Cost tradeoffs in graph embeddings, with applications

Preliminary version
  • Hong Jia-Wei
  • Kurt Mehlhorn
  • Arnold L. Rosenberg
Session 2: F.P. Preparata, Chairman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 115)


An embedding of the graph G in the graph H is a one-to-one association of the vertices of G with the vertices of H. There are two natural measures of the cost of a graph embedding, namely, the dilation-cost of the embedding: the maximum distance in H between the images of vertices that are adjacent in G, and the expansion-cost of the embedding: the ratio of the size of H to the size of G. The main results of this paper illustrate three situations wherein one of these costs can be minimized only at the expense of a dramatic increase in the other cost. The first result establishes the following: there is an embedding of n-node complete ternary trees in complete binary trees with dilation-cost 2 and expansion-cost Θ(nλ) where λ=log3(4/3); but any embedding of these ternary trees in binary trees that has expansion-cost <2 must, infinitely often, have dilation-cost ⩾ (const)log log log n. The second result provides a stronger but less easily stated example of the same type of tradeoff. The third result concerns generic binary trees, that is, complete binary trees into which all n-node binary trees are "efficiently" embeddable. There is a generic binary tree into which all n-node binary trees are embeddable with dilation-cost O(1) and expansion-cost O(nc) for some fixed constant c; if one insists on embeddings whose dilation-cost is exactly 1, then these embeddings must have expansion-cost ω(n(log n)/2); if one insists on embeddings whose expansion-cost is <2, then these embeddings must, infinitely often, have dilation-cost ⩾ (const)log log log n. An interesting application of the polynomial size generic binary tree in the first part of this three-part result is to yield simplified proofs of several results concerning computational systems with an intrinsic notion of "computation tree", such as alternating and nondeterministic Turing machines and context-free grammars.


Generic Tree Binary Tree Turing Machine Host Tree Computation Tree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    R. Aleliunas and A. L. Rosenberg: On embedding rectangular grids in square grids. IBM Report RC-8404, 1980; submitted for publication.Google Scholar
  2. 2.
    A. Borodin, M. J. Fischer, D. Kirkpatrick, N. A. Lynch, M. Tompa: A time-space tradeoff for sorting on nonoblivious machines. Proc. 20th FOCS Symp., 1979, 319–327.Google Scholar
  3. 3.
    J. W. S. Cassels: An Introduction to Diophantine Approximation, Cambridge Tracts in Math. and Math. Physics, No. 45, Cambridge U. Press, Cambridge (1957).Google Scholar
  4. 4.
    A. K. Chandra, D. C. Kozen, L. J. Stockmeyer: Alternation. J. ACM 28 (1981) 114–133.Google Scholar
  5. 5.
    F. R. K. Chung, D. Coppersmith, R. L. Graham: On trees containing all small trees. Typescript, 1979.Google Scholar
  6. 6.
    A. Cobham: The recognition problem for the set of perfect squares. Proc. 7th SWAT Symp., 1966, 78–87.Google Scholar
  7. 7.
    R. A. DeMillo, S. C. Eisenstat, R. J. Lipton: On small universal data structures and related combinatorial problems. Proc. Johns Hopkins Conf. on Inf. Sci. and Syst., 1978, 408–411.Google Scholar
  8. 8.
    J.-W. Hong: On similarity and duality of computation. J. CSS, to appear.Google Scholar
  9. 9.
    J.-W. Hong and A. L. Rosenberg: Graphs that are almost binary trees. Typescript, 1980; submitted for publication; see also Proc. 13th ACM Symp. on Theory of Computing, 1981.Google Scholar
  10. 10.
    H. T. Kung and D. Stevenson: A software technique for reducing the routing time on a parallel computer with a fixed interconnection network. In High Speed Computer and Algorithm Optimization, Academic Press, New York, 1977, pp. 423–433.Google Scholar
  11. 11.
    P. M. Lewis, R. E. Stearns, J. Hartmanis: Memory bounds for recognition of context-free and context-sensitive languages. Proc. 6th Conf. on Switching Circuit Theory and Logical Design, 1965, 191–202.Google Scholar
  12. 12.
    R. J. Lipton, S. C. Eisenstat, R. A. DeMillo: Space and time hierarchies for collections of control structures and data structures. J. ACM 23 (1976) 720–732.Google Scholar
  13. 13.
    R. J. Lipton and R. E. Tarjan: A separator theorem for planar graphs. SIAM J. Appl. Math. 36 (1979) 177–189.Google Scholar
  14. 14.
    R. J. Lipton and R. E. Tarjan: Applications of a planar separator theorem. Proc. 18th FOCS Symp., 1977, 162–170.Google Scholar
  15. 15.
    K. Mehlhorn: Best possible bounds on the weighted path length of optimum binary search trees. SIAM J. Comput. 6 (1977) 235–239.Google Scholar
  16. 16.
    A. L. Rosenberg: Encoding data structures in trees. J. ACM 26 (1979) 668–689.Google Scholar
  17. 17.
    A. L. Rosenberg: Issues in the study of graph embeddings. In Graph-Theoretic Concepts in Computer Science (Proc. of the Int'l Workshop WG80), H. Noltemeier, ed. Lecture Notes in Computer Science, vol. 100, Springer-Verlag, NY, 1981, to appear.Google Scholar
  18. 18.
    A. L. Rosenberg and L. Snyder: Bounds on the costs of data encodings. Math. Syst. Th. 12 (1978) 9–39.Google Scholar
  19. 19.(a)
    W. L. Ruzzo: Tree-size bounded alternation. J. CSS 21 (1980) 218–235Google Scholar
  20. 19.(b)
    see also preliminary version in Proc. 11th ACM Symp. on Theory of Computing, 1979, 352–359.Google Scholar
  21. 20.
    W. J. Savitch: Relationships between nondeterministic and deterministic tape complexities. J. CSS 4 (1970) 177–192.Google Scholar
  22. 21.
    M. Sekanina: On an ordering of the set of vertices of a connected graph. Publ. Fac. Sci. Univ. Brno, No. 412 (1960) 137–142.Google Scholar
  23. 22.
    C. D. Thompson: Area-time complexity for VLSI. Proc. 11th ACM Symp. on Theory of Computing, 1979, 81–88.Google Scholar
  24. 23.
    L. G. Valiant: Universality considerations in VLSI circuits. Univ. of Edinburgh Report CSR-54-80, 1980; to appear in IEEE Trans. Elec. Comp..Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1981

Authors and Affiliations

  • Hong Jia-Wei
    • 1
  • Kurt Mehlhorn
    • 2
  • Arnold L. Rosenberg
    • 3
  1. 1.Peking Municipal Computing CentrePekingChina
  2. 2.Angewandte Mathematik und InformatikUniversität des SaarlandesSaarbrückenFederal Republic of Germany
  3. 3.Mathematical Sciences DepartmentIBM Research CenterYorktown HeightsUSA

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