Determining an Optimal Visualization of Physically Realizable Symbol Maps

Proportional symbol maps are an often used tool to aid cartographers and geo-science professionals to visualize data associated with events (e.g., earthquakes) or geo-positioned statistical data (e.g., population). At specific locations, symbols are placed and scaled so that their areas become proportional to the magnitudes of the events or data. Recent work approaches the problem of drawing these symbols algorithmically and defines metrics to be optimized to attain different kinds of drawings. We focus specifically on optimizing the visualization of physically realizable drawings of opaque disks by maximizing the sum of the visible borders of such disks. As this problem has been proven to be NP-hard, we provide an integer programming model for its solution along with decomposition techniques designed to decrease the size of input instances. We present computational experiments to assess the performance of our model as well as the effectiveness of our decomposition techniques

Determining An Optimal Visualization Of Physically Realizable Symbol Maps

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Proportional symbol maps are an often used tool to aid cartographers and geo-science professionals to visualize data associated with events (e.g., earthquakes) or geo-positioned statistical data (e.g., population). At specific locations, symbols are placed and scaled so that their areas become proportional to the magnitudes of the events or data. Recent work approaches the problem of drawing these symbols algorithmically and defines metrics to be optimized to attain different kinds of drawings. We focus specifically on optimizing the visualization of physically realizable drawings of opaque disks by maximizing the sum of the visible borders of such disks. As this problem has been proven to be NP-hard, we provide an integer programming model for its solution along with decomposition techniques designed to decrease the size of input instances. We present computational experiments to assess the performance of our model as well as the effectiveness of our decomposition techniques. © 2011 IEEE.117124Cons. Nac. Desenvolv. Cient. Tecnol. (CNPq),Coordenacao Aperfeicoamento Pessoal Nivel Superior (CAPES),Fundacao de Amparo a Pesquisa do Estado de Alagoas (FAPEAL),Secr. Estado Cienc., Tecnol. Inovacao (SECTI-AL)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Slocum, T.A., McMaster, R.B., Kessler, F.C., Howard, H.H., (2003) Thematic Cartography and Geographic Visualization, , 2nd ed. Prentice HallCabello, S., Haverkort, H., Van Kreveld, M., Speckmann, B., Algorithmic aspects of proportional symbol maps (2010) Algorithmica, 58 (3), pp. 543-565Kunigami, G., De Rezende, P.J., De Souza, C.C., Yunes, T., Optimizing the layout of proportional symbol maps (2011) Proceedings of ICCSA 2011, 6784, pp. 1-16. , ser. LNCS, B. Murgante, O. Gervasi, A. Iglesias, D. Taniar, and B. Apduhan, Eds. Springer-VerlagKunigami, G., De Rezende, P.J., De Souza, C.C., Yunes, T., (2011) Optimizing the Layout of Proportional Symbol Maps: Polyhedra and Computation, , unpublishedWolsey, L.A., (1998) Integer Programming, , John Wiley and Sons, IncComputational Geometry Algorithms Library, , www.cgal.org(2009) Xpress Optimizer Reference Manual, , Fair Isaac Corporatio

Optimizing the Layout of Proportional Symbol Maps: Polyhedra and Computation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Proportional symbol maps are a cartographic tool to assist in the visualization and analysis of quantitative data associated with specific locations, such as earthquake magnitudes, oil well production, and temperature at weather stations. As the name suggests, symbol sizes are proportional to the magnitude of the physical quantities that they represent. We present two novel integer linear programming (ILP) models to solve this computational geometry problem: how to draw opaque disks on a map so as to maximize the total visible border of all disks. We focus on drawings obtained by layering symbols on top of each other, also known as stacking drawings. We introduce decomposition techniques as well as several families of facet-defining inequalities, which are used to strengthen the ILP models that are supplied to a commercial solver. We demonstrate the effectiveness of our approach through a series of computational experiments using hundreds of instances generated from real demographic and geophysical data sets. To the best of our knowledge, we are the first to use ILP to tackle this problem, and the first to provide provably optimal symbol maps for those data sets.262199207Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)FAEPEX/UNICAMPConselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)CNPq [830510/1999-0, 301732/2007-8, 472504/2007-0, 483177/2009-1, 473867/2010-9, 477692/2012-5]FAPESP [07/52015-0

A hybrid GRASP heuristic to construct effective drawings of proportional symbol maps

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Proportional symbol map is a cartographic tool that employs symbols to represent data associated with specific locations. Each symbol is drawn at the location of an event and its size is proportional to the numerical data collected at that point on the map. The symbols considered here are opaque disks. When two or more disks overlap, part of their boundaries may not be visible and it might be difficult to gauge their size. Therefore, the order in which the disks are drawn affects the visual quality of a map. In this work, we focus on stacking drawings, i.e., a drawing that corresponds to the disks being stacked up, in sequence, starting from the one at the bottom of the stack. We address the Max-Total problem, which consists in maximizing the total visible boundary of all disks. We propose a sophisticated heuristic based on GRASP that includes most of the advanced techniques described in the literature for this procedure. We tested both sequential and parallel implementations on benchmark instances and the comparison against optimal solutions confirms the high quality of our heuristic. To the best of our knowledge, this is the first time a metaheuristic is applied to this problem. (C) 2012 Elsevier Ltd. All rights reserved.40514351447Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAEPEX/UNICAMPFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [2009/17044-5, 2007/52015-0]CNPq [830510/1999-0, 301732/2007-8, 472504/2007-0, 473867/2010-9, 483177/2009-1