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Aspect Ratio Universal Rectangular Layouts (2112.03242v2)

Published 6 Dec 2021 in cs.CG and math.CO

Abstract: A \emph{generic rectangular layout} (for short, \emph{layout}) is a subdivision of an axis-aligned rectangle into axis-aligned rectangles, no four of which have a point in common. Such layouts are used in data visualization and in cartography. The contacts between the rectangles represent semantic or geographic relations. A layout is weakly (strongly) \emph{aspect ratio universal} if any assignment of aspect ratios to rectangles can be realized by a weakly (strongly) equivalent layout. We give combinatorial characterizations for weakly and strongly aspect ratio universal layouts. Furthermore, we describe a quadratic-time algorithm that decides whether a given graph is the dual graph of a strongly aspect ratio universal layout, and finds such a layout if one exists.

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References (38)
  1. A bijection between permutations and floorplans, and its applications. Discret. Appl. Math., 154(12):1674–1684, 2006. doi:10.1016/j.dam.2006.03.018.
  2. Separable d𝑑ditalic_d-permutations and guillotine partitions. Annals of Combinatorics, 14:17–43, 2010. doi:10.1007/s00026-010-0043-8.
  3. The dissection of rectangles into squares. Duke Math. J., 7(1):312–340, 1940. doi:10.1215/S0012-7094-40-00718-9.
  4. Slicible rectangular graphs and their optimal floorplans. ACM Trans. Design Autom. Electr. Syst., 6(4):447–470, 2001. doi:10.1145/502175.502176.
  5. Binary space partitions. In Computational Geometry: Algorithms and Applications, chapter 12, pages 259–281. Springer, 2008. doi:10.1007/978-3-540-77974-2\_12.
  6. Area-universal and constrained rectangular layouts. SIAM J. Comput., 41(3):537–564, 2012. doi:10.1137/110834032.
  7. Stefan Felsner. Lattice structures from planar graphs. Electron. J. Comb., 11(1), 2004. URL: http://www.combinatorics.org/Volume_11/Abstracts/v11i1r15.html.
  8. Stefan Felsner. Rectangle and square representations of planar graphs. In János Pach, editor, Thirty Essays on Geometric Graph Theory, pages 213–248. Springer, 2013. doi:10.1007/978-1-4614-0110-0_12.
  9. Stefan Felsner. Exploiting air-pressure to map floorplans on point sets. J. Graph Algorithms Appl., 18(2):233–252, 2014. doi:10.7155/jgaa.00320.
  10. Éric Fusy. Transversal structures on triangulations: A combinatorial study and straight-line drawings. Discret. Math., 309(7):1870–1894, 2009. doi:10.1016/j.disc.2007.12.093.
  11. Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Trans. Graph., 4(2):74–123, 1985. doi:10.1145/282918.282923.
  12. Box-rectangular drawings of planar graphs. J. Graph Algorithms Appl., 17(6):629–646, 2013. doi:10.7155/jgaa.00309.
  13. Xin He. On finding the rectangular duals of planar triangular graphs. SIAM J. Comput., 22(6):1218–1226, 1993. doi:10.1137/0222072.
  14. Squaring the plane. Am. Math. Mon., 115(1):3–12, 2008. URL: http://www.jstor.org/stable/27642387.
  15. Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theor. Comput. Sci., 172(1-2):175–193, 1997. doi:10.1016/S0304-3975(95)00257-X.
  16. Lutz Kettner. Software design in computational geometry and contour-edge based polyhedron visualization. PhD thesis, ETH Zürich, 1999. doi:10.3929/ethz-a-003861002.
  17. Rectangular duals of planar graphs. Networks, 15(2):145–157, 1985. doi:10.1002/net.3230150202.
  18. Marc J. van Kreveld and Bettina Speckmann. On rectangular cartograms. Comput. Geom., 37(3):175–187, 2007. doi:10.1016/j.comgeo.2006.06.002.
  19. Towards characterizing graphs with a sliceable rectangular dual. In Proc. 23rd Sympos. Graph Drawing and Network Visualization (GD), volume 9411 of LNCS, pages 460–471. Springer, 2015. doi:10.1007/978-3-319-27261-0_38.
  20. László Lovász. Graphs and Geometry, volume 65 of Colloquium Publications. AMS, Providence, RI, 2019.
  21. Combinatorial generation via permutation languages. III. rectangulations. Discret. Comput. Geom., 70(1):51–122, 2023. doi:10.1007/S00454-022-00393-W.
  22. Synthesis and optimization of small rectangular floor plans. Environ. Plann. B Plann. Des., 3(1):37–70, 1976. doi:10.1068/b030037.
  23. Rectangular drawing algorithms. In Handbook on Graph Drawing and Visualization, pages 317–348. Chapman and Hall/CRC, 2013.
  24. Ralph H. J. M. Otten. Automatic floorplan design. In Proc. 19th Design Automation Conference (DAC), pages 261–267. ACM/IEEE, 1982. doi:10.1145/800263.809216.
  25. Rectangular grid drawings of plane graphs. Comput. Geom., 10(3):203–220, 1998. doi:10.1016/S0925-7721(98)00003-0.
  26. Rectangular drawings of plane graphs without designated corners. Comput. Geom., 21(3):121–138, 2002. doi:10.1016/S0925-7721(01)00061-X.
  27. Erwin Raisz. The rectangular statistical cartogram. Geogr. Review, 24(2):292–296, 1934. doi:10.2307/208794.
  28. Nathan Reading. Generic rectangulations. Eur. J. Comb., 33(4):610–623, 2012. doi:10.1016/j.ejc.2011.11.004.
  29. Ingrid Rinsma. Existence theorems for floorplans. PhD thesis, University of Canterbury, Christchurch, New Zealand, 1987.
  30. Ingrid Rinsma. Nonexistence of a certain rectangular floorplan with specified areas and adjacency. Environ. Plann. B Plann. Des., 14(2):163–166, 1987. doi:10.1068/b140163.
  31. Oded Schramm. Square tilings with prescribed combinatorics. Israel J. Math., 84:97–118, 1993. doi:10.1007/BF02761693.
  32. Carsten Thomassen. Plane representations of graphs. In Progress in Graph Theory, pages 43–69. Academic Press Canada, 1984.
  33. William T. Tutte. Squaring the square. Scientific American, 199:136–142, 1958. In Gardner’s ‘Mathematical Games’ column. Reprinted with addendum and bibliography in Martin Gardner, The 2nd Scientific American Book of Mathematical Puzzles & Diversions, pp. 186–209, Simon and Schuster, New York, 1961.
  34. Peter Ungar. On diagrams representing maps. J. London Math. Soc., s1-28(3):336–342, 1953. doi:10.1112/jlms/s1-28.3.336.
  35. Kevin Weiler. Edge-based data structures for solid modeling in a curved surface environment. IEEE Comput. Graph. Appl., 5(1):21–40, 1985. doi:10.1109/MCG.1985.276271.
  36. Floorplans, planar graphs, and layouts. IEEE Trans. Circuits Syst., 35(3):267–278, 1988. doi:10.1109/31.1739.
  37. Floorplan representations: Complexity and connections. ACM Trans. Design Autom. Electr. Syst., 8(1):55–80, 2003. doi:10.1145/606603.606607.
  38. Gary K. H. Yeap and Majid Sarrafzadeh. Sliceable floorplanning by graph dualization. SIAM J. Discret. Math., 8(2):258–280, 1995. doi:10.1137/S0895480191266700.
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Authors (3)
  1. Stefan Felsner (55 papers)
  2. Andrew Nathenson (2 papers)
  3. Csaba D. Tóth (80 papers)
Citations (2)
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