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Parallel Algorithms for Shortest Paths and Related Problems on Trapezoid Graphs

  • F. R. Hsu
  • Yaw-Ling Lin
  • Yin-Te Tsai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1741)

Abstract

In this paper, we consider parallel algorithms for shortest pa- ths and related problems on trapezoid graphs under the CREW PRAM model. Given a trapezoid graph with its corresponding trapezoid dia- gram, we present parallel algorithms solving the following problems: For the single-source shortest path problem, the algorithm runs in O(log n) time using O(n) processors and space. For the all-pair shortest path query problem, after spending O(log n) preprocessing time using O(n log n) space and O(n) processors, the algorithm can answer the query in O(log δ) time using one processor. Here δ denotes the distance between two queried vertices. For the minimum cardinality Steiner set problem, the algorithm runs in O(log n) time using O(n) processors and space.

We also extend our results to the generalized trapezoid graphs. The single-source shortest path problem and the minimum cardinality Steiner set problem on d-trapezoid graphs and circular d-trapezoid graphs can both be solved in O(log n log d) time using O(nd) space and O(d 2 n/log d) processors. The all-pair shortest path query problem on d-trapezoid graphs and circular d-trapezoid graphs can be answered in O(d log δ) time using one processor after spending O(log n log d) preprocessing time using O(nd log n) space and O(d 2 n/log d) processors.

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References

  1. 1.
    S.G. Akl, “Parallel computation: models and methods”, Prentice Hall, Upper Saddle River, New Jersey, 1997.Google Scholar
  2. 2.
    I. Dagan, M.C. Golumbic and R.Y. Pinter, “Trapezoid graphs and their coloring”, Discr. Applied Math.,21:35–46,1988.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    D.Z. Chen, D. T. Lee, R. Sridhar and C. N. Sekharan, “Solving the all-pair shortest path query problem on interval and circular-arc graphs”, Networks, pp. 249–257, 1998.Google Scholar
  4. 4.
    Flotow, “On Powers of m-Trapezoid Graphs”, Discr. Applied Math., 63:187–192, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    H.S. Chao, F.R. Hsu and R.C.T. Lee, “On the Shortest Length Queries for Permutation Graphs”, Proc. Of the 1998 International Computer Symposium, Workshop on Algorithms, NCKU, Taiwan, pp. 132–138, 1998.Google Scholar
  6. 6.
    O.H. Ibarra and Q. Zheng, “An optimal shortest path parallel algorithm for permutation graphs”, J. of Parallel and Distributed Computing, 24:94–99, 1995.CrossRefGoogle Scholar
  7. 7.
    Dieter Kratsch, Ton Kloks and Haiko Müller, “Measuring the vulnerability for classes of intersection graphs”, Discr. Applied Math., 77:259–270, 1997.zbMATHCrossRefGoogle Scholar
  8. 8.
    Y. D. Liang, “Steiner set and connected domination in trapezoid graphs”, Information Processing Letters, 56(2):101–108, 1995.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • F. R. Hsu
    • 1
  • Yaw-Ling Lin
    • 1
  • Yin-Te Tsai
    • 1
  1. 1.Providence UniversityShalu, Taichung HsienTaiwan, Republic of China

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