Optimal Dynamic Embeddings of Complete Binary Trees into Hypercubes
The double-rooted complete binary tree is a complete binary tree where the
root is replaced by an edge. 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 into hypercubes which do not suffer from this disadvantage.
We also present edge-disjoint embeddings of large binary trees with optimal
load and unit dilation. Furthermore, all 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, our
results can be immediately transfered to complete binary trees.
