Advertisement

Fast and Compact Planar Embeddings

  • Leo Ferres
  • José Fuentes
  • Travis Gagie
  • Meng He
  • Gonzalo Navarro
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

There are many representations of planar graphs but few are as elegant as Turán’s (1984): it is simple and practical, uses only four bits per edge, can handle multi-edges and can store any specified embedding. Its main disadvantage has been that “it does not allow efficient searching” (Jacobson, 1989). In this paper we show how to add a sublinear number of bits to Turán’s representation such that it supports fast navigation, thus overcoming this disadvantage. Other data structures for planar embeddings may be asymptotically faster or smaller but ours is simpler, and that can be a theoretical as well as a practical advantage: e.g., we show how our structure can be built efficiently in parallel.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Awerbuch, B., Shiloach, Y.: New connectivity and MSF algorithms for shuffle-exchange network and PRAM. IEEE Trans. Computers 36(10), 1258–1263 (1987)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bader, D.A., Cong, G.: A fast, parallel spanning tree algorithm for symmetric multiprocessors (SMPs). J. Parallel and Distributed Computing 65, 994–1006 (2005)CrossRefMATHGoogle Scholar
  3. 3.
    Barbay, J., Aleardi, L.C., He, M., Munro, J.I.: Succinct representation of labeled graphs. Algorithmica 62, 224–257 (2012)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Biggs, N.: Spanning trees of dual graphs. J. Comb. Theory, Series B, 11: 127–131 (1971)Google Scholar
  5. 5.
    Blandford, D.K., Blelloch, G.E., Kash, I.A.: Compact representations of separable graphs. In: SODA, pp. 679–688 (2003)Google Scholar
  6. 6.
    Blelloch, G.E., Farzan, A.: Succinct representations of separable graphs. In: Amir, A., Parida, L. (eds.) CPM 2010. LNCS, vol. 6129, pp. 138–150. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-13509-5_13 CrossRefGoogle Scholar
  7. 7.
    Aleardi, L.C., Devillers, O., Schaeffer, G.: Succinct representations of planar maps. TCS 408, 174–187 (2008)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Chiang, Y.-T., Lin, C.-C., Lu, H.-I.: Orderly spanning trees with applications. SIAM J. Comp. 34, 924–945 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cong, G., Bader, D.A.: The Euler tour technique and parallel rooted spanning tree. In: ICPP, pp. 448–457 (2004)Google Scholar
  10. 10.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Multithreaded algorithms. In: Introduction to Algorithms, pp. 772–812. MIT Press (2009)Google Scholar
  11. 11.
    Eppstein, D.: Dynamic generators of topologically embedded graphs. In: SODA, pp. 599–608 (2003)Google Scholar
  12. 12.
    Ferres, L., Fuentes-Sepúlveda, J., He, M., Zeh, N.: Parallel construction of succinct trees. In: Bampis, E. (ed.) SEA 2015. LNCS, vol. 9125, pp. 3–14. Springer, Cham (2015). doi: 10.1007/978-3-319-20086-6_1 CrossRefGoogle Scholar
  13. 13.
    Fusy, É., Schaeffer, G., Poulalhon, D.: Dissections, orientations, and trees with applications to optimal mesh encoding and random sampling. TALG 4, 19 (2008)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    He, X., Kao, M., Lu, H.: Linear-time succinct encodings of planar graphs via canonical orderings. SIAM J. Discrete Math. 12, 317–325 (1999)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jacobson, G.: Space-efficient static trees and graphs. In: FOCS, pp. 549–554 (1989)Google Scholar
  16. 16.
    Kao, M., Teng, S., Toyama, K.: An optimal parallel algorithm for planar cycle separators. Algorithmica 14, 398–408 (1995)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Keeler, K., Westbrook, J.: Short encodings of planar graphs and maps. DAM 58, 239–252 (1995)MathSciNetMATHGoogle Scholar
  18. 18.
    Labeit, J., Shun, J., Blelloch, G.E.: Parallel lightweight wavelet tree, suffix array and FM-index construction. In: DCC, pp. 33–42 (2016)Google Scholar
  19. 19.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM J. Applied Math. 36, 177–189 (1979)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Munro, J.I., Nicholson, P.K.: Compressed representations of graphs. In: Encyclopedia of Algorithms, pp. 382–386. Springer (2016)Google Scholar
  21. 21.
    Navarro, G.: Compact Data Structures: A Practical Approach. Cambridge University Press (2016)Google Scholar
  22. 22.
    Riley, T.R., Thurston, W.P.: The absence of efficient dual pairs of spanning trees in planar graphs. Electronic J. Comb. 13 (2006)Google Scholar
  23. 23.
    Shiloach, Y., Vishkin, U.: An o(log n) parallel connectivity algorithm. J. Algorithms 3(1), 57–67 (1982)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Shun, J., Dhulipala, L., Blelloch, G.E.: A simple and practical linear-work parallel algorithm for connectivity. In: SPAA, pp. 143–153 (2014)Google Scholar
  25. 25.
    Turán, A.: On the succinct representation of graphs. DAM 8, 289–294 (1984)MathSciNetMATHGoogle Scholar
  26. 26.
    Tutte, W.T.: A census of planar maps. Canadian J. Math. 15, 249–271 (1963)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Yannakakis, M.: Embedding planar graphs in four pages. JCSS 38, 36–67 (1989)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Leo Ferres
    • 1
  • José Fuentes
    • 2
  • Travis Gagie
    • 3
  • Meng He
    • 4
  • Gonzalo Navarro
    • 2
  1. 1.Faculty of EngineeringUniversidad del DesarrolloConcepciónChile
  2. 2.CeBiB; Department of Computer ScienceUniversity of ChileSantiagoChile
  3. 3.CeBiB; EITDiego Portales UniversitySantiagoChile
  4. 4.Faculty of Computer ScienceDalhousie UniversityHalifaxCanada

Personalised recommendations