Discrete & Computational Geometry

, Volume 50, Issue 3, pp 784–810

# Computing Cartograms with Optimal Complexity

• Md. Jawaherul Alam
• Therese Biedl
• Stefan Felsner
• Michael Kaufmann
• Stephen G. Kobourov
• Torsten Ueckerdt
Article

## Abstract

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. Finally we address the problem of constructing small-complexity cartograms for 4-connected graphs (which are Hamiltonian). We first disprove a conjecture, posed by two set of authors, that any 4-connected maximal planar graph has a one-legged Hamiltonian cycle, thereby invalidating an attempt to achieve a polygonal complexity 6 in cartograms for this graph class. We also prove that it is NP-hard to decide whether a given 4-connected plane graph admits a cartogram with respect to a given weight function.

### Keywords

Planar graphs Geometric representations Contact graphs Cartograms

## Notes

### Acknowledgments

Alam and Kobourov’s research was funded in part by NSF Grants CCF-0545743 and CCF-1115971. Biedl’s research was supported by NSERC. Felsner’s and Ueckerdt’s research was partially supported by EUROGIGA project GraDR and DFG Fe 340/7-2. Kaufmann’s research was partially supported by EUROGIGA project GraDR. The work was done when Ueckerdt was at Technische Universität Berlin.

### References

1. 1.
Alam, M.J., Kobourov, S.G.: Proportional contact representations of 4-connected planar graphs. In: Graph Drawing (GD’12), pp. 211–223. Springer (2013)Google Scholar
2. 2.
Alam, M.J., Biedl, T.C., Felsner, S., Gerasch, A., Kaufmann, M., Kobourov, S.G.: Linear-time algorithms for hole-free rectilinear proportional contact graph representations. In: International Symposium on Algorithms and Computation (ISAAC’11), pp. 281–291. Springer, Berlin (2011)Google Scholar
3. 3.
Alam, M.J., Biedl, T.C., Felsner, S., Kaufmann, M., Kobourov, S.G.: Proportional contact representations of planar graphs. In: Graph Drawing (GD’11), pp. 26–38. Springer, Heidelberg (2012)Google Scholar
4. 4.
Alam, M.J., Biedl, T.C., Felsner, S., Kaufmann, M., Kobourov, S.G., Ueckerdt, T.: Computing cartograms with optimal complexity. In: Symposium on Computational Geometry (SoCG’12), pp. 21–30. Bethesda (2012)Google Scholar
5. 5.
Biedl, T.C., Genc, B.: Complexity of octagonal and rectangular cartograms. In: Canadian Conference on, Computational Geometry (CCCG’05), pp. 117–120. Ontario (2005)Google Scholar
6. 6.
Biedl, T.C., Velázquez, L.E.R.: Orthogonal cartograms with few corners per face. In: Algorithms and Data Structures, Symposium (WADS’11), pp. 98–109. Springer, Berlin (2011)Google Scholar
7. 7.
Buchsbaum, A.L., Gansner, E.R., Procopiuc, C.M., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Trans. Algorithms 4, 28 (2008)
8. 8.
Cederbaum, I.: Analogy between VLSI floorplanning problems and realisation of a resistive network. IEE Circuits Devices Syst. 139(1), 99–103 (1992)
9. 9.
Chen, T., Fan, M.K.H.: On convex formulation of the floorplan area minimization problem. In: Symposium on Physical Design, pp. 124–128. (1998)Google Scholar
10. 10.
Chrobak, M., Payne, T.H.: A linear-time algorithm for drawing a planar graph on a grid. Inf. Process. Lett. 54(4), 241–246 (1995)
11. 11.
de Berg, M., Mumford, E., Speckmann, B.: On rectilinear duals for vertex-weighted plane graphs. Discrete Math. 309(7), 1794–1812 (2009)
12. 12.
de Berg, M., Mumford, E., Speckmann, B.: Optimal BSPs and rectilinear cartograms. Int. J. Comput. Geom. Appl. 20(2), 203–222 (2010)
13. 13.
de Fraysseix, H., Ossona de Mendez, P.: On triangle contact graphs. Comb. Probab. Comput. 3, 233–246 (1994)
14. 14.
de Fraysseix, H., Ossona de Mendez, P.: On topological aspects of orientations. Discrete Math. 229(1–3), 57–72 (2001)Google Scholar
15. 15.
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
16. 16.
Duncan, C.A., Gansner, E.R., Hu, Y.F., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. Algorithmica 63(3), 672–691 (2012)
17. 17.
Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.: Area-universal and constrained rectangular layouts. SIAM J. Comput. 41(3), 537–564 (2012)
18. 18.
He, X.: On floor-plan of plane graphs. SIAM J. Comput. 28(6), 2150–2167 (1999)
19. 19.
Heilmann, R., Keim, D.A., Panse, C., Sips, M.: Recmap: Rectangular map approximations. In: IEEE Symposium on, Information Visualization (INFOVIS’04), pp. 33–40. Minneapolis (2004)Google Scholar
20. 20.
House, D., Kocmoud, C.: Continuous cartogram construction. In: Proceedings of IEEE Visualization, pp. 197–204. Triangle Park (1998)Google Scholar
21. 21.
Izumi, T., Takahashi, A., Kajitani, Y.: Air-pressure model and fast algorithms for zero-wasted-area layout of general floorplan. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E81–A, 857–865 (1998)Google Scholar
22. 22.
Kawaguchi, A., Nagamochi, H.: Orthogonal drawings for plane graphs with specified face areas. In: Theory and Applications of Model of Computation (TAMC’07), pp. 584–594 . Springer, Berlin (2007)Google Scholar
23. 23.
Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig. Math. Phys. Klasse 88, 141–164 (1936)Google Scholar
24. 24.
Koźmiński, K., Kinnen, E.: Rectangular duals of planar graphs. Networks 15, 145–157 (1985)
25. 25.
Leinwand, S.M., Lai, Y.T.: An algorithm for building rectangular floor-plans. In: Design Automation Conference, pp. 663–664. IEEE Press, Piscataway (1984)Google Scholar
26. 26.
Liao, C.C., Lu, H.I., Yen, H.C.: Compact floor-planning via orderly spanning trees. J. Algorithms 48, 441–451 (2003)
27. 27.
Michalek, J., Choudhary, R., Papalambros, P.: Architectural layout design optimization. Eng. Optim. 34(5), 461–484 (2002)
28. 28.
Moh, T.S., Chang, T.S., Hakimi, S.L.: Globally optimal floorplanning for a layout problem. IEEE Trans. Circuits Syst. 43(9), 713–720 (1996)
29. 29.
Nöllenburg, M., Prutkin, R., Rutter, I.: Edge-weighted contact representations of planar graphs. In: Graph Drawing (GD’12), pp. 224–235. Springer (2013)Google Scholar
30. 30.
Raisz, E.: The rectangular statistical cartogram. Geogr. Rev. 24(3), 292–296 (1934)
31. 31.
Rinsma, I.: Nonexistence of a certain rectangular floorplan with specified area and adjacency. Environ. Plan. 14, 163–166 (1987)
32. 32.
Rosenberg, E.: Optimal module sizing in VLSI floorplanning by nonlinear programming. Methods Models Oper. Res. 33, 131–143 (1989)
33. 33.
Schnyder, W.: Embedding planar graphs on the grid. In: Symposium on Discrete Algorithms (SODA’90), pp. 138–148. San Francisco (1990)Google Scholar
34. 34.
Tobler, W.: Thirty five years of computer cartograms. Ann. Assoc. Am. Geogr. 94, 58–73 (2004)
35. 35.
Ueckerdt, T.: Geometric representations of graphs with low polygonal complexity. Ph.D. Thesis, Technische Universität, Berlin (2011)Google Scholar
36. 36.
Ungar, P.: On diagrams representing maps. J. Lond. Math. Soc. 28, 336–342 (1953)
37. 37.
van Kreveld, M.J., Speckmann, B.: On rectangular cartograms. Comput. Geom. 37(3), 175–187 (2007)
38. 38.
Wang, K., Chen, W.K.: Floorplan area optimization using network analogous approach. In: IEEE Transactions Circuits and Systems, pp. 167–170 (1995)Google Scholar
39. 39.
Wimer, S., Koren, I., Cederbaum, I.: Floorplans, planar graphs, and layouts. IEEE Trans. Circuits Syst. 35(3), 267–278 (1988)
40. 40.
Yeap, K.H., Sarrafzadeh, M.: Floor-planning by graph dualization: 2-concave rectilinear modules. SIAM J. Comput. 22, 500–526 (1993)

## Copyright information

© Springer Science+Business Media New York 2013

## Authors and Affiliations

• Md. Jawaherul Alam
• 1
• Therese Biedl
• 2
• Stefan Felsner
• 3
• Michael Kaufmann
• 4
• Stephen G. Kobourov
• 1
• Torsten Ueckerdt
• 5
1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA
2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
3. 3.Institut für MathematikTechnische Universität BerlinBerlinGermany
4. 4.Wilhelm-Schickhard-Institut für InformatikTübingen UniversitätTubingenGermany
5. 5.Fakultät für MathematikKarlsruher Institut für TechnologieKarlsruherGermany