Advertisement

1-Embeddability of 2-Dimensional ℓ1-Rigid Periodic Graphs

  • Norie Fu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8344)

Abstract

The ℓ1 -embedding problem of a graph is the problem to find a map from its vertex set to ℝ d such that the length of the shortest path between any two vertices is equal to the ℓ1-distance between the mapping of the two vertices in ℝ d . The ℓ1-embedding problem partially contains the shortest path problem since an ℓ1-embedding provides the all-pairs shortest paths. While Höfting and Wanke showed that the shortest path problem is NP-hard, Chepoi, Deza, and Grishukhin showed a polynomial-time algorithm for the ℓ1-embedding of planar 2-dimensional periodic graphs. In this paper, we study the ℓ1-embedding problem on ℓ1 -rigid 2-dimensional periodic graphs, for which there are finite representations of the ℓ1-embedding. The periodic graphs form a strictly larger class than planar ℓ1-embeddable 2-dimensional periodic graphs. Using the theory of geodesic fiber, which was originally proposed by Eon as an invariant of a periodic graph, we show an exponential-time algorithm for the ℓ1-embedding of ℓ1-rigid 2-dimensional periodic graphs, including the non-planar ones. Through Höfting and Wanke’s formulation of the shortest path problem as an integer program, our algorithm also provides an algorithm for solving a special class of parametric integer programming.

Keywords

Short Path Planar Graph Static Graph Short Path Problem Complete Subgraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chavey, D.: Tilings by regular polygons – II: A catalog of tilings. Computers & Mathematics with Applications 17, 147–165 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chepoi, V., Deza, M., Grishukhin, V.: Clin d’oeil on L 1-embeddable planar graphs. Discrete Applied Mathematics 80(1), 3–19 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Cohen, E., Megiddo, N.: Recognizing properties of periodic graphs. Applied Geometry and Discrete Mathematics 4, 135–146 (1991)MathSciNetGoogle Scholar
  4. 4.
    Delgado-Friedrichs, O., O’Keeffe, M.: Crystal nets as graphs: Terminology and definitions. Journal of Solid State Chemistry 178, 2480–2485 (2005)CrossRefGoogle Scholar
  5. 5.
    Deza, M., Grishukhin, V., Shtogrin, M.: Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices, ch. 9. World Scientific Publishing Company (2004)Google Scholar
  6. 6.
    Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Springer (1997)Google Scholar
  7. 7.
    Eon, J.-G.: Infinite geodesic paths and fibers, new topological invariants in periodic graphs. Acta Crystallographica Section A 63, 53–65 (2007)MathSciNetGoogle Scholar
  8. 8.
    Feutrier, P.: Parametric integer programming. RAIRO Recherche Opérationnelle 22, 243–268 (1988)Google Scholar
  9. 9.
    Fu, N.: A strongly polynomial time algorithm for the shortest path problem on coherent planar periodic graphs. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 392–401. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Höfting, F., Wanke, E.: Minimum cost paths in periodic graphs. SIAM Journal on Computing 24(5), 1051–1067 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Iwano, K., Steiglitz, K.: Optimization of one-bit full adders embedded in regular structures. IEEE Transaction on Acoustics, Speech and Signal Processing 34, 1289–1300 (1986)CrossRefGoogle Scholar
  12. 12.
    Iwano, K., Steiglitz, K.: Planarity testing of doubly periodic infinite graphs. Networks 18, 205–222 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Karp, R., Miller, R., Winograd, A.: The organization of computations for uniform recurrence equiations. Journal of the ACM 14, 563–590 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Verdoolaege, S.: barvinok: User guide (2007), http://freshmeat.net/projects/barvinok/

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Norie Fu
    • 1
    • 2
  1. 1.National Institute of InformaticsJapan
  2. 2.JST, ERATO, Kawarabayashi Large Graph ProjectJapan

Personalised recommendations