The paper analyzes the performance of the folklore algorithm for constructing exact hopsets, which is a simple randomized algorithm that has been the state-of-the-art for a long time.
The key insights are:
The authors construct a family of n-node weighted undirected graphs where any exact hopset of size O(n) can only reduce the diameter to $\Omega(\sqrt{n})$. This shows that the folklore algorithm, which achieves a diameter bound of $\tilde{O}(\sqrt{n})$, is near-optimal.
The authors achieve this lower bound by allowing the paths in their construction to overlap, in contrast to prior work which required the paths to be edge-disjoint. This relaxation allows them to pack more paths into the construction and obtain the improved lower bound.
The authors also provide a polynomial improvement on the lower bound for shortcut sets, showing that any shortcut set of size O(n) can only reduce the diameter to $\Omega(n^{1/4})$, improving the previous bound of $\Omega(n^{1/6})$.
The authors extend their constructions to provide lower bounds against exact hopsets and shortcut sets of size O(p) for a wider range of parameters p.
Overall, the paper settles the optimality of the folklore algorithm for exact hopsets, confirming that its diameter bound is essentially the best possible.
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by Greg Bodwin,... at arxiv.org 04-23-2024
https://arxiv.org/pdf/2304.02193.pdfDeeper Inquiries