Improve the dependence on p in the dependable spanner construction

Improve the dependence on the edge survival probability p in the edge-count bound for dependable spanners constructed for point sets in R^d under independent edge failures, without degrading the deficiency or hop-length guarantees.

Background

In the presented constructions, the size of the dependable spanner scales as O(n log n) times a factor that includes p^{-4/3} (up to polylogarithmic terms), reflecting the effort needed to preserve short paths despite edge failures occurring independently with probability 1 - p.

The authors ask whether this p-dependence can be further reduced, which would strengthen the robustness-size tradeoff especially in regimes where p is small.