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Efficient parallel recognition of some circular arc graphs, I

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Abstract

We present the first efficient parallel algorithms for recognizing some subclasses of circular arc graphs including γ circular arc graphs and proper interval graphs. These algorithms run in O(log2 n) time withO(n 3) processors on a CRCW PRAM. An intersection representation can also be constructed within the same resource bounds. Furthermore, we propose some new characterizations of Θ circular arc graphs and proper interval graphs.

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References

  1. A. Apostolico and S. E. Hambrusch. Finding maximum cliques on circular-arc graphs.Information Processing Letters,26:209–215, 1987.

    Google Scholar 

  2. M. J. Atallah and D. Z. Chen. An optimal parallel algorithm for the minimum circle-cover problem.Information Processing Letters,32(4):159–165, 1989.

    Google Scholar 

  3. S. Benzer. On the topology of the genetic fine structure.Proceedings of the National Academy of Sciences,45:1607–1620, 1959.

    Google Scholar 

  4. A. A. Bertossi. Finding Hamiltonian circuits in proper interval graphs.Information Processing Letters,17:97–101, 1983.

    Google Scholar 

  5. A. A. Bertossi and M. A. Bonuccelli. Some parallel algorithms on interval graphs.Discrete Applied Mathematics,16:101–111, 1987.

    Google Scholar 

  6. M. A. Bonuccelli. Dominating sets and domatic numbers of circular-arc graphs.Discrete Applied Mathematics,12:203–213, 1985.

    Google Scholar 

  7. K. S. Booth and G. S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms.Journal of Computer and System Sciences,13:335–379, 1976.

    Google Scholar 

  8. L. Boxer and R. Miller. A parallel circle-cover minimization algorithm.Information Processing Letters,32(2):57–60, 1989.

    Google Scholar 

  9. L. Chen. Efficient parallel algorithm for several intersection graphs. InProceedings of the 22nd International Symposium on Circuits and Systems, Portland, Oregon, May 9–11, pp. 973–976. IEEE, New York, 1989.

    Google Scholar 

  10. L. Chen. NC algorithms for circular-arc graphs. In F. Dehne, J.-R. Sack, and N. Santoro, editors,Proceedings of the Workshop on Algorithms and Data Structures, Ottawa, Ontario, August 17–19, pp. 291–302. Lecture Notes in Computer Science, Vol. 382. Springer-Verlag, Berlin, 1989.

    Google Scholar 

  11. L. Chen. Efficient parallel algorithms on circular arcs. In J. Urrutia, editor,Proceedings of the 2nd Canadian Conference in Computational Geometry, Ottawa, Ontario, August 6–10, 1990.

  12. L. Chen and Y. Yesha. Parallel recognition of the consecutive ones property with applications.Journal of Algorithms,12(3): 375–392, 1991.

    Google Scholar 

  13. R. Cole. Parallel merge sort.SIAM Journal on Computing,17(4):770–785, 1988.

    Google Scholar 

  14. R. Cole and U. Vishkin. Faster optimal parallel prefix sums and list ranking.Information and Computation,81(3):334–352, 1989.

    Google Scholar 

  15. F. E. Fich, P. L. Ragde, and A. Wigderson. Relations between concurrent-write models of parallel computation. InProceedings of the 3rd ACM Symposium on Principles of Distributed Computing, Vancouver, B.C., pp. 179–189. Association for Computing Machinery, New York, 1984.

    Google Scholar 

  16. D. R. Fulkerson and O. A. Gross. Incidence matrices and interval graphs.Pacific Journal of Mathematics,15:835–855, 1965.

    Google Scholar 

  17. M. R. Garey, D. S. Johnson, G. L. Miller, and C. H. Papadimitriou. The complexity of coloring circular arcs and chords.SIAM Journal on Algebraic and Discrete Methods,1:216–227, 1980.

    Google Scholar 

  18. F. Gavril. Algorithms on circular-arc graphs.Networks,4:357–369, 1974.

    Google Scholar 

  19. M. C. Golumbic.Algorithmic Graph Theory and Perfect Graphs. Computer Science and Applied Mathematics. Academic Press, New York, 1980.

    Google Scholar 

  20. M. C. Golumbic and P. L. Hammer. Stability in circular-arc graphs.Journal of Algorithms,9:314–318, 1988.

    Google Scholar 

  21. U. I. Gupta, D. T. Lee, and J. Y.-T. Leung. Efficient algorithms for interval graphs and circular-arc graphs.Networks,12:459–467, 1982.

    Google Scholar 

  22. X. He and Y. Yesha. Parallel recognition and decomposition of two terminal series parallel graphs.Information and Computation,75:15–38, 1987.

    Google Scholar 

  23. W.-L. Hsu. Maximum weight clique algorithms for circular-arc graphs and circle graphs.SIAM Journal on Computing,14:224–231, 1985.

    Google Scholar 

  24. W.-L. Hsu and K. Tsai. Linear time algorithms on circular-arc graphs. InProceedings of the 26th Annual Allerton Conference on Communication, Control, and Computing, pp. 842–851. University of Illinois at Urbana-Champaign, 1988.

    Google Scholar 

  25. L. Hubert. Some applications of graph theory and related non-metric techniques to problems of approximate seriation: the case of symmetric proximity measures.British Journal of Mathematical and Statistical Psychology,27:133–153, 1974.

    Google Scholar 

  26. D. S. Johnson. The NP-completeness column: an ongoing guide.Journal of Algorithms,6:434–451, 1985.

    Google Scholar 

  27. S. K. Kim. Optimal Parallel Algorithms on Sorted Circular Arcs. University of Washington, 1989.

  28. C. P. Kruskal, L. Rudolph, and M. Snir. The power of parallel prefix.IEEE Transactions on Computers,34:965–968, 1985.

    Google Scholar 

  29. R. E. Ladner and M. J. Fischer. Parallel prefix computation.Journal of the ACM,27:831–838, 1980.

    Google Scholar 

  30. C. C. Lee and D. T. Lee. On a circle-cover minimization problem.Information Processing Letters,18(2):109–115, 1984.

    Google Scholar 

  31. D. T. Lee, M. Sarrafzadeh, and Y. F. Wu. Minimum cuts for circular-arc graphs.SIAM Journal of Computing,19(6):1041–1050, 1990.

    Google Scholar 

  32. G. S. Lueker and K. S. Booth. A linear time algorithm for deciding interval graph isomorphism.Journal of the ACM,26(2):183–195, 1979.

    Google Scholar 

  33. S. Masuda and K. Nakajima. An optimal algorithm for finding a maximum independence set of a circular-arc graph.SIAM Journal on Computing,17(1):41–52, 1988.

    Google Scholar 

  34. A. Moitra and R. Johnson. PT-optimal algorithms for interval graphs. InProceedings of the 26th Annual Allerton Conference on Communication, Control, and Computing, pp. 274–282. University of Illinois at Urbana-Champaign, 1988.

    Google Scholar 

  35. J. B. Orlin, M. A. Bonuccelli, and D. P. Bovet. AnO(n 2) algorithm for coloring proper circular arc graphs.SIAM Journal of Algebraic Discrete Methods,2:88–93, 1981.

    Google Scholar 

  36. F. S. Roberts. Representations of Indifference Relations. Ph.D. thesis, Stanford University, 1968.

  37. F. S. Roberts. Indifference graphs. In: F. Harary, editor,Proof Techniques in Graph Theory, pp. 139–146. Academic Press, New York, 1969.

    Google Scholar 

  38. D. Sarkar and I. Stojmenovic. An optimal parallel circle-cover algorithm.Information Processing Letters,32(1): 3–6, 1989.

    Google Scholar 

  39. W.-K. Shih and W.-L. Hsu. AnO(n logn+m log logn) maximum weight clique algorithm for circular arc graphs.Information Processing Letters,3(3):129–134, 1989.

    Google Scholar 

  40. Y. Shiloach and U. Vishkin. Finding the maximum, merging, and sorting in a parallel computation model.Journal of Algorithms,2:88–102, 1981.

    Google Scholar 

  41. F. W. Stahl. Circular genetic maps.Journal of Cell Physiology,70(Suppl. 1):1–12, 1967.

    Google Scholar 

  42. K. E. Stouffers. Scheduling of traffic lights—a new approach.Transportation Research,2:199–234, 1968.

    Google Scholar 

  43. W. Trotter and J. Moore. Characterization problems for graphs, partially ordered sets, lattices, and families of sets.Discrete Mathematics,16:361–381, 1976.

    Google Scholar 

  44. A. C. Tucker. Matrix characterization of circular-arc graphs.Pacific Journal of Mathematics,39:535–545, 1971.

    Google Scholar 

  45. A. C. Tucker. Structure theorems for some circular-arc graphs.Discrete Mathematics,7:167–195, 1974.

    Google Scholar 

  46. A. C. Tucker. Coloring a family of circular-arc graphs.SIAM Journal on Applied Mathematics,29:493–502, 1975.

    Google Scholar 

  47. A. C. Tucker. An efficient test for circular-arc graphs.SIAM Journal on Computing,9:1–24, 1980.

    Google Scholar 

  48. M. S. Yu, C. L. Chen, and R. C. T. Lee. Optimal parallel circle-cover and independent set algorithms for circular-arc graphs. In F. Ris and P. M. Kogge, editors,Proceedings of the IEEE International Conference on Parallel Processing, vol. 3, pp. 126–129. IEEE, New York, 1989.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer and Information Science, Ohio State University, 43210, Columbus, OH, USA

    Lin Chen

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  1. Lin Chen
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Additional information

Communicated by D. T. Lee.

Portions of this paper have appeared in preliminary form in theProceedings of the 1989 IEEE international Symposium on Circuits and Systems [9], theProceedings of the 1989 Workshop on Algorithms and Data Structures [10], and theProceedings of the 1990 Canadian Conference on Computational Geometry [11].

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Chen, L. Efficient parallel recognition of some circular arc graphs, I. Algorithmica 9, 217–238 (1993). https://doi.org/10.1007/BF01190897

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  • DOI: https://doi.org/10.1007/BF01190897

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