Cost Trade-offs in Graph Embeddings, with Applications

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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 situaUons 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 O(n~), where ~ = 1og3(4/3); but any embedding of these ternary trees in binary trees that has expansion-cost c < 2 must have dilation-cost G(logloglogn). The second result provides a stronger but less easily stated example of the same type of trade-off. 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 dilauon-cost O(1) and expansion-cost O(n ~) for some fixed constant c; if one insists on embeddings whose dilation-cost is exactly 1, then these embeddings must have expansion-cost f~(n¢~°*~)/~); tf one insists on embeddmgs whose expansion-cost is less than 2, then these embeddings must have dilation cost ~(log log log n) An interesting application of the polynomial size genenc binary tree m the first part of this three-part result is to yield simplified proofs of several results concerning computational systems with an intrinsic nouon of "computation tree," such as alternating and nondeterministic Turing machines and context-free grammars.

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