### 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.

Original language | English (US) |
---|---|

Title of host publication | Proceedings of the Annual Symposium on Computational Geometry |

Pages | 21-30 |

Number of pages | 10 |

DOIs | |

State | Published - 2012 |

Event | 28th Annual Symposuim on Computational Geometry, SCG 2012 - Chapel Hill, NC, United States Duration: Jun 17 2012 → Jun 20 2012 |

### Other

Other | 28th Annual Symposuim on Computational Geometry, SCG 2012 |
---|---|

Country | United States |

City | Chapel Hill, NC |

Period | 6/17/12 → 6/20/12 |

### Fingerprint

### Keywords

- Cartograms
- Contact graphs
- Geometric representations
- Planar graphs

### ASJC Scopus subject areas

- Computational Mathematics
- Geometry and Topology
- Theoretical Computer Science

### Cite this

*Proceedings of the Annual Symposium on Computational Geometry*(pp. 21-30) https://doi.org/10.1145/2261250.2261254

**Computing cartograms with optimal complexity.** / Alam, Md Jawaherul; Biedl, Therese; Felsner, Stefan; Kaufmann, Michael; Kobourov, Stephen G; Ueckerdt, Torsten.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the Annual Symposium on Computational Geometry.*pp. 21-30, 28th Annual Symposuim on Computational Geometry, SCG 2012, Chapel Hill, NC, United States, 6/17/12. https://doi.org/10.1145/2261250.2261254

}

TY - GEN

T1 - Computing cartograms with optimal complexity

AU - Alam, Md Jawaherul

AU - Biedl, Therese

AU - Felsner, Stefan

AU - Kaufmann, Michael

AU - Kobourov, Stephen G

AU - Ueckerdt, Torsten

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

KW - Cartograms

KW - Contact graphs

KW - Geometric representations

KW - Planar graphs

UR - http://www.scopus.com/inward/record.url?scp=84863964949&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84863964949&partnerID=8YFLogxK

U2 - 10.1145/2261250.2261254

DO - 10.1145/2261250.2261254

M3 - Conference contribution

AN - SCOPUS:84863964949

SN - 9781450312998

SP - 21

EP - 30

BT - Proceedings of the Annual Symposium on Computational Geometry

ER -