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

TL;DR

The paper proves near-optimal lower bounds for exact hopsets and shortcut sets, showing that folklore node-sampling remains essentially optimal for exact hopsets with O(n) edges, by constructing graphs where any such hopset yields a hopbound of \\tilde{Ω}(n^{1/2}). It develops overlapping critical-path constructions, decouples edge and direction vectors, and employs symmetry-breaking (ε-shifting for hopsets and edge-vector subsampling for shortcut sets) to sustain long, unique shortest paths while maintaining rigorous lower bounds. In addition, the results generalize to O(p) sized constructions across p ∈ [1, n^2], and extend the shortcut-set bound to \\tilde{Ω}(n^{1/4}) for the same edge budget, improving prior bounds and clarifying the separation between exact and (1+\\epsilon) hopsets. Overall, the findings establish that folklore sampling is near-optimal in wide parameter regimes and provide a unified lower-bound framework for both hopsets and shortcut sets, with potential implications for diameter reduction in weighted, directed graphs.

Abstract

For a graph $G$, a $D$-diameter-reducing exact hopset is a small set of additional edges $H$ that, when added to $G$, maintains its graph metric but guarantees that all node pairs have a shortest path in $G \cup H$ using at most $D$ edges. A shortcut set is the analogous concept for reachability. These objects have been studied since the early '90s due to applications in parallel, distributed, dynamic, and streaming graph algorithms. For most of their history, the state-of-the-art construction for either object was a simple folklore algorithm, based on randomly sampling nodes to hit long paths in the graph. However, recent breakthroughs of Kogan and Parter [SODA '22] and Bernstein and Wein [SODA '23] have finally improved over the folklore diameter bound of $\widetilde{O}(n^{1/2})$ for shortcut sets and for $(1+ε)$-approximate hopsets. For both objects it is now known that one can use $O(n)$ hop-edges to reduce diameter to $\widetilde{O}(n^{1/3})$. The only setting where folklore sampling remains unimproved is for exact hopsets. Can these improvements be continued? We settle this question negatively by constructing graphs on which any exact hopset of $O(n)$ edges has diameter $\widetildeΩ(n^{1/2})$. This improves on the previous lower bound of $\widetildeΩ(n^{1/3})$ by Kogan and Parter [FOCS '22]. Using similar ideas, we also polynomially improve the current lower bounds for shortcut sets, constructing graphs on which any shortcut set of $O(n)$ edges reduces diameter to $\widetildeΩ(n^{1/4})$. This improves on the previous lower bound of $Ω(n^{1/6})$ by Huang and Pettie [SIAM J. Disc. Math. '18]. We also extend our constructions to provide lower bounds against $O(p)$-size exact hopsets and shortcut sets for other values of $p$; in particular, we show that folklore sampling is near-optimal for exact hopsets in the entire range of $p \in [1, n^2]$.

Figures (3)

• Figure 1: State-of-the-art bounds for $O(n)$-size hopsets and shortcut sets, before and after this paper
• Figure 2: Vertex set of the graph $G$ used for our lower bounds against exact hopsets. Each parameter $\varepsilon_i$ is the amount the $i^{th}$ column is shifted upwards in the plane, relative to the previous column; the $\{\varepsilon_i\}$ values are chosen uniformly and independently from the interval $(0, 1)$.
• Figure 3: Our symmetry-breaking strategy for shortcut set lower bounds starts with a large set of convex vectors, but independently subsamples adjacent pairs of convex vectors to generate the edges between adjacent layers. In this picture, there are $4$ edge vectors and $4$ layers, but only two of the edge vectors (in blue) are sampled and available between any given pair of adjacent layers. For clarity, we have only drawn the edges leaving one particular node in each layer.

Theorems & Definitions (43)

• definition 1: Shortcut Sets
• theorem 1: Folklore, UY91
• theorem 2: KP22a
• definition 2: Hopsets
• theorem 3: Folklore
• theorem 4: KP22aBW23
• theorem 5: New
• theorem 6: New
• Corollary 1
• theorem 7