cover algorithm is particularly well suited to obtain the missing hypotheses for a given geometric statement to hold true. We deal with the following problem: to describe the triangles that have at least two bisectors of equal length. The case of two inner bisectors is the well known, XIX century old, Steiner-Lehmus theorem, but the general case of inner and outer bisectors has been only recently addressed. We show how the Gröbner cover method automatically provides, while yielding more insight than through any other method, the conditions for a triangle to have two equal bisectors of whatever kind.2014-07-21T08:52:22Z