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Optimal Folklore Sampling for Exact Hopsets: Confirming the $\sqrt{n}$ Barrier


Core Concepts
The folklore algorithm for constructing exact hopsets on O(n) edges is near-optimal, as any such hopset can only reduce the diameter of the input graph to $\Omega(\sqrt{n})$.
Abstract
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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Deeper Inquiries

Can the gap between the upper and lower bounds for shortcut sets of size O(n) be further narrowed

The gap between the upper and lower bounds for shortcut sets of size O(n) can potentially be further narrowed. The current lower bound for shortcut sets is eΩ(n1/4), while the upper bound is eO(n1/3). This leaves a polynomial gap between the two bounds. By exploring different constructions and techniques, it may be possible to refine the lower bound and bring it closer to the upper bound. Further research and analysis could lead to a more precise understanding of the optimal diameter reduction achievable with O(n)-size shortcut sets.

What is the best achievable diameter bound for exact hopsets in unweighted graphs using O(n) edges

The best achievable diameter bound for exact hopsets in unweighted graphs using O(n) edges is eΩ(n1/2). This lower bound was established in the context of the research presented in the provided text. The construction and analysis demonstrated that any exact hopset of size O(n) would reduce the diameter to eΩ(n1/2) in certain scenarios. This result highlights the limitations and optimal performance of folklore sampling for exact hopsets in unweighted graphs.

Are there applications or settings where the improved lower bounds proven in this paper would have significant practical implications

The improved lower bounds for shortcut sets and exact hopsets proven in the research paper have significant practical implications in various applications. For example, in parallel, distributed, dynamic, or streaming settings where algorithm complexity is influenced by the diameter of the graph, having a better understanding of the optimal reduction achievable with shortcut sets and hopsets can lead to more efficient and effective algorithms. These lower bounds provide insights into the fundamental limits of diameter reduction using a small set of additional edges, which can impact the design and optimization of graph algorithms in real-world scenarios.
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