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Maintaining bridge-connected and biconnected components on-line

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Abstract

We consider the twin problems of maintaining the bridge-connected components and the biconnected components of a dynamic undirected graph. The allowed changes to the graph are vertex and edge insertions. We give an algorithm for each problem. With simple data structures, each algorithm runs inO(n logn +m) time, wheren is the number of vertices andm is the number of operations. We develop a modified version of the dynamic trees of Sleator and Tarjan that is suitable for efficient recursive algorithms, and use it to reduce the running time of the algorithms for both problems toO(mα(m,n)), where α is a functional inverse of Ackermann's function. This time bound is optimal. All of the algorithms useO(n) space.

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References

  1. B. Awerbuch and Y. Shiloach. New connectivity and msf algorithms for shuffle-exchange network and PRAM.IEEE Trans. Comput.,36:1258–1263, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. D. Battista and R. Tamassia. Incremental planarity testing.Proc. 30th IEEE Symposium on Foundations of Computer Science, pages 436–441, 1989.

  3. G. D. Battista and R. Tamassia. On-line graph algorithms with spqr-trees.Proc. 17th Internat. Conf. on Automata, Languages, and Programming (ICALP 1990). Lecture Notes in Computer Science, vol. 443, pages 598–611. Springer-Verlag, Berlin, 1990.

    Google Scholar 

  4. N. Blum. On the single-operation worst-case time complexity of the disjoint set union problem.SIAM J. Comput., 15:1021–1024, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  5. G. A. Cheston. Incremental Algorithms in Graph Theory. Ph.D. thesis, Dept. of Computer Science, University of Toronto, 1976. Technical Report No. 91.

  6. S. Even and Y. Shiloach. An on-line edge deletion problem.J. Assoc. Comput. Mach.,28:1–4, 1981.

    MATH  MathSciNet  Google Scholar 

  7. G. N. Frederickson. Data structures for on-line updating of minimum spanning trees, with applications.SIAM J. Comput.,14:781–798, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. L. Fredman and M. E. Saks. The cell probe complexity of dynamic data structures.Proc. 21st ACM Symposium on Theory of Computing, pages 345–354, Seattle, WA, May 1989.

  9. J. Hopcroft and R. E. Tarjan. Algorithm 447: Efficient algorithms for graph manipulation.Comm. ACM,16:372–378, 1973.

    Article  Google Scholar 

  10. R. Karp and V. Ramachandran. Parallel Algorithms for Shared Memory Machines.Handbook of Theoretical Computer Science, Elsevier, Amsterdam, 1990, pages 869–942.

    Google Scholar 

  11. J. A. La Poutré. Lower bounds for the union-find and split-find problem on pointer machines.Proc. 22nd ACM Symposium on Theory of Computing, pages 34–44, 1990.

  12. J. A. La Poutré, J. van Leeuwen, and M. H. Overmars. Maintenance of 2- and 3-Connected Components of Graphs, Part I: 2- and 3-Edge-Connected Components. Technical Report RUU-CS-90-26, Utrecht University, 1990.

  13. J. H. Reif. A topological approach to dynamic graph connectivity.Inform. Process. Lett.,25:65–70, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. D. Sleator and R. E. Tarjan. A data structure for dynamic trees.J. Comput. System Sci.,26:362–391, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  15. D. D. Sleator and R. E. Tarjan. Self-adjusting binary search trees.J. Assoc. Comput. Mach.,32:652–686, 1985.

    MATH  MathSciNet  Google Scholar 

  16. R. Tamassia. A dynamic data structure for planar graph embedding.Proc. 15th Internat. Conf. on Automata, Languages, and Programming (ICALP1988). Lecture Notes in Computer Science, vol. 317, pages 576–590. Springer-Verlag, Berlin, 1988.

    Google Scholar 

  17. R. Tamassia. Dynamic Data Structures for Two-Dimensional Searching. Ph.D. thesis, Coordinated Science Laboratory, University of Illinois at Urbana-Champagne, 1988. Technical Report ACT-100.

  18. R. E. Tarjan. Depth first search and linear graph algorithms.SIAM J. Comput.,1:146–160, 1972.

    Article  MATH  MathSciNet  Google Scholar 

  19. R. E. Tarjan. Efficiency of a good but not linear set union algorithm.J. Assoc. Comput. Mach.,22:215–225, 1975.

    MATH  MathSciNet  Google Scholar 

  20. R. E. Tarjan. A class of algorithms which require nonlinear time to maintain disjoint sets.J. Comput. System Sci.,18:110–127, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. E. Tarjan.Data Structures and Network Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1983.

    Google Scholar 

  22. R. E. Tarjan. Amortized computational complexity.SIAM J. Algebraic Discrete Methods,6:306–318, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  23. R. E. Tarjan and J. van Leeuwen. Worst-case analysis of set union algorithms.J. Assoc. Comput. Mach.,31:245–281, 1984.

    MATH  MathSciNet  Google Scholar 

  24. R. E. Tarjan and U. Vishkin. An efficient parallel biconnectivity algorithm.SIAM J. Comput.,14:862–874, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  25. J. Westbrook and R. E. Tarjan. Maintaining Bridge-Connected and Biconnected Components On-Line. Technical Report CS-TR-228-89, Dept. of Computer Science, Princeton University, Princeton, NJ, 1989.

    Google Scholar 

  26. A. C. Yao. Should tables be sorted?J. Assoc. Comput. Mach.,28:615–628, 1981.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, Yale University, Yale Station, P.O. Box 2158, 06520-2158, New Haven, CT, USA

    Jeffery Westbrook

  2. Department of Computer Science, Princeton University, 08544, Princeton, NJ, USA

    Robert E. Tarjan

  3. NEC Research Institute, 08540, Princeton, NJ, USA

    Robert E. Tarjan

Authors
  1. Jeffery Westbrook
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  2. Robert E. Tarjan
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Additional information

Communicated by Greg N. Frederickson.

Research at Princeton University supported in part by National Science Foundation Grant DCR-86-05962 and Office of Naval Research Contract N00014-91-J-1463.

This work was partially done while the author was at the Department of Computer Science, Princeton University, Princeton, NJ 08544, USA.

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Westbrook, J., Tarjan, R.E. Maintaining bridge-connected and biconnected components on-line. Algorithmica 7, 433–464 (1992). https://doi.org/10.1007/BF01758773

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  • DOI: https://doi.org/10.1007/BF01758773

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