Discussiones Mathematicae Graph Theory 28(3) (2008) 463-476
doi: 10.7151/dmgt.1420

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S.A. Choudum  and  S. Lavanya

Department of Mathematics
Indian Institute of Technology Madras
Chennai 600 036, India
e-mail: sac@iitm.ac.in
e-mail: s.lavanya@yahoo.com


We inductively describe an embedding of a complete ternary tree Th of height h into a hypercube Q of dimension at most ⎡(1.6)h⎤+1 with load 1, dilation 2, node congestion 2 and edge congestion 2. This is an improvement over the known embedding of Th into Q. And it is very close to a conjectured embedding of Havel [3] which states that there exists an embedding of Th into its optimal hypercube with load 1 and dilation 2. The optimal hypercube has dimension ⎡(log23)h ⎤ ( = ⎡(1.585)h ⎤) or ⎡(log23)h⎤+1.

Keywords: complete ternary trees, hypercube, interconnection network, embedding, dilation, node congestion, edge congestion.

2000 Mathematics Subject Classification: 05C05, 05C90, 68E10.


[1] S.L. Bezrukov, Embedding complete trees into the hypercube, Discrete Math. 110 (2001) 101-119, doi: 10.1016/S0166-218X(00)00256-0.
[2] A.K. Gupta, D. Nelson and H. Wang, Efficient embeddings of ternary trees into hypercubes, Journal of Parallel and Distributed Computing 63 (2003) 619-629, doi: 10.1016/S0743-7315(03)00037-6.
[3] I. Havel, On certain trees in hypercube, in: R. Bodendick and R. Henn, eds, Topics in Combinatorics and Graph Theory (Physica-Verlag, Heidelberg, 1990) 353-358.
[4] X.J. Shen, Q. Hu and W.F. Liang, Embedding k-ary complete trees into hypercubes, Journal of Parallel and Distributed Computing 24 (1995) 100-106, doi: 10.1006/jpdc.1995.1010.
[5] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays, Trees, Hypercubes (Morgan Kauffmann, San Mateo, CA, 1992).
[6] B. Monien and H. Sudbourough, Embedding one interconnection network into another, Computing Supplementary 7 (1990) 257-282.
[7] J. Trdlicka and P. Tvrdík, Embedding of k-ary complete trees into hypercubes with uniform load, Journal of Parallel and Distributed Computing 52 (1998) 120-131, doi: 10.1006/jpdc.1998.1464.
[8] A.Y. Wu, Embedding of tree networks in hypercube, Journal of Parallel and Distributed Computing 2 (1985) 238-249, doi: 10.1016/0743-7315(85)90026-7.

Received 24 October 2007
Revised 12 May 2008
Accepted 12 May 2008