We present a new circular-arc cartogram model in which countries are drawn as polygons with circular arcs instead of straight-line 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 circular-arc cartogram model straight-line 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 circular-arc 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 area-values can be realized as a circular-arc cartogram is an NP-hard problem. Next we describe a heuristic method for constructing circular-arc cartograms, which uses a max-flow computation on the dual graph of the map, along with a computation of the straight skeleton of the underlying polygonal decomposition. Our method is implemented and produces cartograms that, while not yet perfectly accurate, achieve many of the desired areas in our real-world examples.
We consider orthogonal drawings of a plane graph G with specified face areas. For a natural number k, a k-gonal drawing of G is an orthogonal drawing such that the boundary of G is drawn as a rectangle and each inner face is drawn as a polygon with at most k corners whose area is equal to the specified value. In this paper, we show that every slicing graph G with a slicing tree T and a set of specified face areas admits a 10-gonal drawing D such that the boundary of each slicing subgraph that appears in T is also drawn as a polygon with at most 10 corners. Such a drawing D can be found in linear time. Let G=(V,E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of G is a partition of a rectangle into |V| simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E. A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertex-weighted plane triangulated graph G admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant. Furthermore, such a rectilinear cartogram can be constructed in O(nlogn) time where n=|V|. A rectangular layout is a partition of a rectangle into a finite set of interior-disjoint rectangles. Rectangular layouts appear in various applications: as rectangular cartograms in cartography, as floorplans in building architecture and VLSI design, and as graph drawings. Often areas are associated with the rectangles of a rectangular layout and it might hence be desirable if one rectangular layout can represent several area assignments. A layout is area-universal if any assignment of areas to rectangles can be realized by a combinatorially equivalent rectangular layout. We identify a simple necessary and sufficient condition for a rectangular layout to be area-universal: a rectangular layout is area-universal if and only if it is one-sided. More generally, given any rectangular layout L and any assignment of areas to its regions, we show that there can be at most one layout (up to horizontal and vertical scaling) which is combinatorially equivalent to L and achieves a given area assignment. We also investigate similar questions for perimeter assignments. The adjacency requirements for the rectangles of a rectangular layout can be specified in various ways, most commonly via the dual graph of the layout. We show how to find an area-universal layout for a given set of adjacency requirements whenever such a layout exists. In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. 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 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine it into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction of cartograms for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs. In many application domains, data is collected and referenced by its geospatial location. Nowadays, different kinds of maps are used to emphasize the spatial distribution of one or more geospatial attributes. The nature of geospatial statistical data is the highly nonuniform distribution in the real world data sets. This has several impacts on the resulting map visualizations. Classical area maps tend to highlight patterns in large areas, which may, however, be of low importance. Cartographers and geographers used cartograms or value-by-area maps to address this problem long before computers were available. Although many automatic techniques have been developed, most of the value-by-area cartograms are generated manually via human interaction. In this paper, we propose a novel visualization technique for geospatial data sets called RecMap. Our technique approximates a rectangular partition of the (rectangular) display area into a number of map regions preserving important geospatial constraints. It is a fully automatic technique with explicit user control over all exploration constraints within the exploration process. Experiments show that our technique produces visualizations of geospatial data sets, which enhance the discovery of global and local correlations, and demonstrate its performance in a variety of applications We give an algorithm to create orthogonal drawings of 3- connected 3-regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 corners per face.  A rectangular cartogram is a type of map where every region is a rectangle. The size of the rectangles is chosen such that their areas represent a geographic variable (e.g., population). Good rectangular cartograms are hard to generate: The area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map. We present the first algorithms for rectangular cartogram construction. Our algorithms depend on a precise formalization of region adjacencies and build upon existing VLSI layout algorithms. Furthermore, we characterize a non-trivial class of rectangular subdivisions for which exact cartograms can be computed efficiently. An implementation of our algorithms and various tests show that in practice, visually pleasing rectangular cartograms with small cartographic error can be generated effectively. T HE idea of the statistical cartogram occurred to the author when he had occasion to prepare maps of the United States showing the distribution of various economic units, such as steel factories, textile mills, power plants, banks, etc. These maps were far too crowded in the northeast to be useful, while elsewhere, for the most part, they were relatively empty. If a way could be found to increase the scale of the northeastern region and reduce that of the west, distribution could be shown more clearly. Simple distortion of the map would be misleading, but, if we go a step farther, discard altogether the outlines of the country, and give each region a rectangular form of size proportional to the value represented, we arrive at the rectangular statistical cartogram. For purposes of comparison it is essential that a definite system of construction should be followed and identical arrangement should be used whatever values are represented. The system here used starts always with the larger divisions and by "proportionate halving" arrives at the smaller ones. It should be emphasized that the statistical cartogram is not a map. Although it has roughly the proportions of the country and retains as far as possible the relative locations of the various regions, the cartogram is purely a geometrical design to visualize certain statistical facts and to work out certain problems of distribution. Examples of these cartograms are given in the accompanying figures. The division into regions follows the usage of the United States Census Bureau, because only from this source are data available. If natural geographic regions could be used instead, the cartograms would be still more instructive. Given a planar triangulated graph (PTG) G, the problem of constructing a floor-plan F such that G is the dual of F and the boundary of F is rectangular is studied. It is shown that if only zero-concave rectilinear modules (CRM) (or rectangular modules) and 1-CRM (i.e., L-shaped) are allowed, there are PTGs that do not admit any floor-plan. However, if 2-bend modules (e.g., T-shaped and Z-shaped) are also allowed, then every biconnected PTG admits a floor-plan. Thus, the employment of 2-bend modules is necessary and sufficient for graph dualization floor-planning. A linear-time algorithm for constructing a 2-CRM floor-plan of an arbitrary PTG is proposed.