This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
An Efficient Parallel Recognition Algorithm For Bipartite-Permutation Graphs
January 1996 (vol. 7 no. 1)
pp. 3-10

Abstract—In this paper, we present a parallel recognition algorithm for bipartite-permutation graphs. The algorithm can be executed in O(log n) time on the CRCW PRAM if O(n3 /log n) processors are used, or O(log2n) time on the CREW PRAM if O(n3 /log2n) processors are used. Previously, Chen and Yesha have presented another CRCW PRAM algorithm that takes O(log2n) time if O(n3) processors are used. Compared with Chen and Yesha's algorithm, our algorithm requires either less time and fewer processors on the same machine model, or fewer processors on a weaker machine model. Besides, our algorithm can be applied to determine if two bipartite-permutation graphs are isomorphic.

[1] M.J. Atallah,G.K. Manacher, and J. Urrutia,"Finding a minimum independent dominating set in a permutation graph," Discrete Applied Math., vol. 21, pp. 177-183, 1988.
[2] A. Brandstädt and D. Kratsch,"On domination problems for permutation and other graphs," Theoretical Computer Science, vol. 54, pp. 181-198, 1987.
[3] A. Brandstädt,J. Spinrad, and L. Stewart,"Bipartite permutation graphs are bipartite tolerance graphs," Congressus Numerantium, vol. 58, pp. 165-174, 1987.
[4] L. Chen,"Logarithmic time NC algorithms for comparability graphs and circle graphs," Lecture Notes in Computer Science: Advances in Computing and Information, vol. 497, pp. 383-394, 1991.
[5] L. Chen,"An efficient parallel algorithm for testing isomorphism of bipartite permutation graphs," Proc. First Ann. IEEE Symp. Parallel and Distributed Processing, pp. 24-25, 1989.
[6] L. Chen and Y. Yesha,"Efficient parallel algorithms for bipartite permutation graphs," Networks, vol. 22, no. 1, pp. 29-39, 1993.
[7] F.Y. Chin, J. Lam, and I. Chen, "Efficient Parallel Algorithms for Some Graph Problems," Comm. ACM, vol. 25, no. 9, pp. 659-665, 1982.
[8] R. Cole, "Parallel Merge Sort," SIAM J. Computing, vol. 17, pp. 770-785, 1988.
[9] R. Cole and U. Vishkin, "Approximate Parallel Scheduling. Part 1: The Basic Technique with Applications to Optimal Parallel List Ranking in Logarithmic Time," SIAM J. Computing, vol. 18, pp. 128-142, 1988.
[10] M. Farber and J.M. Keil,"Domination in permutation graphs," J. Algorithms, vol. 6, pp. 309-321, 1985.
[11] M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness.New York: W.H. Freeman, 1979.
[12] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs.New York, Academic Press, 1980.
[13] D. Helmbold and E. Mayr,"Perfect graphs and parallel algorithms," Proc. Int'l Conf. Parallel Processing, pp. 853-860, 1986.
[14] D.E. Knuth, The Art of Computer Programming, vol. 1,Addison Wesley, second ed. 1973.
[15] D. Kozen,U.V. Vazirani, and V.V. Vazirani,"NC algorithms for comparability graphs, interval graphs, and testing for unique perfect matching," Proc. Fifth Conf. Foundation of Software Technology and Theoretical Computer Science, New Delhi, pp. 496-503, 1985.
[16] C.L. Liu, Introduction to Combinatorial Mathematics. McGraw-Hill, 1968.
[17] A. Pnueli,A. Lempel, and S. Even,"Transitive orientation of graphs and identification of permutation graphs," Can. J. Math., vol. 23, no. 1, pp. 160-175, 1971.
[18] Y. Shiloach and U. Vishkin,"An O(log n) parallel connectivity algorithm," J. Algorithms, vol. 3, pp. 57-67, 1982.
[19] J. Spinrad,"On comparability and permutation graphs," SIAM J. Computing, vol. 14, no. 3, pp. 658-670, 1985.
[20] J. Spinrad, A. Brandstädt, and L. Stewart, "Bipartite Permutation Graphs," Discrete Applied Mathematics, vol. 18, pp. 279-292, 1987.
[21] K.J. Supowit,"Decomposing a set of points into chains, with applications to permutation and circle graphs," Information Processing Letters, vol. 21, pp. 249-252, 1985.
[22] K.H. Tsai and W.L. Hsu,"Fast algorithms for the dominating set problem on permutation graphs," Lecture Notes in Computer Science: Algorithms, vol. 450, pp. 109-117, 1990.
[23] U. Vishkin,"On efficient parallel strong orientation," Information Processing Letters, vol. 20, pp. 235-240, 1985.
[24] C.W. Yu,"Recognition problems of bipartite-permutation and other related graphs," PhD dissertation, Dept. Computer Science&Information Enginering, National Taiwan Univ., Taipei, Taiwan, Jun. 1993.
[25] C.W. Yu and G.H. Chen,"Parallel algorithms for permutaion graphs," Tech. Report 91-11, Dept. Computer Science and Information Engineering, National Taiwan Univ., Taipei, Taiwan, July 1991.
[26] C.W. Yu and G.H. Chen,"The weighted maximum independent set problem in permutaion graphs," Tech. Report 91-12, Dept. Computer Science and Information Engineering, National Taiwan Univ., Taipei, Taiwan, July 1991.

Index Terms:
Bipartite-permutation graph, graph recognition, graph isomorphism, parallel algorithm, and parallel random access machine.
Citation:
Chang-Wu Yu, Gen-Huey Chen, "An Efficient Parallel Recognition Algorithm For Bipartite-Permutation Graphs," IEEE Transactions on Parallel and Distributed Systems, vol. 7, no. 1, pp. 3-10, Jan. 1996, doi:10.1109/71.481592
Usage of this product signifies your acceptance of the Terms of Use.