Document Open Access Logo

Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

Authors Mikkel Abrahamsen, Stephen Alstrup, Jacob Holm, Mathias Bæk Tejs Knudsen, Morten Stöckel

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2017.128.pdf
  • Filesize: 0.55 MB
  • 14 pages

Document Identifiers

Author Details

Mikkel Abrahamsen
Stephen Alstrup
Jacob Holm
Mathias Bæk Tejs Knudsen
Morten Stöckel

Cite As Get BibTex

Mikkel Abrahamsen, Stephen Alstrup, Jacob Holm, Mathias Bæk Tejs Knudsen, and Morten Stöckel. Near-Optimal Induced Universal Graphs for Bounded Degree Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 128:1-128:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ICALP.2017.128

Abstract

A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U. 

We give upper and lower bounds for the size of induced universal graphs for the family of graphs with n vertices of maximum degree D. Our new bounds improve several previous results except for the special cases where D is either near-constant or almost n/2. For constant even D Butler [Graphs and Combinatorics 2009] has shown O(n^(D/2)) and recently Alon and Nenadov [SODA 2017] showed the same bound for constant odd D. For constant D Butler also gave a matching lower bound. For generals graphs, which corresponds to D = n, Alon [Geometric and Functional Analysis, to appear] proved the existence of an induced universal graph with (1+o(1)) \cdot 2^((n-1)/2) vertices, leading to a smaller constant than in the previously best known bound of 16 * 2^(n/2) by Alstrup, Kaplan, Thorup, and Zwick [STOC 2015].

In this paper we give the following lower and upper bound of

    binom(floor(n/2))(floor(D/2)) * n^(-O(1))

and

    binom(floor(n/2))(floor(D/2)) * 2^(O(sqrt(D log D) * log(n/D))),

respectively, where the upper bound is the main contribution. The proof that it is an induced universal graph relies on a randomized argument. We also give a deterministic upper bound of O(n^k / (k-1)!). These upper bounds are the best known when D <= n/2 - tilde-Omega(n^(3/4)) and either D is even and D = omega(1) or D is odd and D = omega(log n/log log n). In this range we improve asymptotically on the previous best known results by Butler [Graphs and Combinatorics 2009], Esperet, Arnaud and Ochem [IPL 2008], Adjiashvili and Rotbart [ICALP 2014], Alon and Nenadov [SODA 2017], and Alon [Geometric and Functional Analysis, to appear].

Subject Classification

Keywords
  • Adjacency labeling schemes
  • Bounded degree graphs
  • Induced universal graphs
  • Distributed computing

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. S. Abiteboul, H. Kaplan, and T. Milo. Compact labeling schemes for ancestor queries. In Proc. of the 12th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 547-556, 2001. Google Scholar
  2. M. Abrahamsen, S. Alstrup, J. Holm, M. B. T. Knudsen, and M. Stöckel. Near-optimal induced universal graphs for bounded degree graphs. CoRR, abs/1607.04911, 2016. URL: http://arxiv.org/abs/1607.04911.
  3. D. Adjiashvili and N. Rotbart. Labeling schemes for bounded degree graphs. In 41st International Colloquium on Automata, Languages, and Programming (ICALP), pages 375-386, 2014. Google Scholar
  4. N. Alon. private communication, 2016. Google Scholar
  5. N. Alon. Asymptotically optimal induced universal graphs, 2016. [Online; accessed 5-July-2016]. URL: http://www.tau.ac.il/~nogaa/PDFS/induniv1.pdf.
  6. N. Alon and M. Capalbo. Sparse universal graphs for bounded-degree graphs. Random Structures &Algorithms, 31(2):123-133, 2007. Google Scholar
  7. N. Alon and M. Capalbo. Optimal universal graphs with deterministic embedding. In Proc. of the 19th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 373-378, 2008. Google Scholar
  8. N. Alon and R. Nenadov. Optimal induced universal graphs for bounded-degree graphs. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 1149-1157, 2017. URL: http://dx.doi.org/10.1137/1.9781611974782.74.
  9. N. Alon and J. H. Spencer. The probabilistic method. Wiley Publishing, 2000. Google Scholar
  10. S. Alstrup, S. Dahlgaard, and M. B. T. Knudsen. Optimal induced universal graphs and labeling schemes for trees. In Proc. 56th Annual Symp. on Foundations of Computer Science (FOCS), 2015. Google Scholar
  11. S. Alstrup, H. Kaplan, M. Thorup, and U. Zwick. Adjacency labeling schemes and induced-universal graphs. In Proc. of the 47th Annual ACM Symp. on Theory of Computing (STOC), pages 625-634, 2015. Google Scholar
  12. S. Alstrup and T. Rauhe. Small induced-universal graphs and compact implicit graph representations. In Proc. 43rd Annual Symp. on Foundations of Computer Science (FOCS), pages 53-62, 2002. Google Scholar
  13. L. Babai, F. R. K. Chung, P. Erdös R. L. Graham, and J. Spencer. On graphs which contain all sparse graphs. Ann. discrete Math., 12:21-26, 1982. Google Scholar
  14. S. N. Bhatt, F. R. K. Chung, F. T. Leighton, and A. L. Rosenberg. Universal graphs for bounded-degree trees and planar graphs. SIAM J. Discrete Math., 2(2):145-155, 1989. Google Scholar
  15. N. Bonichon, C. Gavoille, and A. Labourel. Short labels by traversal and jumping. In Structural Information and Communication Complexity, pages 143-156. Springer, 2006. Include proof for binary trees and caterpillars. Google Scholar
  16. M. A. Breuer. Coding the vertexes of a graph. IEEE Trans. on Information Theory, IT-12:148-153, 1966. Google Scholar
  17. M. A. Breuer and J. Folkman. An unexpected result on coding vertices of a graph. J. of Mathemathical analysis and applications, 20:583-600, 1967. Google Scholar
  18. S. Butler. Induced-universal graphs for graphs with bounded maximum degree. Graphs and Combinatorics, 25(4):461-468, 2009. URL: http://dx.doi.org/10.1007/s00373-009-0860-x.
  19. G. Chartrand, H. V. Kronk, and C. E. Wall. The point-arboricity of a graph. Israel J. of Mathematics, 6(2):169-175, 1968. URL: http://dx.doi.org/10.1007/BF02760181.
  20. F. R. K. Chung. Universal graphs and induced-universal graphs. J. of Graph Theory, 14(4):443-454, 1990. Google Scholar
  21. F. R. K. Chung and R. L. Graham. On graphs which contain all small trees. J. of combinatorial theory, Series B, 24(1):14-23, 1978. Google Scholar
  22. F. R. K. Chung and R. L. Graham. On universal graphs. Ann. Acad. Sci., 319:136-140, 1979. Google Scholar
  23. F. R. K. Chung and R. L. Graham. On universal graphs for spanning trees. J. London Math. Soc., 27:203-211, 1983. Google Scholar
  24. F. R. K. Chung, R. L. Graham, and N. Pippenger. On graphs which contain all small trees ii. Colloquia Mathematica, pages 213-223, 1976. Google Scholar
  25. E. Cohen, H. Kaplan, and T. Milo. Labeling dynamic XML trees. SIAM J. Comput., 39(5):2048-2074, 2010. URL: http://dx.doi.org/10.1137/070687633.
  26. L. J. Cowen. Compact routing with minimum stretch. J. of Algorithms, 38:170-183, 2001. See also SODA'91. Google Scholar
  27. T. Eilam, C. Gavoille, and D. Peleg. Compact routing schemes with low stretch factor. J. of Algorithms, 46(2):97-114, 2003. URL: http://dx.doi.org/10.1016/S0196-6774(03)00002-6.
  28. P. Erdos and L. Lovász. Problems and results on 3-chromatic hypergraphs and some related questions. Infinite and finite sets, 10(2):609-627, 1975. Google Scholar
  29. L. Esperet, A. Labourel, and P. Ochem. On induced-universal graphs for the class of bounded-degree graphs. Inf. Process. Lett., 108(5):255-260, 2008. URL: http://dx.doi.org/10.1016/j.ipl.2008.04.020.
  30. P. Fraigniaud and C. Gavoille. Routing in trees. In 28^th International Colloquium on Automata, Languages and Programming (ICALP), pages 757-772, 2001. Google Scholar
  31. P. Fraigniaud and A. Korman. On randomized representations of graphs using short labels. In Proc. of the 21st Annual Symp. on Parallelism in Algorithms and Architectures (SPAA), pages 131-137, 2009. URL: http://dx.doi.org/10.1145/1583991.1584031.
  32. P. Fraigniaud and A. Korman. Compact ancestry labeling schemes for XML trees. In Proc. of the 21st annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 458-466, 2010. Google Scholar
  33. C. Gavoille and A. Labourel. Shorter implicit representation for planar graphs and bounded treewidth graphs. In Algorithms-ESA, pages 582-593. Springer, 2007. Google Scholar
  34. C. Gavoille and D. Peleg. Compact and localized distributed data structures. Distributed Computing, 16(2-3):111-120, 2003. URL: http://dx.doi.org/10.1007/s00446-002-0073-5.
  35. S. Kannan, M. Naor, and S. Rudich. Implicit representation of graphs. SIAM J. Disc. Math., 5(4):596-603, 1992. See also STOC'88. Google Scholar
  36. H. Kaplan, T. Milo, and R. Shabo. A comparison of labeling schemes for ancestor queries. In Proc. of the 13th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 954-963, 2002. Google Scholar
  37. A. Korman. Labeling schemes for vertex connectivity. ACM Trans. Algorithms, 6(2):39:1-39:10, 2010. URL: http://dx.doi.org/10.1145/1721837.1721855.
  38. A. Korman and D. Peleg. Labeling schemes for weighted dynamic trees. Inf. Comput., 205(12):1721-1740, 2007. Google Scholar
  39. A. Liebenau and N. Wormald. Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph. arXiv preprint arXiv:1702.08373, 2017. Google Scholar
  40. L. Lovasz. On decomposition of graphs. Studia Sci. Math. Hungar, 1:237-238, 1966. Google Scholar
  41. L. Lovász and M.D. Plummer. Matching Theory. AMS Chelsea Publishing Series. American Mathematical Soc., 2009. Google Scholar
  42. V. V. Lozin and G. Rudolf. Minimal universal bipartite graphs. Ars Comb., 84, 2007. Google Scholar
  43. B. D. McKay and N. C. Wormald. Asymptotic enumeration by degree sequence of graphs of high degree. European Journal of Combinatorics, 11(6):565-580, 1990. Google Scholar
  44. B. D. McKay and N. C. Wormald. Asymptotic enumeration by degree sequence of graphs with degreeso (n 1/2). Combinatorica, 11(4):369-382, 1991. Google Scholar
  45. J. W. Moon. On minimal n-universal graphs. Proc. of the Glasgow Mathematical Association, 7(1):32-33, 1965. Google Scholar
  46. J. W. Moon. Topics on tournaments. Holt, Rinehart and Winston, 1968. Google Scholar
  47. J. H. Müller. Local structure in graph classes. PhD thesis, Georgia Institute of Technology, 1988. Google Scholar
  48. D. Peleg. Informative labeling schemes for graphs. In Proc. 25th Symp. on Mathematical Foundations of Computer Science, pages 579-588, 2000. Google Scholar
  49. R. Rado. Universal graphs and universal functions. Acta. Arith., 9:331-340, 1964. Google Scholar
  50. N. Santoro and R. Khatib. Labeling and implicit routing in networks. The computer J., 28:5-8, 1985. Google Scholar
  51. M. Thorup and U. Zwick. Approximate distance oracles. J. of the ACM, 52(1):1-24, 2005. See also STOC'01. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact