Doubling Dimension Independent Exponent

Background: The update time complexity for dynamic light spanners in doubling metrics is currently polynomial in $\log\Phi$ with an exponent dependent on the doubling dimension ($d$).

Question / Future Work: Can the runtime bound for dynamic light spanners be improved such that the exponent of $\log\Phi$ (or $\log n$) is independent of the doubling dimension $d$? Ideally, the goal is a runtime of $O(\log n)$ with an exponent of $O(d)$.