Abstract
The ℓ1 -embedding problem of a graph is the problem to find a map from its vertex set to ℝd 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 ℝd. 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Cohen, E., Megiddo, N.: Recognizing properties of periodic graphs. Applied Geometry and Discrete Mathematics 4, 135–146 (1991)
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)
Feutrier, P.: Parametric integer programming. RAIRO Recherche Opérationnelle 22, 243–268 (1988)
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)
Höfting, F., Wanke, E.: Minimum cost paths in periodic graphs. SIAM Journal on Computing 24(5), 1051–1067 (1995)
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)
Iwano, K., Steiglitz, K.: Planarity testing of doubly periodic infinite graphs. Networks 18, 205–222 (1988)
Karp, R., Miller, R., Winograd, A.: The organization of computations for uniform recurrence equiations. Journal of the ACM 14, 563–590 (1967)
Verdoolaege, S.: barvinok: User guide (2007), http://freshmeat.net/projects/barvinok/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Fu, N. (2014). ℓ1-Embeddability of 2-Dimensional ℓ1-Rigid Periodic Graphs. In: Pal, S.P., Sadakane, K. (eds) Algorithms and Computation. WALCOM 2014. Lecture Notes in Computer Science, vol 8344. Springer, Cham. https://doi.org/10.1007/978-3-319-04657-0_32
Download citation
DOI: https://doi.org/10.1007/978-3-319-04657-0_32
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-04656-3
Online ISBN: 978-3-319-04657-0
eBook Packages: Computer ScienceComputer Science (R0)