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)}.
Hk ofk=O(logn) shortcuts of total size P i |Hi|=O(|H|logn) such that •H⊆E∪H 1 ∪ · · · ∪H k
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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)}.