D. S. Hirschberg
Abstract
We present a parallel algorithm which uses n2 processors to find the connected components of an undirected graph with n vertices in time O(log2n). An O(log2n) time bound also can be achieved using only n⌈n/⌈log2n⌉⌉ processors. The algorithm can be used to find the transitive closure of a symmetric Boolean matrix. We assume that the processors have access to a common memory. Simultaneous access to the same location is permitted for fetch instructions but not for store instructions.
- 1 Aho, AN., Hopcroft, J.E., and Ullman, J.D. The Design and Analysis o f Computer Algorithms, Addison-Wesley, Reading, Mass., 1974. Google Scholar
Digital Library
- 2 Batcher, K.E. Sorting networks and their applications. Proc. AFIPS 1968 SJCC, Vol. 32, AFIPS Press, Montvale, N.J., pp. 307- 314.Google Scholar
- 3 Baudet, G., and Stevenson, D. Optimal sorting algorithms for parallel computers. 1EEE Trans. Comptrs. C-27 (Jan. 1978), 84-87.Google Scholar
- 4 Brent, R.P. The parallel evaluation of general arithmetic expressions. J. ACM 21, (April 1974), 201-206. Google ScholarDigital Library
- 5 Chandra, A.K. Maximal parallelism in matrix multiplication. IBM Tech. Rep. RC6193, Sept. 1976.Google Scholar
- 6 Csanky, L. Fast parallel matrix inversion algorithms, SlAM J. Comping. 5 (Dec. 1976), 618-623.Google ScholarCross Ref
- 7 Hirschberg, D.S. Fast parallel sorting algorithms. Comm. ACM 21, 8 (Aug. 1978), 657-661. Google ScholarDigital Library
- 8 Hyafil, L., and Kung, H.T. The complexity of parallel evaluation of linear recurrences. J. ACM 24, (July 1977), 513-521. Google ScholarDigital Library
- 9 Levitt, K.N., and Kautz, W.H. Cellular arrays for the solution of graph problems. Comm. ACM 15, (Sept. 1972), 789-801. Google ScholarDigital Library
- 10 Munro, I., }nd Paterson, M. Optimal algorithms for parallel polynomial evaluation. J. Comp. Syst. Sci. 7 (1973), 183-198.Google ScholarDigital Library
- 11 Pan, V.Y. Strassen's algorithm is not optimal. Proc. 19th Annu. Symp. on Foundations of Comptr. Sci., 1978, pp. 166-176.Google Scholar
- 12 Preparata, F.P. New parallel-sorting schemes. IEEE Trans. Comptrs. C-27 (July 1978), 669-673.Google ScholarDigital Library
- 13 Preparata, F.P., and Sarwate, D.V. An improved parallel processor bound in fast matrix inversion. Inf. Proc. Letters 7 (April 1978), 148-150.Google ScholarCross Ref
- 14 Reghbati, E., and Corneil, D.C. Parallel computations in graph theory. SlAM J. Comping. 7 (May 1978), 230-237.Google Scholar
- 15 Savage, C.D. Parallel algorithms for graph theoretic problems. Ph.D. Th., U. of Illinois, Urbana, I11., Aug. 1977.Google Scholar
- 16 Thompson, C.D., and Kung, H.T. Sorting on a mesh-connected parallel computer. Comm. A CM 20, 4 (April 1977), 263-271. Google ScholarDigital Library
Index Terms
- Computing connected components on parallel computers
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