Parallel ear decomposition search (EDS) and st-numbering in graphs
The linear time serial algorithm for planarity testing of [LEC-67] uses the linear time serial algorithm of [ET-76] for st-numbering. This st-numbering algorithm is based on depth-first search (DFS). A known conjecture states that DFS, which is a key technique in designing serial algorithms, is not amenable to poly-log time parallelism using "around linearly" (or even polynomially) many processors. The first contribution of this paper is a general method for searching efficiently in parallel undirected graphs, called ear-decomposition search (EDS).
The second contribution demonstrates the applicability of this search method. We present an efficient parallel algorithm for st-numbering in a biconnected graph. The algorithm is quite subtle and runs in logarithmic time using a linear number of processors on a concurrent-read concurrent-write (CRCW) PRAM. An efficient parallel algorithm for the problem did not exist before. It was not even known to be in NC.
KeywordsParallel Algorithm Internal Vertex Numbering Algorithm Adjacency List Serial Algorithm
Unable to display preview. Download preview PDF.
- [A-85]Anderson, R., "A Parallel Algorithm for the Maximal Path Problem", Proc. 17th ACM Symp. on Theory of Computing (1985), pp. 33–37.Google Scholar
- [CV-85]Cole, R. and Vishkin, U., "Deterministic Coin Tossing and Accelerating Cascades: Micro and Macro Techniques for Designing Parallel Algorithms", Proc. 18th ACM Symp. on Theory of Computing (1986), to appear.Google Scholar
- [E-79]Even, S., "Graph Algorithms", Computer Science Press, 1979.Google Scholar
- [ET-76]Even, S., and Tarjan, R.E., "Computing an st-numbering", Th. Comp. Science 2 (1976), pp. 339–344.Google Scholar
- [IR-84]Itai, A. and Rodeh, M., "The Multi-Tree Approach to Reliability in Distributed Networks", Proc. 25th Symp. on Foundations of Comp. Science (1984), pp. 137–147.Google Scholar
- [JS-82]Ja'Ja' J. and Simon, J., "Parallel Algorithms in Graph Theory: Planarity Testing", SIAM J. of Computing 11 (1982), pp. 314–328.Google Scholar
- [LEC-67]Lempel, A., Even, S. and Cederbaum, I., "An Algorithm for Planarity Testing of Graphs", Theory of Graphs, Int. Symp., Rome, July 1966. P. Rosenstiehl, Ed., Gordon and Breach, NY (1967), pp. 215–232.Google Scholar
- [L-85]Lovasz, L., "Computing Ears and Branchings in Parallel", Proc. 26th Symp. on Foundations of Comp. Science (1985), pp. 464–467.Google Scholar
- [MSV-86]Maon, Y., Schieber, B. and Vishkin, U., "Parallel Ear Decomposition Search (EDS) and ST-Numbering in Graphs", TR 46/86 Dept. of Computer Science, Tel Aviv University, 1986.Google Scholar
- [R-85]Reif, J.H., "An Optimal Parallel Algorithm for Integer Sorting", Proc. 26th Symp. on Foundations of Comp. Science (1985), pp. 496–503.Google Scholar
- [SV-82]Shiloach, Y. and Vishkin, U., "An O (log n) Parallel Connectivity Algorithm", J. of Algorithms 3 (1982), pp. 57–63.Google Scholar
- [TV-85]Tarjan, R.E. and Vishkin, U., "An Efficient Parallel Biconnectivity Algorithm", SIAM J. of Computing 14 (1985), pp. 862–874.Google Scholar
- [V-83]Vishkin, U., "Synchronous Parallel Computation — a Survey", TR #71, Dept. of Computer Science, Courant Inst., NYU, 1983.Google Scholar
- [V-85]Vishkin, U., "On Efficient Parallel Strong Orientation", Information Proc. Letters 20 (1985), pp. 235–240.Google Scholar
- [W-32]Whitney, H., "Non-separable and Planar Graphs", Trans. Amer. Math. Soc. 34 (1932), pp. 339–362.Google Scholar