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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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)
Karp, R., Li, S.-Y.: Two special cases of the assignment problem. Discrete Mathematics 13, 129–142 (1975)
Barequet, G., Dickerson, M., Goodrich, M.: Voronoi diagrams for convex polygon-offset distance functions. Discrete and Computational Geormetry 25(2), 271–291 (2001)
Chavey, D.: Tilings by regular polygons – ii: A catalog of tilings. Computers & Mathematics with Applications 17, 147–165 (1989)
Chepoi, V., Deza, M., Grishukhin, V.: Clin d’oeil on L 1-embeddable planar graphs. Discrete Applied Mathematics 80(1), 3–19 (1997)
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)
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)
Cohen, E., Megiddo, N.: Recognizing properties of periodic graphs. Applied Geometry and Discrete Mathematics 4, 135–146 (1991)
Călinescu, G., Dumitrescu, A., Pach, J.: Reconfigurations in graphs and grids. SIAM Jounal on Discrete Mathematics 22, 124–138 (2008)
Delgado-Friedrichs, O., O’Keeffe, M.: Crystal nets as graphs: Terminology and definitions. Journal of Solid State Chemistry 178, 2480–2485 (2005)
Deza, M., Grishukhin, V., Shtogrin, M.: Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices, ch.9. World Scientific Publishing Company (2004)
Deza, M., Laurent, M.: Geometry of Cuts and Metrics. Springer (1997)
Eon, J.-G.: Infinite geodesic paths and fibers, new topological invariants in periodic graphs. Acta Crystallographica Section A 63, 53–65 (2007)
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)
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)
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)
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)
Karp, R., Miller, R., Winograd, A.: The organization of computations for uniform recurrence equiations. Journal of the ACM 14, 563–590 (1967)
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)
Vaidya, P.M.: Geometry helps in matching. SIAM Journal on Computing 18, 1201–1225 (1989)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
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)