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)


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.


Galois Group Algebraic Number Computation Tree Galois Theory Irreducible Polynomial 
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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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