Certified shortcuts are precisely the ones constructible in near-linear time, and no almost-linear-size certified shortcut can reduce diameter below n^{1/4-o(1)}.
Set the left tree of the recursive split to be the new right subtree, and return the old root and the right tree of the recursive split
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Better Diameter Bounds for Efficient Shortcuts and a Structural Criterion for Constructiveness
Certified shortcuts are precisely the ones constructible in near-linear time, and no almost-linear-size certified shortcut can reduce diameter below n^{1/4-o(1)}.