Advertisement

Algorithmica

, Volume 69, Issue 1, pp 92–116 | Cite as

Compact Navigation and Distance Oracles for Graphs with Small Treewidth

  • Arash Farzan
  • Shahin Kamali
Article

Abstract

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 enumeration argument, which is of interest in its own right, we show the space requirement of the oracle is optimal to within lower order terms for all graphs with n vertices and treewidth k. 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-pairs 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 3log3 k) time. Particularly, for the class of graphs of popular interest, graphs of bounded treewidth (where k is constant), the distances are reported in constant worst-case time.

Keywords

Graph decomposition Treewidth Navigation oracles Distance oracles 

Notes

Acknowledgements

We are thankful to Magnus Wahlstrom for helpful discussions.

References

  1. 1.
    Aleardi, L.C., Devillers, O., Schaeffer, G.: Succinct representation of triangulations with a boundary. In: Proceedings of the 9th Workshop on Algorithms and Data Structures Google Scholar
  2. 2.
    Barbay, J., Aleardi, L.C., He, M., Munro, J.I.: Succinct representation of labeled graphs. Algorithmica 62(1–2), 224–257 (2012) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Baswana, S., Gaur, A., Sen, S., Upadhyay, J.: Distance oracles for unweighted graphs: Breaking the quadratic barrier with constant additive error. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Part I Google Scholar
  4. 4.
    Blandford, D.K., Blelloch, G.E., Kash, I.A.: Compact representations of separable graphs. In: Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms Google Scholar
  5. 5.
    Blelloch, G.E., Farzan, A.: Succinct representations of separable graphs. In: Proceedings 21st Conference on Combinatorial Pattern Matching Google Scholar
  6. 6.
    Bodirsky, M., Giménez, O., Kang, M., Noy, M.: Enumeration and limit laws for series-parallel graphs. Eur. J. Comb. 28(8), 2091–2105 (2007) CrossRefMATHGoogle Scholar
  7. 7.
    Bodlaender, H.L.: NC-algorithms for graphs with small treewidth. In: Proceedings of the 14th International Workshop on Graph-Theoretic Concepts in Computer Science Google Scholar
  8. 8.
    Bodlaender, H.L.: Treewidth: Characterizations, applications, and computations. In: Proceedings of the 32nd International Workshop on Graph-Theoretic Concepts in Computer Science Google Scholar
  9. 9.
    Bodlaender, H.L.: A tourist guide through treewidth. Acta Cybern. 11, 1–23 (1993) MATHMathSciNetGoogle Scholar
  10. 10.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Bodlaender, H.L., Fellows, M.R., Warnow, T.: Two strikes against perfect phylogeny. In: Proceedings of the 19th International Colloquium on Automata, Languages and Programming Google Scholar
  12. 12.
    Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms 18, 238–255 (1995) CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Bouchitté, V., Kratsch, D., Müller, H., Todinca, I.: On treewidth approximations. Discrete Appl. Math. 136, 183–196 (2004) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Castelli Aleardi, L., Devillers, O., Schaeffer, G.: Succinct representations of planar maps. Theor. Comput. Sci. 408, 174–187 (2008) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Chan, T.M.: All-pairs shortest paths for unweighted undirected graphs in o(mn) time. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithm Google Scholar
  16. 16.
    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: Proceedings of the 25th International Colloquium on Automata, Languages and Programming Google Scholar
  17. 17.
    de Fluiter, B.: Algorithms for graphs of small treewidth. PhD thesis, Utrecht University (1997) Google Scholar
  18. 18.
    Deo, N., Krishnamoorthy, M.S., Langston, M.A.: Exact and approximate solutions for the gate matrix layout problem. IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 6(1), 79–84 (1987) CrossRefGoogle Scholar
  19. 19.
    Dorn, B., Hüffner, F., Krüger, D., Niedermeier, R., Uhlmann, J.: Exploiting bounded signal flow for graph orientation based on cause-effect pairs. In: Proceedings of the First International ICST Conference on Theory and Practice of Algorithms in Computer Systems Google Scholar
  20. 20.
    Dujmovic, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. Discrete Comput. Geom. 37, 641–670 (2007) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Erdös, P., Rényi, A.: Asymmetric graphs. Acta Math. Hung. 14, 295–315 (1963) MATHGoogle Scholar
  22. 22.
    Farzan, A.: Succinct representation of trees and graphs. PhD thesis, School of Computer Science, University of Waterloo (2009) Google Scholar
  23. 23.
    Farzan, A., Munro, J.I.: A uniform approach towards succinct representation of trees. In: Proceedings of the 11th Scandinavian Workshop on Algorithm Theory Google Scholar
  24. 24.
    Farzan, A., Munro, J.I.: Succinct representations of arbitrary graphs. In: Proceedings of the 16th European Symposium on Algorithms (2008) Google Scholar
  25. 25.
    Farzan, A., Raman, R., Rao, S.S.: Universal succinct representations of trees? In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming, Part I Google Scholar
  26. 26.
    Ferragina, P., Nitto, I., Venturini, R.: On compact representations of all-pairs-shortest-path-distance matrices. Theor. Comput. Sci. 411(34–36), 3293–3300 (2010) CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Fredman, M.L., Willard, D.E.: Surpassing the information theoretic bound with fusion trees Google Scholar
  28. 28.
    Gavoille, C., Hanusse, N.: On compact encoding of pagenumber k graphs. Discrete Math. Theor. Comput. Sci. 10(3), 23–34 (2008) MathSciNetGoogle Scholar
  29. 29.
    Gavoille, C., Labourel, A.: Shorter implicit representation for planar graphs and bounded treewidth graphs. In: Proceedings of the 15th European Symposium on Algorithms Google Scholar
  30. 30.
    Grossi, R., Gupta, A., Vitter, J.S.: High-order entropy-compressed text indexes. In: Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms Google Scholar
  31. 31.
    Harary, F., Robinson, R.W., Schwenk, A.J.: Twenty-step algorithm for determining the asymptotic number of trees of various species: corrigenda. J. Aust. Math. Soc. 41(A), 325 (1986) CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    He, M., Munro, J.I., Rao, S.S.: Succinct ordinal trees based on tree covering. In: Proceedings of the 34th International Colloquium on Automata, Languages and Programming, Part I Google Scholar
  33. 33.
    Jacobson, G.: Space-efficient static trees and graphs Google Scholar
  34. 34.
    Jansson, J., Sadakane, K., Sung, W.-K.: Ultra-succinct representation of ordered trees with applications. J. Comput. Syst. Sci. 78(2), 619–631 (2012) CrossRefMATHMathSciNetGoogle Scholar
  35. 35.
    Kannan, S., Naor, M., Rudich, S.: Implicit representation of graphs. SIAM J. Discrete Math. 5(4), 596–603 (1992) CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Keeler, K., Westbrook, J.: Short encodings of planar graphs and maps. Discrete Appl. Math. 58, 239–252 (1995) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Munro, J.I.: Succinct data structures. Electron. Notes Theor. Comput. Sci. 91, 3 (2004) CrossRefGoogle Scholar
  38. 38.
    Munro, J.I., Raman, V.: Succinct representation of balanced parentheses, static trees and planar graphs. In: Proceedings of the 38th Symposium on Foundations of Computer Science Google Scholar
  39. 39.
    Otter, R.: The number of trees. Ann. Math. 49(3), 583–599 (1948) CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Patrascu, M.: Succincter. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science Google Scholar
  41. 41.
    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) CrossRefMathSciNetGoogle Scholar
  42. 42.
    Sadakane, K., Navarro, G.: Fully-functional succinct trees. In: Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms Google Scholar
  43. 43.
    Thorup, M.: All structured programs have small tree width and good register allocation. Inf. Comput. 142, 159–181 (1998) CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Thorup, M., Zwick, U.: Approximate distance oracles. J. ACM 52(1), 1–24 (2005) CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Turán, G.: On the succinct representation of graphs. Discrete Appl. Math. 8, 289–294 (1984) CrossRefMATHMathSciNetGoogle Scholar
  46. 46.
    Zwick, U.: Exact and approximate distances in graphs—a survey. In: Proceedings of the 9th European Symposium on Algorithms Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany
  2. 2.David Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations