Cost tradeoffs in graph embeddings, with applications: Preliminary version

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.