Abstract
We consider the problem of designing space efficient solutions for representing triangle meshes. Our main result is a new explicit data structure for compactly representing planar triangulations: if one is allowed to permute input vertices, then a triangulation with n vertices requires at most 4n references (5n references if vertex permutations are not allowed). Our solution combines existing techniques from mesh encoding with a novel use of minimal Schnyder woods. Our approach extends to higher genus triangulations and could be applied to other families of meshes (such as quadrangular or polygonal meshes). As far as we know, our solution provides the most parsimonious data structures for triangulations, allowing constant time navigation in the worst case. Our data structures require linear construction time, and all space bounds hold in the worst case. We have implemented and tested our results, and experiments confirm the practical interest of compact data structures.
This work is supported by ERC (agreement “ERC StG 208471 - ExploreMap”).
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.
References
Alumbaugh, T.J., Jiao, X.: Compact array-based mesh data structures. In: Proc. of the 14th Intern. Meshing Roundtable (IMR), pp. 485–503 (2005)
Barbay, J., Castelli-Aleardi, L., He, M., Munro, J.I.: Succinct representation of labeled graphs. Algorithmica (to appear, 2011); preliminary version in ISAAC 2007
Baumgart, B.G.: Winged-edge polyhedron representation. Technical report, Stanford (1972)
Baumgart, B.G.: A polyhedron representation for computer vision. In: AFIPS National Computer Conference, pp. 589–596 (1975)
Benoit, D., Demaine, E.D., Munro, J.I., Raman, R., Raman, V., Rao, S.S.: Representing trees of higher degree. Algorithmica 43(4), 275–292 (2005)
Blanford, D., Blelloch, G., Kash, I.: Compact representations of separable graphs. In: SODA, pp. 342–351 (2003)
Boissonnat, J.-D., Devillers, O., Pion, S., Teillaud, M., Yvinec, M.: Triangulations in CGAL. Comp. Geometry 22, 5–19 (2002)
Brehm, E.: 3-orientations and Schnyder-three tree decompositions. Master’s thesis, Freie Universitaet Berlin (2000)
Campagna, S., Kobbelt, L., Seidel, H.P.: Direct edges - a scalable representation for triangle meshes. Journal of Graphics Tools 3(4), 1–12 (1999)
Castelli-Aleardi, L., Devillers, O., Mebarki, A.: Catalog Based Representation of 2D triangulations. Internat. J. Comput. Geom. Appl. 21(4), 393–402 (2011)
Castelli-Aleardi, L., Devillers, O., Schaeffer, G.: Succinct Representation of Triangulations with a Boundary. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 134–145. Springer, Heidelberg (2005)
Castelli-Aleardi, L., Devillers, O., Schaeffer, G.: Succinct representations of planar maps. Theor. Comput. Sci. 408(2-3), 174–187 (2008)
Castelli-Aleardi, L., Devillers, O.: Explicit array-based compact data structures for triangulations. INRIA research report 7736 (2011), http://hal.archives-ouvertes.fr/inria-00623762/
Castelli-Aleardi, L., Fusy, E., Lewiner, T.: Schnyder woods for higher genus triangulated surfaces, with applications to encoding. Discr. & Comp. Geom. 42(3), 489–516 (2009)
Chuang, R.C.-N., Garg, A., He, X., Kao, M.-Y., Lu, H.-I.: Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 118–129. Springer, Heidelberg (1998)
de Fraysseix, H., Ossona de Mendez, P.: On topological aspects of orientations. Disc. Math. 229, 57–72 (2001)
Felsner, S.: Convex drawings of planar graphs and the order dimension of 3-polytopes. Order 18, 19–37 (2001)
Guibas, L.J., Stolfi, J.: Primitives for the manipulation of general subdivisions and computation of Voronoi diagrams. ACM Trans. Graph. 4(2), 74–123 (1985)
Gurung, T., Laney, D., Lindstrom, P., Rossignac, J.: SQUAD: Compact representation for triangle meshes. Comput. Graph. Forum 30(2), 355–364 (2011)
Gurung, T., Rossignac, J.: SOT: compact representation for tetrahedral meshes. In: Proc. of the ACM Symp. on Solid and Physical Modeling, pp. 79–88 (2009)
Gurung, T., Luffel, M., Lindstrom, P., Rossignac, J.: LR: compact connectivity representation for triangle meshes. ACM Trans. Graph. 30(4), 67 (2011)
Kallmann, M., Thalmann, D.: Star-vertices: a compact representation for planar meshes with adjacency information. Journal of Graphics Tools 6, 7–18 (2002)
Munro, J.I., Raman, V.: Succinct representation of balanced parentheses and static trees. SIAM J. on Computing 31(3), 762–776 (2001)
Poulalhon, D., Schaeffer, G.: Optimal coding and sampling of triangulations. Algorithmica 46, 505–527 (2006)
Rossignac, J.: Edgebreaker: Connectivity compression for triangle meshes. Transactions on Visualization and Computer Graphics 5, 47–61 (1999)
Schnyder, W.: Embedding planar graphs on the grid. In: SoDA, pp. 138–148 (1990)
Snoeyink, J., Speckmann, B.: Tripod: a minimalist data structure for embedded triangulations. In: Workshop on Comput. Graph Theory and Combinatorics (1999)
Yamanaka, K., Nakano, S.: A compact encoding of plane triangulations with efficient query supports. Inf. Process. Lett. 110, 803–809 (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Castelli Aleardi, L., Devillers, O. (2011). Explicit Array-Based Compact Data Structures for Triangulations. In: Asano, T., Nakano, Si., Okamoto, Y., Watanabe, O. (eds) Algorithms and Computation. ISAAC 2011. Lecture Notes in Computer Science, vol 7074. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25591-5_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-25591-5_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25590-8
Online ISBN: 978-3-642-25591-5
eBook Packages: Computer ScienceComputer Science (R0)