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Efficient Algorithms for the Hamiltonian Problem on Distance-Hereditary Graphs

  • Sun-yuan Hsieh
  • Chin-wen Ho
  • Tsan-sheng Hsu
  • Ming-tat Ko
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2387)

Abstract

In this paper, we first present an O(|V| + |E|)-time sequential algorithm to solve the Hamiltonian problem on a distance-hereditary graph G = (V, E). This algorithm is faster than the previous best result which takes O(|V|2) time. Let T d (|V|, |E|) and P d (|V|, |E|) denote the parallel time and processor complexities, respectively, required to construct a decomposition tree of a distance-hereditary graph on a PRAM model M d . We also show that this problem can be solved in O(T d (|V|, |E|) + log|V|) time using O(P d (|V|, |E|) + (|V| + |E|)/log|V|) processors on M d . Moreover, if G is represented by its decomposition tree form, the problem can be solved optimally in O(log |V|) time using O((|V| + |E|)/log|V|) processors on an EREW PRAM.

Keywords

Steiner Tree Hamiltonian Cycle Decomposition Tree Hamiltonian Problem Tree Contraction 
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.

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References

  1. 1.
    K. Abrahamson, N. Dadoun, D. G. Kirkpatrick, and T. Przytycka, A simple parallel tree contraction algorithm, Journal of Algorithms, 10:287–302, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    H. J. Bandelt and H. M. Mulder, Distance-hereditary graphs, Journal of Combinatorial Theory Series B, 41(1):182–208, 1989.MathSciNetGoogle Scholar
  3. 3.
    A. Brandstädt and F. F. Dragan, A linear time algorithm for connected γ-domination and Steiner tree on distance-hereditary graphs, Networks, 31:177–182, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    M. S. Chang, S. Y. Hsieh, and G. H. Chen, Dynamic programming on distance-hereditary graphs, Proceedings of 7th International Symposium on Algorithms and Computation (ISAAC’97), LNCS 1350, pp. 344–353, 1997.Google Scholar
  5. 5.
    B. Courcelle, J. A. Makowsky, and U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory of Computing Systems, 33:125–150, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. D’atri and M. Moscarini, Distance-hereditary graphs, steiner trees, and connected domination, SIAM Journal on Computing, 17(3):521–538, 1988.CrossRefMathSciNetGoogle Scholar
  7. 7.
    F. F. Dragan, Dominating cliques in distance-hereditary graphs, Algorithm Theory-SWAT’94-4th Scandinavian Workshop on Algorithm Theory, LNCS 824, Springer, Berlin, pp. 370–381, 1994.Google Scholar
  8. 8.
    M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic press, New York, 1980.zbMATHGoogle Scholar
  9. 9.
    M. C. Golumbic and U. Rotics, On the clique-width of perfect graph classes, WG’99, LNCS 1665, pp. 135–147, 1999.Google Scholar
  10. 10.
    P. L. Hammer and F. Maffray, Complete separable graphs, Discrete Applied Mathematics, 27(1):85–99, 1990.CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    E. Howorka, A characterization of distance-hereditary graphs, Quarterly Journal of Mathematics (Oxford), 28(2):417–420, 1977.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    S.-y. Hsieh, C. W. Ho, T.-s. Hsu, M. T. Ko, and G. H. Chen, Efficient parallel algorithms on distance-hereditary graphs, Parallel Processing Letters, 9(1):43–52, 1999.CrossRefMathSciNetGoogle Scholar
  13. 13.
    S.-y. Hsieh, C. W. Ho, T.-s. Hsu, M. T. Ko, and G. H. Chen, Characterization of Efficiently Solvable Problems on Distance-Hereditary Graphs, Proceedings of 9th International Symposium on Algorithms and Computation (ISAAC’98), LNCS 1533, pp. 257–266, 1998.Google Scholar
  14. 14.
    S.-y. Hsieh, C. W. Ho, T.-s. Hsu, M. T. Ko, and G. H. Chen, A faster implementation of a parallel tree contraction scheme and its application on distance-hereditary graphs, Journal of Algorithms, 35:50–81, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    S.-y. Hsieh, Parallel decomposition of distance-hereditary graphs, Proceedings of the 4th International ACPC Conference Including Special Tracks on Parallel Numerics (ParNum’99) and Parallel Computing in Image Processing, Video Processing, and Multimedia (ACPC’99), LNCS 1557, pp. 417–426, 1999.Google Scholar
  16. 16.
    R. W. Hung, S. C. Wu, and M. S. Chang, Hamiltonian cycle problem on distance-hereditary graphs, manuscript.Google Scholar
  17. 17.
    H. Müller and F. Nicolai, Polynomial time algorithms for Hamiltonian problems on bipartite distance-hereditary graphs, Information Processing Letters, 46:225–230, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Falk Nicolai, Hamiltonian problems on distance-hereditary graphs, Technique report, Gerhard-Mercator University, Germany, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Sun-yuan Hsieh
    • 1
  • Chin-wen Ho
    • 2
  • Tsan-sheng Hsu
    • 3
  • Ming-tat Ko
    • 3
  1. 1.Department of Computer Science and Information EngineeringNational Cheng Kung UniversityTainanTaiwan
  2. 2.Department of Computer Science and Information EngineeringNational Central UniversityChung-LiTaiwan
  3. 3.Institute of Information ScienceAcademia SinicaTaipeiTaiwan

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