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Abstract—A tree T is labeled when the n vertices are distinguished from one another by names such as v1,v2⋯,vn. Two labeled trees are considered to be distinct if they have different vertex labels even though they might be isomorphic. According to Cayley's tree formula, there are nn−2 labeled trees on n vertices. Prüfer used a simple way to prove this formula and demonstrated that there exists a mapping between a labeled tree and a number sequence. From his proof, we can find a naive sequential algorithm which transfers a labeled tree to a number sequence and vice versa. However, it is hard to parallelize. In this paper, we shall propose an O(log n) time parallel algorithm for constructing a labeled tree by using O(n) processors and O(n log n) space on the EREW PRAM computational model.
1. M.J. Atallah, R. Cole, and M.T. Goodrich, "Cascading Divide-and-Conquer: A Technique for Designing Parallel Algorithms," SIAM J. Computing, vol. 18, no. 3, pp. 499-532, June 1989.
2. M.J. Atallah, and S.R. Kosaraju, "An Efficient Algorithm for Maxdominance, with Applications," Algorithmica, vol. 4, pp. 221-236, 1889.
3. M.T. De Berg, S. Carlsson, and M.H. Overmars, "A General Approach to Dominance in the Plane," J. Algorithms, vol. 13, pp. 274-296, 1992.
4. A. Cayley, "A Theorem on Trees," Quarterly J. Math., vol. 23, pp. 376-378, 1989.
5. R. Cole, "Parallel Merge Sort," Proc. 27th IEEE Symp. Foundations of Computer Science, pp. 511-516, 1986.
6. R. Gould, Graph Theory. Benjamin Cummings. 1988.
7. F. Harary, Graph Theory.Reading, Mass.: Addison-Wesley, 1969.
8. E. Horowitz, and S. Sahni, Fundamentals of Data Structures in Pascal, third edition. Computer Science Press, 1990.
9. J. Jájá, Introduction to Parallel Algorithms. Addison-Wesley, 1992.
10. C.P. Kruskal, L. Rudolph, and M. Snir, "The Power of Parallel Prefix," IEEE Trans. Computers, vol. 34, pp. 965-968, 1985.
11. J.W. Moon, Counting Labeled Trees.Montreal: Canadian Mathematical Congress, 1970.
12. H. Prüfer, "Neuer Beweis eines satzes über Permutationen," Archiv der Mathematik und Physik, vol. 27, pp. 742-744, 1918