Abstract
In order to find a 2-factor of a graph, there exists a O(n1.5) deterministic algorithm [7] and a O(n3) randomized algorithm [14]. In this paper, we propose novel O(nlog3nloglogn) algorithms to find a 2-factor, if one exists, of a graph in which all n vertices have degree 4 or less. Such graphs are actually dual graphs of quadrilateral and tetrahedral meshes. A 2-factor of such graphs implicitly defines a linear ordering of the mesh primitives in the form of strips. Further, by introducing a few additional primitives, we reduce the number of tetrahedral strips to represent the entire tetrahedral mesh and represent the entire quad surface using a single quad strip.
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Asano, T., Asano, T., Imai, H.: Partitioning a polygonal region into trapezoids. J. ACM 33(2), 290–312 (1986)
Bogomjakov, A., Gotsman, C.: Universal rendering sequences for transparent vertex caching of progressive meshes. Comput. Graph. Forum 21(2), 137–148 (2002)
Bose, P., Toussaint, G.T.: No quadrangulation is extremely odd. In: International Symposium on Algorithms and Computation, pp. 372–381 (1995)
Conn, H.E., O’Rourke, J.: Minimum weight quadrilateralization in O(n3logn) time. In: Proceedings of the 28th Allerton Conference on Communication, Control, and Computing, pp. 788–797 (1990)
Diaz-Gutierrez, P., Bhushan, A., Gopi, M., Pajarola, R.: Constrained Strip Generation and Management for Efficient Interactive 3D Rendering. In: Proceedings of the International Conference on Computer Graphics (2005)
Evans, F., Skiena, S., Varshney, A.: Optimizing triangle strips for fast rendering. In: Proceedings of IEEE Visualization, pp. 319–326. IEEE Press, New York (1996)
Gibbons, A.: Algorithmic graph theory. Cambridge University Press (1985)
Gopi, M., Eppstein, D.: Single strip triangulation of manifolds with arbitrary topology. Comput. Graph. Forum (EUROGRAPHICS) 23(3), 371–379 (2004)
Heighway, E.: A mesh generator for automatically subdividing irregular polygons into quadrilaterals. IEEE Trans. Magnet. 19(6), 2535–2538 (1983)
King, D., Wittenbrink, C.M., Wolters, H.J.: An architecture for interactive tetrahedral volume rendering. Technical report HPL-2000-121 (R.3), HP Laboratories, Palo Alto, CA (2001)
Mallon, P.N., Boo, M., Amor, M., Bruguera, J.: Compression and on the fly rendering using tetrahedral concentric strips. Technical report, University of Santiago de Compostela, Spain (2004)
Mukhopadhyay, A., Jing, Q.: Encoding Quadrilateral Meshes. In: 15th Canadian Conference on Computational Geometry (2003)
Pajarola, R., Rossignac, J., Szymczak, A.: Implant sprays: compression of progressive tetrahedral mesh connectivity. In: Proceedings of IEEE Visualization, pp. 299–305. IEEE Press (1999)
Pandurangan, G.: On a simple randomized algorithm for finding a 2-factor in sparse graphs. Inf. Process. Lett. 95(1), 321–327 (2005)
Pascucci, V.: Isosurface computation made simple: Hardware acceleration, adaptive refinement and tetrahedral stripping. In: Joint EUROGRAPHICS – IEEE TCVG Symposium on Visualization (2004)
Peng, J., Kim, C.S., Kuo, C.C.J.: Technologies for 3d mesh compression: A survey. Technical report, preprint (2005)
Petersen, J.P.C.: Die theorie der regularen graphs (The Theory of Regular Graphs). Acta Math. 15, 193–220 (1891)
Sommer, O., Ertl, T.: Geometry and rendering optimization for the interactive visualization of crash-worthiness simulations. In: Proceedings of the Visual Data Exploration and Analysis Conference in IT&T/SPIE Electronic Imaging, pp. 124–134 (2000)
Szymczak, A., Rossignac, J.: Grow & Fold: Compression of tetrahedral meshes. In: 5th Symposium on Solid Modeling, pp. 54–64 (1999)
Taubin, G.: Constructing hamiltonian triangle strips on quadrilateral meshes. In: International Workshop on Visualization and Mathematics, IBM Research Technical Report RC-22295 (2002)
Thorup, M.: Near-optimal fully-dynamic graph connectivity. In: STOC ’00: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 343–350. ACM Press, New York (2000). DOI http://doi.acm.org/10.1145/335305.335345
Ueng, S.K.: Out-of-core encoding of large tetrahedral meshes. In: Volume Graphics, pp. 95–102 (2003)
Vanecek, P., Svitak, R., Kolingerova, I., Skala, V.: Quadrilateral meshes stripification. Technical report, University of West Bohemia, Czech Republic (2004)
Xiang, X., Held, M., Mitchell, J.S.B.: Fast and effective stripification of polygonal surface models. In: Proceedings of the Symposium on Interactive 3D Graphics, pp. 71–78. ACM Press, New York (1999)
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Diaz-Gutierrez, P., Gopi, M. Quadrilateral and tetrahedral mesh stripification using 2-factor partitioning of the dual graph. Visual Comput 21, 689–697 (2005). https://doi.org/10.1007/s00371-005-0336-9
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DOI: https://doi.org/10.1007/s00371-005-0336-9