Strengthen the tree-cover embedding (Lemma 5.11) for hyperbolic spanners

Strengthen the tree-based embedding result (Lemma ‘verylargedist’) by either reducing the number of trees needed to cover distances or by constructing a Steiner (1+ε)-spanner with O(d·n) edges for distances o((1/ε)·log(1/ε)) in hyperbolic space.

Background

The paper leverages a union of O(d) trees derived from hyperbolic quadtrees to obtain sparse Steiner spanners and quasi-isometric embeddings at large scales (Lemma ‘verylargedist’). This is central to achieving near-linear edge bounds when points are sufficiently separated.

Improving this lemma—either by using fewer trees or by pushing the regime where O(d·n)-edge Steiner (1+ε)-spanners are possible to distances below (1/ε)·log(1/ε)—would tighten the bounds and expand the applicability of the authors’ framework.