Computing Cartograms with Optimal Complexity

We show how to compute cartograms with worst-case optimal polygonal complexity. Specifically we study rectilinear duals which are side-contact representations of a planar graph G with vertices represented by simple rectilinear polygons and adjacencies represented by a non-trivial 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 of the corresponding vertex. We show how to construct an area-universal rectilinear dual, i.e., one that can realize a cartogram with any pre-specified set of weights for the vertices of the graph. In a series of papers the polygonal complexity of such representations for maximal planar graphs was reduced in several steps from the initial 40 to 34, then to 12 and very recently to the currently best known 10, while it has been known that 8-sided polygons are sometimes necessary. Our construction uses only 8-sided polygons and hence is optimal in terms of polygonal...

• Type: Journal
• Year: 2012
• Where: CORR
• Authors: Md. Jawaherul Alam, Therese C. Biedl, Stefan Felsner, Michael Kaufmann, Stephen G. Kobourov, Torsten Ueckerdt
• Keywords: CORR 2012, Education, Maximal Planar Graphs, Sided Polygons, Time Representations