COMPUTATIONAL GEOMETRY - ALGORITHMS

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Fig. 14. FIFA World Cup 2010.

6 A PPLICATION

We created a proof-of-concept implementation of the majority of ouralgorithms in Java. Our program supports centroid and wedge inter-vals, computes the optimal symbol sizes using the dynamic program-ming algorithm, and uses the force-based approach to optimize thesymbol placements. It also implements the extensions described inSection 5. All �gures in this paper were created by our program.

Our implementation is quite fast: all maps in this paper were com-puted in under a tenth of a second. Our implementation works forthicknessK ≤ 15. Maps with K = 15 take several minutes to com-pute;K ≤ 6 for all maps in this paper. We have tested our implemen-tation with up to 100 symbols on a single necklace, but we expect it towork for many more symbols, especially when using multiple neck-laces. The thickness of geographic maps rarely exceeds 10 and if thisis the case, then it is generally advisable to use several necklaces.

Our data stems from various sources: the European Com-mission Eurostat Program (ec.europa.eu/ eurostat ),Worldmapper (www.worldmapper.org ), the Swiss Federal Of�ceof Energy (www.bfe.admin.ch ), and Statistics Netherlands(www.cbs.nl ). We use the ISO 3166-1 alpha-2 and alpha-3 two-and three-letter country codes to label our maps.

7 D ISCUSSION

The major advantage of necklace maps is their clear and unclutteredappearance. The linear ordering of the symbols along the necklacesmakes it easy to estimate and compare symbol sizes correctly. Neck-lace maps visualize data sets well which are not proportional to regionsizes and which do not contain data for all regions of the input map.

Our algorithm computes necklace maps of high quality: the relativespatial position of symbols and regions captures the spatial relationbetween regions and necklaces well and enables users to quickly asso-ciate a symbol with the correct region. Our symbols are disjoint andappear along the necklaces in an order which mirrors the neighbor re-lations of their regions. Furthermore, our optimization approach yieldsnecklace maps whose symbols are as large as possible given the input.

There is a clear trade-o�between symbols sizes and the spatial re-lation of symbols and regions which is also in�uenced by the num-ber and distribution of symbols per necklace. Generally speaking it isbetter to use several necklaces instead o� ust one necklace. We aremapping two-dimensional data onto a one-dimensional domain. In-

tuitively, several necklaces increase the chances that a symbol can beplaced close to its region and scaled to an appropriate size. In theextreme, many necklaces will bring us back to symbol maps. How-ever, necklace maps with few necklaces still preserve their structuredappearance while allowing for good symbol placement and sizing.

Clearly the shape and the exact location of the necklaces is veryimportant for a good necklace map. Currently we create our necklacesby hand and add them to the input map. In the future we hope to beable to automatically generate suitable necklaces for given input mapsand data sets, however, this seems to be a quite challenging algorith-mic problem on its own. Also, our necklace maps currently lack alegend as it is commonly seen with symbols maps. Hence users can atthe moment only judge relative sizes of symbols but cannot read ab-solute values o�the maps. Finally, it would be interesting to exploreanimated necklace maps for time-varying data.

ACKNOWLEDGMENTS

The authors wish to thank Marc van Kreveld and Jack van Wijk forhelpful discussions on the topic of this paper, and Wouter Meulemans,Frank Razenberg, Andre van Renssen, and Jeaphianne van Rijn forhelp with the creation of the input maps. Bettina Speckmann andKevin Verbeek are supported by the Netherlands Organisation for Sci-enti�c Research (NWO) under project no. 639.022.707.

R EFERENCES

[1] M. A. Bekos, M. Kaufmann, A. Symvonis, and A. Wol�. Boundary la-beling: Models and ef�cient algorithms for rectangular maps.Computa-tional Geometry: Theory and Applications, 36(3):215–236, 2007.

[2] S. Cabello, H. Haverkort, M. van Kreveld, and B. Speckmann. Algorith-mic aspects of proportional symbol maps.Algorithmica, 2010. to appear.

[3] B. Dent. Cartography - thematic map design. McGraw-Hill, 5th edition,1999.

[4] D. Dorling. Area Cartograms: their Use and Creation, volume 59 ofConcepts and Techniques in Modern Geography. University of East An-glia, Environmental Publications, Norwich, 1996.

[5] J. A. Dougenik, N. R. Chrisman, and D. R. Niemeyer. An algorithm toconstruct continous area cartograms.Professional Geographer, 3:75–81,1985.

[6] H. Edelsbrunner and E. Waupotitsch. A combinatorial approach to car-tograms. Computational Geometry: Theory and Applications, 7:343–360, 1997.

Cartogram showing Europeanpopulation as area

Bettina Speckmann is an associate professor at the department of math-ematics and computer science of the Eindhoven University of Technology (the Netherlands). She received her diploma degree in mathematics from WWU Münster (Germany) in 1996 and her PhD in computer science from the University of British Columbia (Canada) in 2001. She spent two years as a postdoc at the Institute for Theoretical Computer Science of ETH Zurich (Switzerland) and became an assistant professor at TU Eindhoven in 2003. Since March 2010 she is a member of The Young Academy of the Royal Netherlands Academy of Arts and Sciences (KNAW), and since March 2011 a member of the Global Young Academy. Bettina’s research interests include the design and analysis of algorithms and data structures, discrete and computational geometry, applications of computational geometry to geo- graphic information systems, automated cartography, and graph drawing.

by step sequence of instructions that performs a well defined task. It is natural enough to be understandable by humans, but precise and formal enough to be converted easily into a computer program.

Computational geometers focus on algorithms for geometry problems that come from many areas: robotics, databases, computer graphics, molecular biology and geographic information systems. Bettina Speckmann’s research group produced algorithms to generate the maps below.

Computers generate more numbers about what is going on in our world than we can take in. Fortunately, they can also help us present numbers in ways that we can interpret visually -- as maps and diagrams. Computers, however, have no spatial insight; they need computer scientists to develop geometric algorithms that can collect the numbers and create meaningful, interpretable results.

Computer scientists develop algorithms that enable computers to efficiently and reliably work with data. An algorithm is a step

RESEARCH ONTHEMATIC MAPSMaps are one of the most efficient ways to communicate information. They help people to make decisions in navigation, spatial plan-ning, or risk and disaster management. Maps also communicate geopolitical information, and generally aid the process of public opinion and consensus building. Decision makers and the greater public benefit from high quality up to-date, on demand, and online map production, which necessarily has to be fully automated.

Thematic maps are special purpose maps for conveying specific information. They usually focus on a single theme and visualize such diverse topics as the gross domestic product per country, the effects of pollution on the water quality, or the migration patterns of animals. Thematic maps can be found on webpages, in newspapers, infrastructural reports, or biological studies, to name a few.

Flow MapsAn example of Thematic Maps are Flow Maps. Flow Maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of flow between a source and a target. Good flow maps reduce visual clutter by merg-ing (bundling) lines smoothly and by avoid-ing self-intersections. Most flow maps are still drawn by hand and only few automated meth-ods exist. Bettina Speckmann presented a new algorithmic method that uses edge-bundling and computes crossing-free flows of high visual quality.

This method is based on spiral trees, defined mathematically by Bettina Speckmann and her group. Spiral trees naturally cluster targets and smoothly bundle lines. To draw one, they start with a thin spiral tree bypassing obstructions, which is then thickened, then smoothed using mathematical optimization tools that prevent overlap.

RESEARCH ONMOTION DATAOver the past years the availability of devices that canbe used to track moving objects GPS systems, mobilephones, surveillance cameras, RFID tags, and morehas increased dramatically, leading to an explosivegrowth in data about moving objects. Objects beingtracked range from migratory birds to delivery trucks,sea turtles to sports players, hurricanes to suspected terrorists. Naturally the goal is not only to track

objects but to determine typical or unusual patterns of behavior by doing trajectory analysis: computing similarity, clustering, classifying, simplifying, segment-ing, and detecting the effects of time and space. The movements of animals, people and vehicles are embedded in a geographic context, which both enables and limits movement: cars can move on roads and turtles ride ocean currents, but people

cannot walk on water and wolves cannot cross a wide river gorge.Most analysis algorithms for trajectories have so far ignored context, except for the display of results. Hence one of the big challenges for the coming years is to capture the essential properties of context and develop efficient analysis algorithms that are context aware.

Dutch Prize for ICT Research 2011Computer Science Research on Models, Maps and Motion

IPN ICT Research Platform Netherlands IPN Is the national platform for ICT research in the Netherlands. ICT research has an enormous impact on our everyday lives. We are often not aware that we make use of the results of ICT research dozens of times each day. IPN strives to give ICT research in the Netherlands a stronger position and more visible. By coordinating Dutch ICT efforts IPN functions as the national contact point for policy makers, politi-cians, industry and other groups in society. Researchers in computer science and closely related disci-plines, such as electronics, communication and signal processing, are increasingly working together. IPN encourages this movement towards a single ICT research field and is actively contributing to this. For more information: www.ictonderzoek.net

NWO PHYSICAL SCIENCES NWO (Netherlands Organisation for Scientific Research) enhances the quality and innovative nature of excellent scientific research in the Netherlands. The NWO division Physical Sciences strengthens the scientific environment of the disciplines Astronomy, Mathematics and Computer Science by: allocating grants for research, facilities and equipment; developing research directions via research agendas and programmes; cooperating with partners from the public and private sector; stimulating multidisciplinary and international projects; creating career opportunities for talented researchers. For more information: www.nwo.nl/ew

CopyrightNWO/Exacte Wetenschappen.

DesignSmidswater, Den Haag/Amsterdam/Breda

Movements of individual lesser black-backed gulls breeding on the island Texel and tracked with UvA-BiTS (www.uva-bits.nl)

University of Amsterdam & Royal Netherlands Institute for Sea Research

Bettina Speckmann

Flow map showingScottish whiskey exports

Necklace map of 2010World Cup soccer success

Symbol map of deathsfrom major earthquakes

Three important steps in the construction of a flow map