# Cost tradeoffs in graph embeddings, with applications

## 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(*n*^{λ}) where λ=log_{3}(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*(*n*^{c}) 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## References

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