Efficient embeddings of ternary trees into hypercubes

Abstract

In this paper we present efficient graph embeddings for complete k-ary trees into boolean hypercubes. In particular, we describe an efficient embedding of a complete ternary tree ( k=3) of height h into a hypercube, which achieves dilation 3 and expansion Θ(1.0104… h ). The previously best-known result is dilation 2 and expansion Θ(1.333… h ). Our embedding achieves exponentially better expansion at the cost of an increase of 1 in the dilation. We also describe efficient embeddings of k-ary trees into hypercubes when k=2 p∗3 q for some p, q>0 such that the embeddings achieve small constant dilation.