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Computing Cartograms with Optimal Complexity
"... In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons and edges are represented by sidecontact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a prespecified weight of the corresponding ve ..."
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In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons and edges are represented by sidecontact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a prespecified weight of the corresponding vertex. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8sided polygons, which is optimal in terms of polygonal complexity as 8sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine the representation into an areauniversal rectangular layout in linear time. The exact cartogram can be computed from the areauniversal rectangular layout with numerical iteration, or can be approximated with a hillclimbing heuristic. We also describe an alternative construction for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8sided rectilinear polygons are necessary, by constructing a nontrivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is onelegged, as in outerplanar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outerplanar graphs.
Proportional Contact Representations of Planar Graphs
"... Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a pointcontact or a sidecontact between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex w ..."
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Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a pointcontact or a sidecontact between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2segment graphs, which include maximal outerplanar graphs and partial 2trees. 1
Circulararc cartograms
"... We present a new circulararc cartogram model in which countries are drawn as polygons with circular arcs instead of straightline segments. Given a political map and values associated with each country in the map, a cartogram is a distorted map in which the areas of the countries are proportional ..."
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We present a new circulararc cartogram model in which countries are drawn as polygons with circular arcs instead of straightline segments. Given a political map and values associated with each country in the map, a cartogram is a distorted map in which the areas of the countries are proportional to the corresponding values. In the circulararc cartogram model straightline segments can be replaced by circular arcs in order to modify the areas of the polygons, while the corners of the polygons remain fixed. The countries in circulararc cartograms have the aesthetically pleasing appearance of clouds or snowflakes, depending on whether their edges are bent outwards or inwards. This makes it easy to determine whether a country has grown or shrunk, just by its overall shape. We show that determining whether a given map and given areavalues can be realized as a circulararc cartogram is an NPhard problem. Next we describe a heuristic method for constructing circulararc cartograms,
Quantitative Measures for Cartogram Generation Techniques Submission No. # 327
"... Cartograms are used to visualize geographically distributed data by scaling the regions of a map (e.g., US states) such that their areas are proportional to some data associated with them (e.g., population). Thus the cartogram computation problem can be considered as a map deformation problem where ..."
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Cartograms are used to visualize geographically distributed data by scaling the regions of a map (e.g., US states) such that their areas are proportional to some data associated with them (e.g., population). Thus the cartogram computation problem can be considered as a map deformation problem where the input is a planar polygonal map M and an assignment of some positive weight for each region. The goal is to create a deformed map M′, where the area of each region realizes the weight assigned to it (no cartographic error) while the overall map remains readable and recognizable (e.g., the topology, relative positions and shapes of the regions remain as close to those before the deformation as possible). Although several such measures of cartogram quality are wellknown, different cartogram generation methods optimize different features and there is no standard set of quantitative metrics. In this paper we define such a set of seven quantitative measures, designed to evaluate how faithfully a cartogram represents the desired weights and to estimate the readability of the final representation. We then study several cartogramgeneration algorithms and compare them in terms of these quantitative measures. 1.