Improve epsilon dependence from 1/epsilon^d to 1/epsilon^{d-1} in Euclidean dependable spanners

Determine whether the dependence on the stretch parameter epsilon in the edge-count bound of the dependable (1+epsilon)-spanner for point sets in R^d under independent edge failures with survival probability p can be improved from order 1/epsilon^d to order 1/epsilon^{d-1}, up to polylogarithmic factors.

Background

The paper constructs dependable (1+epsilon)-spanners for n points in R^d that, under independent edge failures with survival probability p, preserve short paths for almost all pairs, while using O(C n log n) edges with C ≈ epsilon^{-d} p^{-4/3} (up to polylogarithmic factors).

This epsilon dependence arises from the use of locality-sensitive orderings (LSOs) whose size scales roughly as epsilon^{-d}. The authors note recent advances on LSOs suggesting that an epsilon^{-(d-1)} dependence might be achievable, motivating the question whether the overall spanner size can match this improved dependence.