Computational geometry

Cartographic maps have been shown to provide cognitive benefits when interpreting data in relation to a geographic location. In visualization, the term map‐like describes techniques that incorporate characteristics of cartographic maps in their representation of abstract data. However, the field of map‐like visualization is vast and currently lacks a clear classification of the existing techniques. Moreover, choosing the right technique to support a particular visualization task is further complicated, as techniques are scattered across different domains, with each considering different characteristics as map‐like. In this paper, we give an overview of the literature on map‐like visualization and provide a hierarchical classification of existing techniques along two general perspectives: imitation and schematization of cartographic maps. Each perspective is further divided into four principal categories that group common map‐like techniques along the visual primitives they affect. We further discuss this classification from a task‐centered view and highlight open research questions.

The State of the Art in Map-Like Visualization

We prove two art gallery theorems in which the guards must guard one another in addition to the gallery. A set G of points (the guards) in a simple closed polygon (the art gallery) is a guarded guard set provided (i) every point in the polygon is visible to some point in G; and (ii) every point in G is visible to some other point in G. We prove that a polygon with n sides always has a guarded guard set of cardinality ⌊(3n - 1)/7⌋ and that this bound is sharp (n ≥ 5); our result corrects an erroneous formula in the literature. We also use a coloring argument to give an entirely new proof that the corresponding sharp function for orthogonal polygons is ⌊n/3⌋ for n ≥ 6; this result was originally established by induction by Hernandez-Penalver.

/pdf/art-gallery-theorems-for-guarded-guards-32gabu1p17.pdf

Art gallery theorems for guarded guards

In this paper we consider a variation of the Art Gallery Problem. A set of points G in a polygon Pn is a connected guard set for Pn provided that is a guard set and the visibility graph of the set of guards G in Pn is connected. We use a coloring argument to prove that the minimum number of connected guards which are necessary to watch any polygon with n sides is ⌊(n - 2)/2⌋. This result was originally established by induction by Hernandez-Penalver [3]. From this result it easily follows that if the art gallery is orthogonal (each interior angle is 90° or 270°), then the minimum number of connected guards is n/2 - 2.

A coloring algorithm for finding connected guards in art galleries

In this paper we study the Anchored Graph Drawing AGD problem: Given a planar graph G, an initial placement for its vertices, and a distance d, produce a planar straight-line drawing of G such that each vertex is at distance at most d from its original position.

We show that the AGD problem is NP-hard in several settings and provide a polynomial-time algorithm when d is the uniform distance L ∞ and edges are required to be drawn as horizontal or vertical segments.

/pdf/anchored-drawings-of-planar-graphs-3994bpb6mh.pdf

Anchored Drawings of Planar Graphs

The problem of finding a minimum area polygonization for a given set of points in the plane, Minimum Area Polygonization (MAP) is NP-hard. Due to the complexity of the problem we aim at the development of algorithms to obtain approximate solutions. In this work, we suggest di?erent strategies in order to minimize the polygonization area.We propose algorithms to search for approximate solutions for MAP problem. We present an experimental study for a set of instances for MAP problem.

/pdf/approaching-minimum-area-polygonization-4pkoyov10u.pdf

Approaching minimum area polygonization

Globally optimal triangulations and pseudo-triangulations are difficult to be found by deterministic methods as, for most type of criteria, no polynomial algorithm is known. In this work, we consider the Minimum Weight Triangulation (MWT) and Minimum Weight Pseudo-Triangulation (MWPT) problems of a given set of n points in the plane. This paper shows how the Ant Colony Optimization (ACO) metaheuristic can be used to find high quality triangulations and pseudo-triangulations of minimum weight. For the experimental study presented here we have created a set of instances for MWT and MWPT problems since no reference to benchmarks for these problems were found in the literature. Through the experimental evaluation, we assess the applicability of the ACO metaheuristic for MWT and MWPT problems considering greedy and Simulated Annealing algorithms.

Approximations on Minimum Weight Triangulations and Minimum Weight Pseudo-Triangulations Using Ant Colony Optimization Metaheuristic

Globally optimal triangulations are difficult to be found by deterministic methods as, for most type of criteria, no polynomial algorithm is known. In this work, we consider the Minimum Weight Triangulation (MWT) problem of a given set of n points in the plane. This paper shows how the Ant Colony Optimization (ACO) metaheuristic can be used to find high quality triangulations. For the experimental study we have created a set of instances for MWT problem since no reference to benchmarks for these problems were found in the literature. Through the experimental evaluation, we assess the applicability of the ACO metaheuristic for MWT problem.

/pdf/using-aco-metaheuristic-for-mwt-problem-468521ltgx.pdf

Using ACO Metaheuristic for MWT Problem

/pdf/evaluacion-de-metricas-en-redes-de-computadoras-okbrur94zg.pdf

Evaluación de métricas en redes de computadoras

In this paper we concentrate on an unexplored graph drawing problem: the one arising when each vertex of the graph represents a geographical region. For this problem we first define new aesthetic criteria. Secondly, several force-directed algorithms for finding layouts that satisfy those specific criteria are proposed, which successfully obtain much better drawings for this problem than the previous algorithms.

/pdf/network-drawing-with-geographical-constraints-on-vertices-3tm7pf56vb.pdf

Gregorio Hernández Peñalver

Papers

Approaching minimum area polygonization

Approximations on Minimum Weight Triangulations and Minimum Weight Pseudo-Triangulations Using Ant Colony Optimization Metaheuristic

Using ACO Metaheuristic for MWT Problem

Evaluación de métricas en redes de computadoras

Network drawing with geographical constraints on vertices