Abstract
We consider several graph embedding problems which have applications in parallel and distributed computing and which have been unsolved so far. Our major result is that the complete binary tree can be embedded into the square grid of the same size with almost optimal dilation (up to a very small factor). To achieve this, we first state an embedding of the complete binary tree into the line with optimal dilation.
The work of these authors was supported by grant Mo 285/4-1 from the German Research Association (DFG).
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References
M.J. Atallah, S.R. Kosaraju, ”Optimal simulations between mesh-connected arrays of processors”, Journal of the ACM, vol. 35 (1988), pp. 635–650.
R. Aleliunas, A.L. Rosenberg, ”On Embedding Rectangular Grids in Square Grids”, IEEE Transactions on Computers, vol. C-31 (1982), pp. 907–913.
P.Z. Chinn, J. Chvátalová, A.K. Dewedney, and N.E. Gibbs. The bandwidth problem for graphs and matrices — a survey. Journal of Graph Theory, 6:223–254, 1982.
F.R.K. Chung, ”Labelings of Graphs”, in Selected Topics in Graph Theory III, edited by L.W. Beineke and R.J. Wilson, Academic Press, (1988) pp. 151–168.
J.A. Ellis, ”Embedding Rectangular Grids into Square Grids”, Proceedings of the 3rd Aegean Workshop on Computing (AWOC): VLSI Algorithms and Architectures (1988), LNCS 319, pp. 181–190.
M.J. Fischer, M.S. Paterson, ”Optimal Tree Layout”, Proceedings of the 12th ACM Symposium on the Theory of Computing (1980), pp. 177–189.
M.R. Garey, D.S. Johnson, ”Computers and Intractability”, W.H. Freeman, New York, 1979.
D. Gordon, ”Efficient Embeddings of Binary Trees in VLSI Arrays”, IEEE Transactions on Computers, vol. C-36 (1987), pp. 1009–1018.
J. Haralambides, F. Makedon, B. Monien, ”Approximation Algorithms for the Bandwidth Minimization Problem for Caterpillar Graphs”, Proceedings of the 2nd IEEE Symposium on Parallel and Distributed Processing, (1990), pp. 301–307.
B. Monien, ”The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete”, SIAM J. Alg. and Discrete Methods, 7 (1986), pp. 505–512.
B. Monien, ”Simulating Binary Trees on X-trees”, Proceedings of the 3rd A CM Symposium on Parallel Algorithms and Architectures (1991).
B. Monien, I.H. Sudborough, ”Embedding one Interconnection Network in Another”, Computing Suppl. 7 (1990), pp. 257–282.
mS. Paterson, W.L. Ruzzo, L. Snyder, ”Bounds on minimax edge length for complete binary trees”, Proceedings of the 13th ACM Symposium on the Theory of Computing (1981), pp. 293–299.
A.L. Rosenberg, ”Graph embeddings 1988: Recent breakthroughs, new directions”, Proceedings of the 3rd Aegean Workshop on Computing (AWOC): VLSI Algorithms and Architectures (1988), LNCS 319, pp. 160–169.
W.L. Ruzzo, L. Snyder, ”Minimum Edge Length Planar Embeddings of Trees”, in Kung, Sproull, Steele: VLSI Systems and Computations, Computer Science Press (1981), pp. 119–123.
A.D. Singh, H.Y. Youn, ”Near Optimal Embedding of Binary Tree Architectures in VLSI”, Proceedings of the 8th International Conference on Distributed Computing Systems (1988), pp. 86–93.
J.D. Ullman, ”Computational Aspects of VLSI”, Computer Science Press, 1984.
P. Zienicke, ”Embedding of Treelike Graphs into 2-dimensional Meshes”, Proceedings of the 16th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 90).
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© 1992 Springer-Verlag Berlin Heidelberg
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Heckmann, R., Klasing, R., Monien, B., Unger, W. (1992). Optimal embedding of complete binary trees into lines and grids. In: Schmidt, G., Berghammer, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 1991. Lecture Notes in Computer Science, vol 570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55121-2_3
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DOI: https://doi.org/10.1007/3-540-55121-2_3
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