Proximity and Motion Planning on ℓ1-Rigid Planar Periodic Graphs

  • Norie Fu
  • Akihiro Hashikura
  • Hiroshi Imai
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8110)


Motivated by an application to nanotechnology, Voronoi diagrams on periodic graphs with few orbits under translations and a motion planning problem on ℓ1-embeddable Archimedean tilings have been investigated by Fu, Hashikura, Imai and Moriyama. In this paper, through the investigations on the geodesic fibers defined originally as invariants on periodic graphs by Eon, we show fast geometric algorithms for Voronoi diagrams and the motion planning on ℓ1-rigid planar periodic graphs.


Short Path Distance Function Motion Planning Voronoi Diagram Quotient Graph 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abe, M., Sugimoto, Y., Namikawa, T., Morita, K., Oyabu, N., Morita, S.: Drift-compensated data acquisition performed at room temperature with frequecy modulation atomic force microscopy. Applied Physics Letters 90, 203103 (2007)CrossRefGoogle Scholar
  2. 2.
    Karp, R., Li, S.-Y.: Two special cases of the assignment problem. Discrete Mathematics 13, 129–142 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barequet, G., Dickerson, M., Goodrich, M.: Voronoi diagrams for convex polygon-offset distance functions. Discrete and Computational Geormetry 25(2), 271–291 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chavey, D.: Tilings by regular polygons – ii: A catalog of tilings. Computers & Mathematics with Applications 17, 147–165 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chepoi, V., Deza, M., Grishukhin, V.: Clin d’oeil on L 1-embeddable planar graphs. Discrete Applied Mathematics 80(1), 3–19 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chew, L.P., Drysdale, R.L.: Voronoi diagrams based on convex distance functions. In: Proceedings of the First Annual Symposium on Computational Geometry, pp. 235–244 (1985)Google Scholar
  7. 7.
    Chung, S.J., Hahn, T., Klee, W.E.: Nomenclature and generation of three-periodic nets: The vector method. Acta Crystallographica Section A 40, 42–50 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cohen, E., Megiddo, N.: Recognizing properties of periodic graphs. Applied Geometry and Discrete Mathematics 4, 135–146 (1991)MathSciNetGoogle Scholar
  9. 9.
    Călinescu, G., Dumitrescu, A., Pach, J.: Reconfigurations in graphs and grids. SIAM Jounal on Discrete Mathematics 22, 124–138 (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Delgado-Friedrichs, O., O’Keeffe, M.: Crystal nets as graphs: Terminology and definitions. Journal of Solid State Chemistry 178, 2480–2485 (2005)CrossRefGoogle Scholar
  11. 11.
    Deza, M., Grishukhin, V., Shtogrin, M.: Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices, ch.9. World Scientific Publishing Company (2004)Google Scholar
  12. 12.
    Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Springer (1997)Google Scholar
  13. 13.
    Eon, J.-G.: Infinite geodesic paths and fibers, new topological invariants in periodic graphs. Acta Crystallographica Section A 63, 53–65 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Fu, N., Hashikura, A., Imai, H.: Proximity and motion planning on l 1-embeddable tilings. In: Proceedings of the Eighth International Symposium on Voronoi Diagrams in Science and Engineering, pp. 150–159 (2011)Google Scholar
  16. 16.
    Fu, N., Imai, H., Moriyama, S.: Voronoi diagrams on periodic graphs. In: Proceedings of the Seventh International Symposium on Voronoi Diagrams in Science and Engineering, pp. 189–198 (2010)Google Scholar
  17. 17.
    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
  18. 18.
    Karp, R., Miller, R., Winograd, A.: The organization of computations for uniform recurrence equiations. Journal of the ACM 14, 563–590 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Klein, R., Wood, D.: Voronoi diagrams based on general metrics in the plane. In: Cori, R., Wirsing, M. (eds.) STACS 1988. LNCS, vol. 294, pp. 281–291. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  20. 20.
    Vaidya, P.M.: Geometry helps in matching. SIAM Journal on Computing 18, 1201–1225 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Norie Fu
    • 1
  • Akihiro Hashikura
    • 1
  • Hiroshi Imai
    • 1
  1. 1.Department of Computer ScienceUniversity of TokyoJapan

Personalised recommendations