Compact Navigation and Distance Oracles for Graphs with Small Treewidth

  • Arash Farzan
  • Shahin Kamali
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)


Given an unlabeled, unweighted, and undirected graph with n vertices and small (but not necessarily constant) treewidth k, we consider the problem of preprocessing the graph to build space-efficient encodings (oracles) to perform various queries efficiently. We assume the word RAM model where the size of a word is Ω(logn) bits.

The first oracle, we present, is the navigation oracle which facilitates primitive navigation operations of adjacency, neighborhood, and degree queries. By way of an enumerate argument, which is of independent interest, we show the space requirement of the oracle is optimal to within lower order terms for all treewidths. The oracle supports the mentioned queries all in constant worst-case time. The second oracle, we present, is an exact distance oracle which facilitates distance queries between any pair of vertices (i.e., an all-pair shortest-path oracle). The space requirement of the oracle is also optimal to within lower order terms. Moreover, the distance queries perform in O(k 2log3 k) time. Particularly, for the class of graphs of our interest, graphs of bounded treewidth (where k is constant), the distances are reported in constant worst-case time.


Lower Order Term Tree Decomposition Neighborhood Query Tree Vertex Distance Query 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barbay, J., Aleardi, L.C., He, M., Ian Munro, J.: Succinct representation of labeled graphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 316–328. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Blandford, D.K., Blelloch, G.E., Kash, I.A.: Compact representations of separable graphs. In: Proceedings of 14th ACM-SIAM Symposium on Discrete Algorithms, SODA 2003, pp. 679–688 (2003)Google Scholar
  3. 3.
    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)CrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L.: NC-algorithms for graphs with small treewidth. In: van Leeuwen, J. (ed.) WG 1988. LNCS, vol. 344, pp. 1–10. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybernetica 11, 1–23 (1993)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L.: Treewidth: Characterizations, applications, and computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Chan, T.M.: All-pairs shortest paths for unweighted undirected graphs in o(mn) time. In: Proc. 17th ACM-SIAM Symposium on Discrete Algorithm, SODA 2006, pp. 514–523 (2006)Google Scholar
  8. 8.
    Dujmovic, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. Discrete Comput. Geom. 37, 641–670 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Farzan, A.: Succinct Representation of Trees and Graphs. PhD thesis, School of Computer Science, University of Waterloo (2009)Google Scholar
  10. 10.
    Farzan, A., Ian Munro, J.: Succinct representations of arbitrary graphs. In: Halperin, D., Mehlhorn, K. (eds.) Esa 2008. LNCS, vol. 5193, pp. 393–404. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Gavoille, C., Hanusse, N.: On Compact Encoding of Pagenumber k Graphs. Discrete Mathematics & Theoretical Computer Science 10(3), 23–34 (2008)MathSciNetGoogle Scholar
  12. 12.
    Gavoille, C., Labourel, A.: Shorter implicit representation for planar graphs and bounded treewidth graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 582–593. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Grossi, R., Gupta, A., Vitter, J.S.: High-order entropy-compressed text indexes. In: Proc. 14th ACM-SIAM Symposium on Discrete Algorithms, SODA 2003, pp. 841–850 (2003)Google Scholar
  14. 14.
    Harary, F., Robinson, R.W., Schwenk, A.J.: Twenty-step algorithm for determining the asymptotic number of trees of various species: Corrigenda. Journal of the Australian Mathematical Society 41(A), 325 (1986)CrossRefzbMATHGoogle Scholar
  15. 15.
    He, M., Ian Munro, J., Srinivasa Rao, S.: Succinct Ordinal Trees Based on Tree Covering. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 509–520. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  16. 16.
    Nitto, I., Venturini, R.: On compact representations of all-pairs-shortest-path-distance matrices. In: Ferragina, P., Landau, G.M. (eds.) CPM 2008. LNCS, vol. 5029, pp. 166–177. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Raman, R., Raman, V., Satti, S.R.: Succinct indexable dictionaries with applications to encoding k-ary trees, prefix sums and multisets. ACM Trans. Algorithms 3(4), 43 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Arash Farzan
    • 1
  • Shahin Kamali
    • 2
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.Cheriton School of Computer ScienceUniversity of WaterlooCanada

Personalised recommendations