Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-24T15:05:57.423Z Has data issue: false hasContentIssue false
  • English
  • Français

Parallel Algorithms for Maximal Cliques in Circle Graphs and Unrestricted Depth Search

Published online by Cambridge University Press:  23 June 2010

E. N. Cáceres ,
S. W. Song  and
J. L. Szwarcfiter
E. N. Cáceres
Affiliation:
Universidade Federal de Mato Grosso do Sul, Faculdade de Computação, 79070-900 Campo Grande, MS, Brazil; [email protected]
S. W. Song
Affiliation:
Universidade de São Paulo, Instituto de Matemática e Estatística, 05508-900 São Paulo, SP, Brazil; visiting professor of Universidade Federal do ABC; [email protected]
J. L. Szwarcfiter
Affiliation:
Universidade Federal do Rio de Janeiro, Instituto de Matemática, Núcleo de Computação Eletrônica and COPPE, 21.945-970 Rio de Janeiro, RJ, Brazil; [email protected]
Get access

Abstract

We present parallel algorithms on the BSP/CGM model, with p processors, to count and generate all the maximal cliques of a circle graph with n vertices and m edges. To count the number of all the maximal cliques, without actually generating them, our algorithm requires O(log p) communication rounds with O(nm/p) local computation time. We also present an algorithm to generate the first maximal clique in O(log p) communication rounds with O(nm/p) local computation, and to generate each one of the subsequent maximal cliques this algorithm requires O(log p) communication rounds with O(m/p) local computation. The maximal cliques generation algorithm is based on generating all maximal paths in a directed acyclic graph, and we present an algorithm for this problem that uses O log (p) communication rounds with O(m/p) local computation for each maximal path. We also show that the presented algorithms can be extended to the CREW PRAM model.

Keywords

BSP/CGM algorithmPRAM algorithmcircle graphmaximal cliqueunrestricted depth search.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 44 , Issue 3 , July 2010 , pp. 293 - 311
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

C.E.R. Alves, E.N. Cáceres, A.A. Castro Jr, S.W. Song and J.L. Szwarcfiter, Efficient Parallel Implementation of Transitive Closure of Digraphs, in 10th European PVM/MPI Users' Group Conference. Edited by J. Dongarra, D. Laforenza and S. Orlando, Springer Verlag, Berlin (2003) 126–133.
R. Anderson and E. Mayr, Parallelism and Greedy Algorithms. Technical Report STAN-CS-84-1003, Computer Science Department, Stanford University Bonn (1984).
Bouchet, A., Reducing Prime Graphs and Recognizing Circle Graphs. Combinatorica 7 (1987) 243–254. CrossRef
E.N. Cáceres, S.W. Song and J.L. Szwarcfiter, A Coarse-Grained Parallel Algorithm for Maximal Cliques in Circle Graphs, in Proc. The 2001 International Conference on Computational Science. Springer Verlag, Berlin (2001) 638–647.
E.N. Cáceres, S.W. Song and J.L. Szwarcfiter, A Parallel Unrestricted Depth Search Algorithm, in Proc. 2001 International Conference on Parallel and Distributed Processing Techniques and Applications (2001) 521–526.
E.N. Cáceres, S.W. Song and J.L. Szwarcfiter, A Parallel Algorithm for Transitive Closure, in Proc. 14th IASTED International Conference on Parallel and Distributed Computing and Systems. IASTED, Zurich (2002) 116–118.
Chan, A. and Dehne, F., Note, A on Coarse Grained Parallel Integer Sorting. Parallel Process. Lett. 9 (1999) 533–538. CrossRef
Chiba, N. and Nishizeki, T., Arboricity and subgraphs listing algorithms. SIAM J. Comput. 14 (1985) 210–223. CrossRef
E. Dahlhaus and M. Karpinski, A Fast Parallel Algorithm for Computing all Maximal Cliques in a Graph and the Related Problems, in Proc. Scandinavian Workshop on Algorithm Theory – SWAT (1988) 139–144.
F. Dehne (Ed.), Coarse grained parallel algorithms. Algorithmica 24 (1999) 173–426.
Dehne, F., Ferreira, A., Cáceres, E., Song, S.W. and Roncato, A., Efficient Parallel Graph Algorithms For Coarse Grained Multicomputers and BSP. Algorithmica 33 (2002) 183–200. CrossRef
Ferreira, A. and Schabanel, N., A randomized BSP/CGM algorithm for the maximal independent set. Parallel Process. Lett. 9 (2000) 411–422. CrossRef
Gabor, C.P., Hsu, W.L. and Supowit, K.J., Recognizing Circle Graphs in Polynomial Time. J. Assoc. Comput. Mach. 36 (1989) 435–474. CrossRef
Ghosh, R.K. and Bhattacharjee, G.P., Parallel Search Al, Agorithm for Directed Acyclic Graphs. BIT 24 (1984) 134–150. CrossRef
E. Gioan, C. Paul, M. Tedder and D. Corneil, Quasi-linear circle graph recognition. Technical Report, University of Toronto (2009).
M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs. Academic Press (1980).
R. Gupta, J. Walrand and O. Goldschmidt, Maximal Cliques in Unit Disk Graphs: Polynomial Approximation, in Proc. International Network Optimization Conference (INOC), Lisbon (2005).
Ho, C.W. and Lee, R.C.T., Efficient Parallel Algorithms for Finding Maximal Cliques, Clique Trees, and Minimum Coloring on Chordal Graphs. Inf. Process. Lett. 28 (1988) 301–309. CrossRef
E. Jennings and L. Motyckova, A Distributed Algorithm for finding All Maximal Cliques in a Network Graph, in Proc. of the 1st Latin American Symposium on Theoretical Informatics (LATIN'92) (1992) 281–293.
R.M. Karp and V. Ramachandran, Parallel Algorithms for Shared-Memory Machines. Edited by J. van Leeuwen Handbook of Theoretical Computer Science. MIT Press/Elsevier (1990) 869–941.
Klein, P.N., Efficient Parallel Algorithms for Chordal Graphs. SIAM J. Comput. 25 (1996) 797–827. CrossRef
K. Makino and T. Uno, New Algorithms for Enumerating All Maximal Cliques, in Proc. Scandinavian Workshop on Algorithm Theory – SWAT (2004) 260–272.
Moon, J.W. and Moser, L., On cliques in graphs. Israel J. Math 3 (1965) 23–28. CrossRef
Naji, W., Graphes des Cordes, Caractérisation et Reconnaissance. Disc. Math. 54 (1985) 329–337. CrossRef
Naor, J., Naor, M. and Schäffer, A.A., Fast Parallel Algorithms for Chordal Graphs. SIAM J. Comput. 18 (1989) 327–349. CrossRef
F. Protti, F.M.G. França and J.L. Szwarcfiter, On Computing All Maximal Cliques Distributedly, in Proc. Workshop on Parallel Algorithms for Irregularly Structured Problems (IRREGULAR) (1997) 37–48.
Spinrad, J.P., Recognition of Circle Graphs. J. Algor. 16 (1994) 264–282. CrossRef
J.L. Szwarcfiter and M. Barroso, Enumerating the Maximal Cliques of Circle Graph, Graph Theory, Combinatorics, Algorithms and Applications. edited by F.R.K. Chung, R.L. Graham and D.F. Hsu, SIAM Publications (1991) 511–517.
Tarjan, R.E., Depth First Search and Linear Graph Algorithms. SIAM J. Comput. 1 (1972) 146–160. CrossRef
Tsukiyama, S., Ide, M., Arujoshi, H. and Ozaki, H., New Al, Agorithm for Generating the Maximal Independent Sets. SIAM J. Comput. 6 (1997) 505–517. CrossRef
Valiant, L.G., Bridging Model, A for Parallel Computation. Communication of the ACM 33 (1990) 103–111. CrossRef
Wang, C.S. and Chang, R.S., Parallel Maximal Cliques Al, Agorithm for Interval Graphs with Applications. J. Inf. Sci. Eng. 13 (1997) 649–663.
Y. Zhang, Parallel Algorithms for Problems Involving Directed Graphs. Ph.D. thesis, Department of Computer Science, Drexel University (1990).
Y. Zhang, F.N. Abu-Khzam, N.E. Baldwin, E.J. Chesler, N.F. Samatova and M.A. Langston, Genome-Scale Computational Approaches to Memory-Intensive Applications in Systems Biology. Proceedings of SC, Seattle, Washington (2005).