DSpace Collection:
http://hdl.handle.net/2117/3197
2014-12-18T14:28:32ZNecklaces, 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.2014-07-21T08:52:22ZSingularities of non-redundant manipulators: a short account and a method for their computation in the planar case
http://hdl.handle.net/2117/22508
Title: Singularities of non-redundant manipulators: a short account and a method for their computation in the planar case
Authors: Bohigas Nadal, Oriol; Manubens Ferriol, Montserrat; Ros Giralt, Lluís
Abstract: The study of the singularity set is of utmost utility in understanding the local and global behavior of a manipulator. After reviewing the mathematical conditions that characterize this set, and their kinematic and geometric interpretation, this paper shows how these conditions can be formulated in an amenable manner in planar manipulators, allowing the definition of a conceptually-simple method for isolating the set exhaustively, even in higher-dimensional cases. As a result, the method delivers a collection of boxes bounding the location of all points of the set, whose accuracy can be adjusted through a threshold parameter. Such boxes can then be projected to the input or output coordinate spaces, obtaining informative diagrams, or portraits, on the global motion capabilities of the manipulator. Examples are included that show the application of the method to simple manipulators, and to a complex mechanism that would be difficult to analyze using common-practice procedures.2014-04-03T17:06:36ZNordhaus-Gaddum bounds for locating domination
http://hdl.handle.net/2117/21023
Title: Nordhaus-Gaddum bounds for locating domination
Authors: Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
Abstract: A dominating set S of graph G is called metric-locating–dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S. If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S, then it is said to be locating–dominating. Locating, metric-locating–dominating and locating–dominating sets of minimum cardinality are called β-codes, η-codes and λ-codes, respectively. A Nordhaus–Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its complement View the MathML source. In this paper, we present some Nordhaus–Gaddum bounds for the location number β, the metric-location–domination number η and the location–domination number λ. Moreover, in each case, the graph family attaining the corresponding bound is fully characterized.2013-12-17T11:50:01ZSome structural, metric and convex properties of the boundary of a graph
http://hdl.handle.net/2117/20898
Title: Some structural, metric and convex properties of the boundary of a graph
Authors: Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Seara Ojea, Carlos
Abstract: Let
u;v
be two vertices of a connected graph
G
. The vertex
v
is
said to be a
boundary vertex
of
u
if no neighbor of
v
is further away
from
u
than
v
. The boundary of a graph is the set of all its boundary
vertices. In this work, we present a number of properties of the
boundary of a graph under diÆerent points of view: (1) a realization
theorem involving diÆerent types of boundary vertex sets: extreme
set, periphery, contour, and the whole boundary; (2) the contour is a
monophonic set; and (3) the cardinality of the boundary is an upper
bound for both the metric dimension and the determining number
of a graph2013-12-03T12:14:54ZLocating-dominating codes: Bounds and extremal cardinalities
http://hdl.handle.net/2117/20810
Title: Locating-dominating codes: Bounds and extremal cardinalities
Authors: Cáceres, Jose; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. Luz
Abstract: In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our contributions on λ-codes and η-codes concerning bounds, extremal values and realization theorems.2013-11-27T11:55:50ZFlow computations on imprecise terrains
http://hdl.handle.net/2117/20799
Title: Flow computations on imprecise terrains
Authors: Driemel, Anne; Haverkort, Herman; Löffler, Maarten; Silveira, Rodrigo Ignacio
Abstract: We study water flow computation on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water
flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along
the edges of a prede ned graph, for example a grid or a triangulation. In both cases each
vertex has an imprecise elevation, given by an interval of possible values, while its (x; y)-coordinates are fi xed. For the fi rst model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, or a set of vertices, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time.2013-11-27T09:28:28ZPlanning singularity-free paths on closed-chain manipulators
http://hdl.handle.net/2117/20248
Title: Planning singularity-free paths on closed-chain manipulators
Authors: Bohigas Nadal, Oriol; Henderson, Michael E.; Ros Giralt, Lluís; Manubens Ferriol, Montserrat; Porta Pleite, Josep Maria
Abstract: This paper provides an algorithm for computing singularity-free paths on closed-chain manipulators. Given two
non-singular configurations of the manipulator, the method attempts to connect them through a path that maintains a
minimum clearance with respect to the singularity locus at all points, which guarantees the controllability of the manipulator everywhere along the path. The method can be applied to non-redundant manipulators of general architecture, and it is resolution-complete. It always returns a path whenever one exists at a given resolution, or determines path non-existence
otherwise. The strategy relies on defining a smooth manifold that maintains a one-to-one correspondence with the singularity-free
C-space of the manipulator, and on using a higher-dimensional continuation technique to explore this manifold systematically from one configuration, until the second configuration is found.
If desired, the method can also be used to compute an exhaustive atlas of the whole singularity-free component reachable from a
given configuration, which is useful to rapidly resolve subsequent planning queries within such component, or to visualize the
singularity-free workspace of any of the manipulator coordinates.
Examples are included that demonstrate the performance of the method on illustrative situations.2013-10-01T13:29:52ZConnecting Red Cells in a Bicolour Voronoi Diagram
http://hdl.handle.net/2117/18614
Title: Connecting Red Cells in a Bicolour Voronoi Diagram
Authors: Abellanas, Manuel; Bajuelos, Antonio L.; Canales, Santiago; Claverol Aguas, Mercè; Hernández, Gregorio; Pereira de Matos, Inés
Abstract: Let S be a set of n + m sites, of which n are red and have weight wR, and m are blue and weigh wB. The objective of this paper
is to calculate the minimum value of the red sites’ weight such that the union of the red Voronoi cells in the weighted Voronoi diagram of S is a connected region. This problem is solved for the multiplicativelyweighted
Voronoi diagram in O((n+m)2 log(nm)) time and for both the additively-weighted and power Voronoi diagram in O(nmlog(nm)) time2013-04-04T13:14:09ZMedian trajectories
http://hdl.handle.net/2117/17768
Title: Median trajectories
Authors: Buchin, Kevin; Buchin, Maike; Kreveld, Marc van; Löffler, Maarten; Silveira, Rodrigo Ignacio; Wenk, Carola; Wiratma, Lionov
Abstract: We investigate the concept of a median among a set of trajectories. We establish criteria that a “median trajectory” should meet, and present two different methods to construct a median for a set of input trajectories. The first method is very simple, while the second method is more complicated and uses homotopy with respect to sufficiently large faces in the arrangement formed by the trajectories. We give algorithms for both methods, analyze the worst-case running time, and show that under certain assumptions both methods can be implemented efficiently. We empirically compare the output of both methods on randomly generated trajectories, and evaluate whether the two methods yield medians that are according to our intuition. Our results suggest that the second method, using homotopy, performs considerably better.2013-02-14T14:40:06ZOn the metric dimension of infinite graphs
http://hdl.handle.net/2117/17651
Title: On the metric dimension of infinite graphs
Authors: Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz
Abstract: A set of vertices Sresolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph G is the minimum cardinality of a resolving set. In this paper we study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs.
Description: Infinite graph;
Locally finite graph;
Resolving set;
Metric dimension;
Cartesian product2013-02-12T12:08:09Z