An efficient NC algorithm for finding Hamiltonian cycles in dense directed graphs

  • Martin Fürer
  • Balaji Raghavachari
Parallel Algorithms (Session 10)
Part of the Lecture Notes in Computer Science book series (LNCS, volume 510)


Let G be a directed graph with n vertices such that whenever there is no arc from any vertex u to another vertex v, then the sum of the outdegree of u and the indegree of v is at least n. It is known that such a graph G always contains a Hamiltonian cycle. We show that such a cycle can be computed with a linear number of processors in O(log3n) time on a CREW PRAM.


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  1. 1.
    A. Aggarwal and R.J. Anderson, A Random NC-algorithm for Depth First Search, Combinatorica 8 (1988) 1–12.Google Scholar
  2. 2.
    A. Aggarwal, R.J. Anderson and M. Kao, Parallel Depth-First Search in General Directed Graphs, Proc. 21st ACM STOC (1989) 297–308.Google Scholar
  3. 3.
    S.G. Akl, The Design and Analysis of Parallel Algorithms, Prentice-Hall, 1989.Google Scholar
  4. 4.
    R.J. Anderson, A Parallel Algorithm for the Maximal Path Problem, Combinatorica 7 (1987) 315–326.Google Scholar
  5. 5.
    A. Awerbuch, A. Israeli and Y. Shiloach, Finding Euler Circuits in Logarithmic Parallel Time, Proc. 16th ACM STOC (1984) 249–257.Google Scholar
  6. 6.
    J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, American Elsevier, New York 1976.Google Scholar
  7. 7.
    N. Chiba and T. Nishizeki, The Hamiltonian Cycle Problem is Linear-Time Solvable for Four-connected Planar Graphs, J. Algorithms 10 (1989) 187–211.Google Scholar
  8. 8.
    E. Dahlhaus, P. Hajnal and M. Karpinski, Optimal Parallel Algorithm for the Hamiltonian Cycle Problem on Dense Graphs, 29th Annual Symp. on Foundations of Comp. Sci. (1988) 186–193.Google Scholar
  9. 9.
    G.A. Dirac, Some Theorems on Abstract Graphs, Proc. Lond. Math. Soc. 2 (1952), 69–81.Google Scholar
  10. 10.
    A.M. Frieze, Parallel Algorithms for Finding Hamiltonian Cycles in Random Graphs, Inf. Proc. Lett. 25 (1987) 111–117.Google Scholar
  11. 11.
    M. Fürer and B. Raghavachari, An efficient NC approximation algorithm for edgecoloring graphs with applications to maximal matching, In preparation.Google Scholar
  12. 12.
    M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, W.H. Freeman, 1979.Google Scholar
  13. 13.
    M. Gondran and M. Minoux, Graphs and Algorithms, John Wiley & Sons, 1979.Google Scholar
  14. 14.
    A. Israeli and Y. Shiloach, An Improved Parallel Algorithm for Maximal Matching, Inf. Proc. Lett. 22 (1986) 57–60.Google Scholar
  15. 15.
    B. Jackson, Hamilton Cycles in Regular 2-Connected Graphs, J. Comb. Theory Ser. B 29 (1980) 27–46.Google Scholar
  16. 16.
    R.M. Karp and V.L. Ramachandran, A Survey of Parallel Algorithms for Shared Memory Machines, Handbook of Theoretical Computer Science, edited by J. van Leeuwen, MIT Press, 1990.Google Scholar
  17. 17.
    S. Khuller, On Computing Graph Closures, Inf. Proc. Lett. 31 (1989) 249–255.Google Scholar
  18. 18.
    G.F. Lev, N. Pippenger and L.G. Valiant, A Fast Parallel Algorithm for Routing in Permutation Networks, IEEE Transactions on Computers C-30 (1981) 93–100.Google Scholar
  19. 19.
    M. Meyniel, Une condition suffisante d'existence d'un circuit Hamiltonien dans un graphe orienté, J. Comb. Theory Ser. B 14 (1973) 137–147.Google Scholar
  20. 20.
    O. Ore, Note on Hamiltonian Circuits, Amer. Math. Monthly 67 (1960) 55.Google Scholar
  21. 21.
    D. Soroker, Fast Parallel Algorithms for Finding Hamiltonian Paths and Cycles in a Tournament, Report no. UCB/CSD 87/309, Univ. of California, Berkeley 1986.Google Scholar
  22. 22.
    K. Takamizawa, T. Nishizeki and N. Saito, An O(p 3) Algorithm for Finding Hamiltonian Cycle in Certain Digraphs, J. Inf. Proc. 3 (1980) 68–72.Google Scholar
  23. 23.
    W.T. Tutte, A Theorem on Planar Graphs, Trans. Am. Math. Soc. 82 (1956) 99–116.Google Scholar
  24. 24.
    D.R. Woodall, Sufficient Conditions for Circuits in Graphs, Proc. Lond. Math. Soc. 24 (1972) 739–755.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Martin Fürer
    • 1
  • Balaji Raghavachari
    • 1
  1. 1.Computer Science DepartmentPennsylvania State UniversityUniversity Park

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