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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)

Abstract

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.

Keywords

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.

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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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