Floor-planning by graph dualization: 2-concave rectilinear modules
SIAM Journal on Computing
A linear-time algorithm for drawing a planar graph on a grid
Information Processing Letters
On convex formulation of the floorplan area minimization problem
ISPD '98 Proceedings of the 1998 international symposium on Physical design
Continuous cartogram construction
Proceedings of the conference on Visualization '98
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
SIAM Journal on Computing
An algorithm for building rectangular floor-plans
DAC '84 Proceedings of the 21st Design Automation Conference
Compact floor-planning via orderly spanning trees
Journal of Algorithms
RecMap: Rectangular Map Approximations
INFOVIS '04 Proceedings of the IEEE Symposium on Information Visualization
Computational Geometry: Theory and Applications
Rectangular layouts and contact graphs
ACM Transactions on Algorithms (TALG)
Area-universal rectangular layouts
Proceedings of the twenty-fifth annual symposium on Computational geometry
Orthogonal drawings for plane graphs with specified face areas
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
Orthogonal cartograms with few corners per face
WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
Optimal Polygonal Representation of Planar Graphs
Algorithmica - Special Issue: Theoretical Informatics
Proportional contact representations of planar graphs
GD'11 Proceedings of the 19th international conference on Graph Drawing
Linear-time algorithms for hole-free rectilinear proportional contact graph representations
ISAAC'11 Proceedings of the 22nd international conference on Algorithms and Computation
Drawing planar 3-trees with given face areas
Computational Geometry: Theory and Applications
On representing graphs by touching cuboids
GD'12 Proceedings of the 20th international conference on Graph Drawing
Proportional contact representations of 4-connected planar graphs
GD'12 Proceedings of the 20th international conference on Graph Drawing
Edge-Weighted contact representations of planar graphs
GD'12 Proceedings of the 20th international conference on Graph Drawing
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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.