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Transactions on Computational Science XX pp 72–91Cite as

Proximity and Motion Planning on ℓ1-Rigid Planar Periodic Graphs

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Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 8110))

Abstract

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.

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References

  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)

    Article  Google Scholar 

  2. Karp, R., Li, S.-Y.: Two special cases of the assignment problem. Discrete Mathematics 13, 129–142 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barequet, G., Dickerson, M., Goodrich, M.: Voronoi diagrams for convex polygon-offset distance functions. Discrete and Computational Geormetry 25(2), 271–291 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chavey, D.: Tilings by regular polygons – ii: A catalog of tilings. Computers & Mathematics with Applications 17, 147–165 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chepoi, V., Deza, M., Grishukhin, V.: Clin d’oeil on L 1-embeddable planar graphs. Discrete Applied Mathematics 80(1), 3–19 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cohen, E., Megiddo, N.: Recognizing properties of periodic graphs. Applied Geometry and Discrete Mathematics 4, 135–146 (1991)

    MathSciNet  Google Scholar 

  9. Călinescu, G., Dumitrescu, A., Pach, J.: Reconfigurations in graphs and grids. SIAM Jounal on Discrete Mathematics 22, 124–138 (2008)

    Article  MATH  Google Scholar 

  10. Delgado-Friedrichs, O., O’Keeffe, M.: Crystal nets as graphs: Terminology and definitions. Journal of Solid State Chemistry 178, 2480–2485 (2005)

    Article  Google Scholar 

  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. Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Springer (1997)

    Google Scholar 

  13. Eon, J.-G.: Infinite geodesic paths and fibers, new topological invariants in periodic graphs. Acta Crystallographica Section A 63, 53–65 (2007)

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  18. Karp, R., Miller, R., Winograd, A.: The organization of computations for uniform recurrence equiations. Journal of the ACM 14, 563–590 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  20. Vaidya, P.M.: Geometry helps in matching. SIAM Journal on Computing 18, 1201–1225 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Tokyo, Japan

    Norie Fu, Akihiro Hashikura & Hiroshi Imai

Authors
  1. Norie Fu
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  2. Akihiro Hashikura
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  3. Hiroshi Imai
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Editor information

Editors and Affiliations

  1. University of Calgary, Calgary, AB, Canada

    Marina L. Gavrilova

  2. CloudFabriQ Ltd., Warnford Court, 29 Throgmorton Street, EC2N 2AT, London, UK

    C. J. Kenneth Tan

  3. Rutgers University, New Brunswick, NJ, USA

    Bahman Kalantari

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Fu, N., Hashikura, A., Imai, H. (2013). Proximity and Motion Planning on ℓ1-Rigid Planar Periodic Graphs. In: Gavrilova, M.L., Tan, C.J.K., Kalantari, B. (eds) Transactions on Computational Science XX. Lecture Notes in Computer Science, vol 8110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41905-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-41905-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-41904-1

  • Online ISBN: 978-3-642-41905-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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