Optimal Dynamic Edge-Disjoint Embeddings of Complete Binary Trees into Hypercubes

The double-rooted complete binary tree is a complete binary tree where the path (of length ) between the children of the root is replaced by a path of length . It is folklore that the double-rooted complete binary tree is a spanning tree of the hypercube of the same size. Unfortunately, the usual construction of an embedding of a double-rooted complete binary tree into the hypercube does not provide any hint how this embedding can be extended if each leaf spawns two new leaves. In this paper, we present simple dynamic embeddings of double-rooted complete binary trees and, therefore, of complete binary trees into hypercubes which do not suffer from this disadvantage. We also give an edge-disjoint embedding with optimal load and unit dilation such that each hypercube vertex is an image of at most one vertex of a level. Moreover, these embeddings can be efficiently implemented on the hypercube itself such that the embedding of each new level of leaves can be computed in constant time. Since complete binary trees are similar to double-rooted complete binary trees, we can transfer our results immediately to complete binary trees.