Graph-Theoretic Solutions to Computational Geometry Problems

  • David Eppstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5911)

Abstract

Many problems in computational geometry are not stated in graph-theoretic terms, but can be solved efficiently by constructing an auxiliary graph and performing a graph-theoretic algorithm on it. Often, the efficiency of the algorithm depends on the special properties of the graph constructed in this way. We survey the art gallery problem, partition into rectangles, minimum-diameter clustering, rectilinear cartogram construction, mesh stripification, angle optimization in tilings, and metric embedding from this perspective.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P.K., Efrat, A., Sharir, M.: Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput. 29(3), 912–953 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ageev, A.A.: A triangle-free circle graph with chromatic number 5. Discrete Mathematics 152, 295–298 (1996)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aggarwal, A., Imai, H., Katoh, N., Suri, S.: Finding k points with minimum diameter and related problems. J. Algorithms 12(1), 38–56 (1991)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. In: Theory and Practice of Combinatorics. North-Holland Mathematics Studies, vol. 60, pp. 9–12 (1982)Google Scholar
  5. 5.
    Arkin, E.M., Held, M., Mitchell, J.S.B., Skiena, S.S.: Hamiltonian triangulations for fast rendering. The Visual Computer 12(9), 429–444 (1996)Google Scholar
  6. 6.
    Bandelt, H.-J., Chepoi, V., Eppstein, D.: Combinatorics and geometry of finite and infinite squaregraphs. Electronic preprint arxiv:0905.4537 (2009)Google Scholar
  7. 7.
    Biedl, T.C., Bose, P., Demaine, E.D., Lubiw, A.: Efficient algorithms for Petersen’s matching theorem. J. Algorithms 38, 110–134 (2001)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Computational Geometry Theory and Applications 9(1–2), 3–24 (1998)MATHMathSciNetGoogle Scholar
  9. 9.
    Carlson, J., Eppstein, D.: Trees with convex faces and optimal angles. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 77–88. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Chazelle, B.: Triangulating a simple polygon in linear time. Discrete & Computational Geometry 6(1), 485–524 (1991)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Cheng, Y., Iyengar, S.S., Kashyap, R.L.: A new method of image compression using irreducible covers of maximal rectangles. IEEE Trans. Software Engineering 14(5), 651–658 (1988)CrossRefGoogle Scholar
  12. 12.
    Chepoi, V., Dragan, F., Vaxès, Y.: Center and diameter problem in planar quadrangulations and triangulations. In: Proc. 13th Annu. ACM–SIAM Symp. on Discrete Algorithms (SODA 2002), pp. 346–355. ACM Press, New York (2002)Google Scholar
  13. 13.
    Chvátal, V.: A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, Series B 18, 39–41 (1975)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Chvátal, V.: Perfectly orderable graphs. In: Berge, C., Chvátal, V. (eds.) Topics in Perfect Graphs. Annals of Discrete Mathematics, vol. 21, pp. 63–68. North-Holland, Amsterdam (1984)CrossRefGoogle Scholar
  15. 15.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Mathematics 86, 165–177 (1990)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    de Bruijn, N.G.: Algebraic theory of Penrose’s non-periodic tilings of the plane. Indagationes Mathematicae 43, 38–66 (1981)Google Scholar
  17. 17.
    Deering, M.: Geometry compression. In: Proc. 22nd Conf. Computer Graphics and Interactive Techniques (SIGGRAPH), pp. 13–20 (1995)Google Scholar
  18. 18.
    Dey, T.K.: Improved bounds for planar k-sets and related problems. Discrete & Computational Geometry 19(3), 373–382 (1998)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood Cliffs (1998)Google Scholar
  20. 20.
    Díaz-Gutiérrez, P.: Using graph algorithms for geometry processing on surfaces. PhD thesis, Univ. of California, Irvine (2009)Google Scholar
  21. 21.
    Engel, K.: Optimal matrix-segmentation by rectangles. Discrete Applied Mathematics 157(9), 2015–2030 (2009)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Eppstein, D.: Geometric lower bounds for parametric matroid optimization. Discrete & Computational Geometry 20, 463–476 (1998)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Eppstein, D.: Spanning trees and spanners. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, ch. 9, pp. 425–461. Elsevier, Amsterdam (2000)CrossRefGoogle Scholar
  24. 24.
    Eppstein, D.: Algorithms for drawing media. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 173–183. Springer, Heidelberg (2005)Google Scholar
  25. 25.
    Eppstein, D.: The traveling salesman problem for cubic graphs. J. Graph Algorithms and Applications 11(1), 61–81 (2007)MATHMathSciNetGoogle Scholar
  26. 26.
    Eppstein, D.: Testing bipartiteness of geometric intersection graphs. ACM Trans. Algorithms 5(2), 15 (2009)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Eppstein, D., Erickson, J.: Iterated nearest neighbors and finding minimal polytopes. Discrete & Computational Geometry 11(3), 321–350 (1994)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Eppstein, D., Falmagne, J.-C., Ovchinnikov, S.: Media Theory. Springer, Heidelberg (2007)Google Scholar
  29. 29.
    Eppstein, D., Gopi, M.: Single-strip triangulation of manifolds with arbitrary topology. Eurographics Forum 23(3), 371–379 (2004); Proc. 25th Conf. Eur. Assoc. for Computer Graphics (EuroGraphics 2004)Google Scholar
  30. 30.
    Eppstein, D., Mumford, E.: Orientation-constrained rectangular layouts. In: Dehne, F., et al. (eds.) WADS 2009. LNCS, vol. 5664, pp. 266–277. Springer, Heidelberg (2009)Google Scholar
  31. 31.
    Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.A.B.: Area-universal rectangular layouts. In: Proc. 25th ACM Symp. Computational Geometry, pp. 267–276 (2009)Google Scholar
  32. 32.
    Eppstein, D., Wortman, K.: Optimal angular resolution for face-symmetric drawings. Electronic preprint arxiv:0907.5474 (2009)Google Scholar
  33. 33.
    Eppstein, D., Wortman, K.: Optimal embedding into star metrics. In: Dehne, F., et al. (eds.) WADS 2009. LNCS, vol. 5664, pp. 290–301. Springer, Heidelberg (2009)Google Scholar
  34. 34.
    Evans, F., Skiena, S.S., Varshney, A.: Optimizing triangle strips for fast rendering. In: Proc. 7th IEEE Conf. Visualization, pp. 319–326 (1996)Google Scholar
  35. 35.
    Ferrari, L., Sankar, P.V., Sklansky, J.: Minimal rectangular partitions of digitized blobs. Computer Vision, Graphics, and Image Processing 28(1), 58–71 (1984)CrossRefGoogle Scholar
  36. 36.
    Fisk, S.: A short proof of Chvátal’s watchman theorem. Journal of Combinatorial Theory, Series B 24(3), 374 (1978)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Fomin, F., Lokshtanov, D., Saurabh, S.: An exact algorithm for minimum distortion embedding. In: Paul, C., Habib, M. (eds.) WG 2009. LNCS, vol. 5911. Springer, Heidelberg (2009)Google Scholar
  38. 38.
    Garg, A., Tamassia, R.: A new minimum cost flow algorithm with applications to graph drawing. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 201–216. Springer, Heidelberg (1997)Google Scholar
  39. 39.
    Hannenhalli, S., Hubbell, E., Lipshutz, R., Pevzner, P.A.: Combinatorial algorithms for design of DNA arrays. In: Chip Technology. Advances in Biochemical Engineering/Biotechnology, vol. 77, pp. 1–19. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  40. 40.
    Hershberger, J., Suri, S.: Finding tailored partitions. J. Algorithms 12(3), 431–463 (1991)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Hopcroft, J.E., Karp, R.M.: An n5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput. 2(4), 225–231 (1973)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Imai, H., Asano, T.: Efficient algorithms for geometric graph search problems. SIAM J. Comput. 15, 478–494 (1986)MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Imai, H., Iwano, K.: Efficient sequential and parallel algorithms for planar minimum cost flow. In: Asano, T., Imai, H., Ibaraki, T., Nishizeki, T. (eds.) SIGAL 1990. LNCS, vol. 450, pp. 21–30. Springer, Heidelberg (1990)Google Scholar
  44. 44.
    Iwama, K., Nakashima, T.: An improved exact algorithm for cubic graph TSP. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 108–117. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  45. 45.
    Junger, M., Mutzel, P.: Graph Drawing Software. Springer, Heidelberg (2004)Google Scholar
  46. 46.
    Kahn, J., Klawe, M., Kleitman, D.: Traditional galleries require fewer watchmen. SIAM Journal on Algebraic and Discrete Methods 4(2), 194–206 (1983)MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Kalinowski, T.: A dual of the rectangle-segmentation problem for binary matrices. Electronic J. Combinatorics 16(1), R89 (2009)MathSciNetGoogle Scholar
  48. 48.
    Karp, R.M., Orlin, J.B.: Parametric Shortest Path Algorithms with an Application to Cyclic Staffing. Technical Report OR 103-80, MIT Operations Research Center (1980)Google Scholar
  49. 49.
    Kőnig, D.: Gráfok és mátrixok. Matematikai és Fizikai Lapok 38, 116–119 (1931)Google Scholar
  50. 50.
    Leighton, T.: Complexity Issues in VLSI. Foundations of Computing Series. MIT Press, Cambridge (1983)Google Scholar
  51. 51.
    Li, G., Zhang, H.: A rectangular partition algorithm for planar self-assembly. In: Proc. IEEE/RSJ Int. Conf. Intelligent Robots and Systems, pp. 3213–3218 (2005)Google Scholar
  52. 52.
    Linial, N.: Finite metric spaces–combinatorics, geometry and algorithms. In: Proc. International Congress of Mathematicians, Beijing, vol. 3, pp. 573–586 (2002)Google Scholar
  53. 53.
    Lipski Jr., W.: Finding a Manhattan path and related problems. Networks 13, 399–409 (1983)MATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Lipski Jr., W.: An O(nlogn) Manhattan path algorithm. Information Processing Letters 19, 99–102 (1984)MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Lipski Jr., W., Lodi, E., Luccio, F., Mugnai, C., Pagli, L.: On two-dimensional data organization II. Fundamenta Informaticae 2, 245–260 (1979)MATHMathSciNetGoogle Scholar
  56. 56.
    Malitz, S., Papakostas, A.: On the angular resolution of planar graphs. SIAM J. Discrete Mathematics 7, 172–183 (1994)MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Miller, G.: Flow in planar graphs with multiple sources and sinks. SIAM J. Comput. 24(5), 1002–1017 (1995)MATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Mumford, E.: Drawing Graphs for Cartographic Applications. PhD thesis, Technische Universiteit Eindhoven (2008)Google Scholar
  59. 59.
    Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific, Singapore (2004)MATHGoogle Scholar
  60. 60.
    Ohtsuki, T.: Minimum dissection of rectilinear regions. In: Proc. IEEE Int. Symp. Circuits and Systems, pp. 1210–1213 (1982)Google Scholar
  61. 61.
    O’Rourke, J.: Art Gallery Theorems and Algorithms. Oxford University Press, Oxford (1987)MATHGoogle Scholar
  62. 62.
    Patel, K.: Computer-aided decomposition of geometric contours into standardized areas. Computer-Aided Design 9(3), 199–203 (1977)CrossRefGoogle Scholar
  63. 63.
    Petersen, J.P.C.: Die theorie der regularen graphs. Acta Mathematica 15, 193–220 (1891)CrossRefMathSciNetGoogle Scholar
  64. 64.
    Raisz, E.: The rectangular statistical cartogram. Geographical Review 24(2), 292–296 (1934)CrossRefGoogle Scholar
  65. 65.
    Sack, J.-R., Urrutia, J.: Polygon decomposition. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 491–518. Elsevier, Amsterdam (1999)Google Scholar
  66. 66.
    Savage, C.: Parallel Algorithms for Graph Theoretic Problems. PhD thesis, University of Illinois, Urbana-Champaign (1977)Google Scholar
  67. 67.
    Shamos, M.I., Hoey, D.: Closest-point problems. In: Proc. 16th IEEE Symp. Foundations of Computer Science, pp. 151–162 (1975)Google Scholar
  68. 68.
    Tait, P.G.: Listing’s Topologie. Philosophical Magazine (5th ser.) 17, 30–46 (1884)Google Scholar
  69. 69.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)MATHCrossRefMathSciNetGoogle Scholar
  70. 70.
    Tutte, W.T.: On Hamiltonian circuits. Journal of the London Mathematical Society (2nd ser.) 21(2), 98–101 (1946); Reprinted in Scientific Papers, vol. II, pp. 85–98MATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Vaidya, P.M.: Geometry helps in matching. SIAM J. Comput. 18(6), 1201–1225 (1989)MATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    van Kreveld, M., Speckmann, B.: On rectangular cartograms. Computational Geometry Theory and Applications 37(3), 175–187 (2007)MATHMathSciNetGoogle Scholar
  73. 73.
    Xiang, X., Held, M., Mitchell, J.S.B.: Fast and effective stripification of polygonal surface models. In: Proc. Symp. Interactive 3D Graphics, pp. 71–78 (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • David Eppstein
    • 1
  1. 1.Computer Science DepartmentUniversity of CaliforniaIrvine

Personalised recommendations