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Algorithmica

, Volume 15, Issue 5, pp 448–466 | Cite as

Searching among intervals and compact routing tables

  • G. N. Frederickson
Article

Abstract

Shortest paths in weighted directed graphs are considered within the context of compact routing tables. Strategies are given for organizing compact routing tables so that extracting a requested shortest path will takeo(k logn) time, wherek is the number of edges in the path andn is the number of vertices in the graph. The first strategy takesO (k+logn) time to extract a requested shortest path. A second strategy takes Θ(k) time on average, assuming alln(n−1) shortest paths are equally likely to be requested. Both strategies introduce techniques for storing collections of disjoint intervals over the integers from 1 ton, so that identifying the interval within which a given integer falls can be performed quickly.

Key words

All pairs shortest paths Analysis of algorithms Compact routing table Graph decomposition Outerplanar graph Predecessor finding Succinct encoding 

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References

  1. [AHU]
    A. V. Aho, J. E. Hopcroft, and J. D. Ullman.The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Massachusetts, 1974.Google Scholar
  2. [CLR]
    T. H. Cormen, C. E. Leiserson, and R. L. Rivest.Introduction to Algorithms. McGraw-Hill, New York, 1990.Google Scholar
  3. [DP]
    N. Deo and C. Pang. Shortest-path algorithms: taxonomy and annotation.Networks, 14:275–323, 1984.Google Scholar
  4. [D]
    E. W. Dijkstra. A note on two problems in connexion with graphs.Numer. Math., 1:269–271, 1959.Google Scholar
  5. [DPZ]
    H. N. Djidjev, G. E. Pantziou, and C. D. Zaraliagis. Computing shortest paths and distances in planar graphs.Proceedings of the International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, Vol. 510, pp. 327–338. Springer-Verlag, Berlin, 1991.Google Scholar
  6. [Fl]
    R. W. Floyd. Algorithm 97: shortest path.Comm. ACM, 5:345, 1962.Google Scholar
  7. [Fs1]
    G. N. Frederickson. Implicit data structures for the dictionary problem.J. Assoc. Comput. Mach., 30:80–94, 1983.Google Scholar
  8. [Fs2]
    G. N. Frederickson. Implicit data structures for weighted elements.Inform. and Control, 66:61–82, 1985.Google Scholar
  9. [Fs3]
    G. N. Frederickson. Fast algorithms for shortest paths in planar graphs, with applications.SIAM J. Comput., 16:1004–1022, 1987.Google Scholar
  10. [Fs4]
    G. N. Frederickson. Planar graph decomposition and all pairs shortest paths.J. Assoc. Comput. Mack., 38:162–204, 1991.Google Scholar
  11. [Fs5]
    G. N. Frederickson. Searching among intervals and compact routing tables.Proceedings of the International Colloquium on Automata, Languages and Programming. Lecture Notes in Computer Science, Vol. 709, pp. 28–39. Springer-Verlag, Berlin, 1993.Google Scholar
  12. [Fs6]
    G. N. Frederickson. Using cellular embeddings in solving all pairs shortest paths problems.J. Algorithms, 19:45–85, 1995.Google Scholar
  13. [FJ]
    G. N. Frederickson and R. Janardan. Designing networks with compact routing tables.Algorithmica, 3:171–190, 1988.Google Scholar
  14. [Fm]
    M. L. Fredman. New bounds on the complexity of the shortest path problem.SlAM J. Comput., 5:83–89, 1976.Google Scholar
  15. [FKS]
    M. L. Fredman, J. Komlos, and E. Szemeredi. Storing a sparse table withO(1) worst case access.J. Assoc. Comput. Mach., 31:538–544, 1984.Google Scholar
  16. [FT]
    M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms.J. Assoc. Comput. Mach., 34:596–615, 1987.Google Scholar
  17. [H]
    F. Harary.Graph Theory. Addison-Wesley, Reading, Massachusetts, 1969.Google Scholar
  18. [MS]
    J. I. Munro and H. Suwanda. Implicit data structures for fast search and update.J. Comput. System Sci., 21:236–250, 1980.Google Scholar
  19. [SK]
    N. Santoro and R. Khatib. Labelling and implicit routing in networks.Comput. J., 28:5–8, 1985.Google Scholar
  20. [TY]
    R. E. Tarjan and A. C.-C. Yao. Storing a sparse table.Comm. ACM, 21:606–611, 1979.Google Scholar
  21. [vBKZ]
    P. Van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue.Math. Systems Theory, 10:99–127, 1977.Google Scholar
  22. [vLT]
    J. van Leeuwen and R. B. Tan. Computer networks with compact routing tables. In G. Rozenberg and A. Salomaa, editors,The Book of L, pp. 259–273. Springer-Verlag, New York, 1986.Google Scholar
  23. [Wa]
    S. Warshall. A theorem on boolean matrices.J. Assoc. Comput. Mach., 9:11–12, 1962.Google Scholar
  24. [Wi]
    D. E. Willard. Log-logarithmic worst-case range queries are possible in space Θ(n).Inform. Process. Lett., 17:81–84, 1983.Google Scholar
  25. [Y]
    A. C.-C. Yao. Should tables be sorted?J. Assoc. Comput. Mach., 28:615–628, 1981.Google Scholar

Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • G. N. Frederickson
    • 1
  1. 1.Department of Computer SciencePurdue UniversityWest LafayetteUSA

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