On efficient entreeings

Abstract

A data encoding is a formal model of how a logical data structure is mapped into or represented in a physical storage structure. Both structures are complete trees in this paper, and we encode the logical or guest tree in the leaves of the physical or host tree giving a restricted class of encodings called entreeings. The cost of an entreeing is the total amount that the edges of the guest tree are stretched or dilated when they are replaced by shortest paths in the host tree. We are particularly interested in the asymptotic average cost of families of similar entreeings.

Our investigation is a continuation of the study initiated by Rosenberg et al. In particular, the paper contains the following results.

• 1. We refute a conjecture of Rosenberg et al. that a particular family of entreeings of binary guests into binary hosts is optimal.

• 2. We provide an efficient family of entreeings fork-ary guests intok-ary hosts, fork≧2.

• 3. We provide an efficient family of entreeings ofk-ary guests into binary hosts, fork≧3.

• 4. We provide a new simple lower-bound technique that can be applied to the entreeings in part 2 to prove that they are very close to optimal. Moreover, it can be adapted for the entreeings of part 3, in which case we are able to show near optimality whenk is sufficiently large.