Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Graph-Theoretic Concepts in Computer Science

WG 1994: Graph-Theoretic Concepts in Computer Science pp 232–241Cite as

The maximal f-dependent set problem for planar graphs is in NC

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 903))

Abstract

The maximal f-dependent set (Max-f-DS) problem is the following problem: Given a graph G=(V, E) and a nonnegative integer-valued function f defined on V, find a maximal subset U of V such that no vertex u∈U has degree>f(u) in the subgraph induced by U. Whether the problem is in NC (or RNC) or not is an open question. Concerning this question, only a rather trivial result due to Diks, Garrido, and Lingas is known up to now, which says that the problem can be solved in NC if the maximum value of f is poly-logarithmic in the input size [Proceedings of the 2nd International Symposium on Algorithms, LNCS 557 (1991) 385–395]. In this paper, we show a nontrivial interesting result that the Max-f-DS problem for planar graphs can be solved in O(log5 n) time with O(n) processors on a CRCW PRAM, where n is the input size.

This is a preview of subscription content, log in via an institution.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N. Alon, L. Babai, and A. Itai, A Fast and Simple Randomized Parallel Algorithm for the Maximal Independent Set Problem, J. Algorithms 7 (1986) 567–583.

    Google Scholar 

  2. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications (North-Holland, New York, 1980).

    Google Scholar 

  3. Z.-Z. Chen and X. He, Parallel Algorithms for Maximal Cycle-Free Sets, Submitted for publication.

    Google Scholar 

  4. M. Chrobak and M. Yung, Fast Algorithms for Edge-Coloring Planar Graphs, J. Algorithms 10 (1986) 35–51.

    Google Scholar 

  5. K. Diks, O. Garrido and A. Lingas, Parallel Algorithm for Finding maximal k-Dependent Sets and Maximal f-Matchings, in: Proc. 2nd International Symp. on Algorithms, Lecture Notes in Computer Science, Vol. 557 (Springer, Berlin, 1991) 385–395.

    Google Scholar 

  6. A. Gibbons and W. Rytter, Efficient Parallel Algorithms (Cambridge University Press, Cambridge, 1988).

    Google Scholar 

  7. A.V. Goldberg, S.A. Plotkin and G.E. Shannon, Parallel Symmetry-Breaking in Sparse Graphs, in: Proc. 19th ACM Symp. on Theory of Computing (ACM, 1987) 315–324.

    Google Scholar 

  8. M. Goldberg and T. Spencer, A New Parallel Algorithm for the Maximal Independent Set Problem, SIAM J. Comput. 18 (1989) 419–427.

    Google Scholar 

  9. M. Goldberg and T. Spencer, Constructing a Maximal Independent Set in Parallel, SIAM J. Disc. Math. 2 (1989) 322–328.

    Google Scholar 

  10. A. Israeli and A. Itai, A Fast and Simple Randomized Parallel Algorithm for Maximal Matching, Inform. Process. Lett. 22 (1986) 77–80.

    Google Scholar 

  11. A. Israeli and Y. Shiloach, An Improved Maximal Matching Parallel Algorithm, Inform. Process. Lett. 22 (1986) 57–60.

    Google Scholar 

  12. R.M. Karp and V. Ramachandran, Parallel Algorithms for Shared Memory Machines, in: J. van Leeuwen ed., Handbook of Theoretical Computer Science Vol. A (Elsevier, Amsterdam, 1990) 868–941.

    Google Scholar 

  13. R.M. Karp and A. Wigderson, A Fast Parallel Algorithm for the Maximal Independent Set Problem, J. ACM 32 (1985) 762–773.

    Google Scholar 

  14. L. Lovász and M.D. Plummer, Matching Theory, Annals of Discrete Mathematics (29), North-Holland Mathematics Studies 121 (Elsevier Science Publisher B.V., 1986).

    Google Scholar 

  15. M. Luby, A Simple Parallel Algorithm for the Maximal Independent Set Problem, SIAM J. Comput. 15 (1986) 1036–1053.

    Google Scholar 

  16. D. Pearson and V. V. Vazirani, Efficient Sequential and Parallel Algorithms for Maximal Bipartite Sets, J. Algorithms 14 (1993), 171–179.

    Google Scholar 

  17. T. Shoudai and S. Miyano, Using Maximal Independent Sets to Solve Problems in Parallel, in: Proc. 17th International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, Vol. 570 (Springer, Berlin, 1991) 126–134.

    Google Scholar 

  18. L.G. Valiant, Parallel Computation, in: Proc. 7th IBM Symposium on Mathematical Foundations of Computer Science (1982) 173–189.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tokyo Denki University Hatoyama, 350-03, Saitama, Japan

    Zhi -Zhong Chen

Authors
  1. Zhi -Zhong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ernst W. Mayr Gunther Schmidt Gottfried Tinhofer

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chen, Z.Z. (1995). The maximal f-dependent set problem for planar graphs is in NC. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 1994. Lecture Notes in Computer Science, vol 903. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59071-4_51

Download citation

  • DOI: https://doi.org/10.1007/3-540-59071-4_51

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59071-2

  • Online ISBN: 978-3-540-49183-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation