The Galois Complexity of Graph Drawing: Why Numerical Solutions Are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

  • Michael J. Bannister
  • William E. Devanny
  • David Eppstein
  • Michael T. Goodrich
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

Abstract

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.

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References

  1. 1.
    Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions. In: 12th Symp. on Computational Geometry (SoCG), pp. 319–328 (1996)Google Scholar
  2. 2.
    Hopcroft, J.E., Kahn, P.J.: A paradigm for robust geometric algorithms. Algorithmica 7, 339–380 (1992)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Tutte, W.T.: How to draw a graph. Proc. London Math. Soc. 3, 743–767 (1963)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Software: Practice and Experience 21, 1129–1164 (1991)Google Scholar
  5. 5.
    Kamada, T., Kawai, S.: An algorithm for drawing general undirected graphs. Information Processing Letters 31, 7–15 (1989)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Koren, Y.: Drawing graphs by eigenvectors: theory and practice. Computers & Mathematics with Applications 49, 1867–1888 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kruskal, J.B., Seery, J.B.: Designing network diagrams. In: Proc. First General Conf. on Social Graphics, pp. 22–50 (1980)Google Scholar
  8. 8.
    Koebe, P.: Kontaktprobleme der Konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88, 141–164 (1936)Google Scholar
  9. 9.
    Bajaj, C.: The algebraic complexity of shortest paths in polyhedral spaces. In: Proc. 23rd Allerton Conf. on Communication, Control and Computing, pp. 510–517 (1985)Google Scholar
  10. 10.
    Carufel, J.L.D., Grimm, C., Maheshwari, A., Owen, M., Smid, M.: A Note on the unsolvability of the weighted region shortest path problem. In: Booklet of Abstracts of the 28th European Workshop on Computational Geometry, pp. 65–68 (2013)Google Scholar
  11. 11.
    Bajaj, C.: The algebraic degree of geometric optimization problems. Discrete Comput. Geom. 3, 177–191 (1988)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Nister, D., Hartley, R., Stewenius, H.: Using Galois theory to prove structure from motion algorithms are optimal. In: IEEE Conf. Computer Vision & Pattern Recog., pp. 1–8 (2007)Google Scholar
  13. 13.
    Varfolomeev, V.V.: Galois groups of the Heron–Sabitov polynomials for inscribed pentagons. Mat. Sb. 195, 3–16 (2004); Translation in Sb. Math. 195, 149–162 (2004)Google Scholar
  14. 14.
    Brightwell, G., Scheinerman, E.: Representations of planar graphs. SIAM J. Discrete Math. 6, 214–229 (1993)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Ben-Or, M.: Lower bounds for algebraic computation trees. In: Proc. 15th Annu. Symp. Theory of Computing, pp. 80–86 (1983)Google Scholar
  16. 16.
    Yao, A.C.: Lower bounds for algebraic computation trees of functions with finite domains. SIAM J. Comput. 20, 655–668 (1991)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Shoup, V.: A Computational Introduction to Number Theory and Algebra. Cambridge Univ. Press (2009)Google Scholar
  18. 18.
    Baker, R.C., Harman, G.: Shifted primes without large prime factors. Acta Arith. 83, 331–361 (1998)MATHMathSciNetGoogle Scholar
  19. 19.
    Cox, D.A.: Galois Theory. 2nd edn. Pure and Applied Mathematics. Wiley (2012)Google Scholar
  20. 20.
    Jacobson, N.: Basic Algebra I, 2nd edn. Dover Books on Mathematics. Dover (2012)Google Scholar
  21. 21.
    Stäckel, P.: Arithmetische Eigenschaften ganzer Funktionen (Fortsetzung.). J. Reine Angew. Math. 148, 101–112 (1918)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Michael J. Bannister
    • 1
  • William E. Devanny
    • 1
  • David Eppstein
    • 1
  • Michael T. Goodrich
    • 1
  1. 1.Department of Computer ScienceUniversity of CaliforniaIrvineUSA

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