For an n-vertex directed graph $G = (V,E)$, a $\beta$-\emph{shortcut set} $H$ is a set of additional edges $H \subseteq V \times V$ such that $G \cup H$ has the same transitive closure as $G$, and for every pair $u,v \in V$, there is a $uv$-path in $G \cup H$ with at most $\beta$ edges. A natural generalization of shortcut sets to distances is a $(\beta,\epsilon)$-\emph{hopset} $H \subseteq V \times V$, where the requirement is that $H$ and $G \cup H$ have the same shortest-path distances, and for every $u,v \in V$, there is a $(1+\epsilon)$-approximate shortest path in $G \cup H$ with at most $\beta$ edges. There is a large literature on the tradeoff between the size of a shortcut set / hopset and the value of $\beta$. We highlight the most natural point on this tradeoff: what is the minimum value of $\beta$, such that for any graph $G$, there exists a $\beta$-shortcut set (or a $(\beta,\epsilon)$-hopset) with $O(n)$ edges? Not only is this a natural structural question in its own right, but shortcuts sets / hopsets form the core of many distributed, parallel, and dynamic algorithms for reachability / shortest paths. Until very recently the best known upper bound was a folklore construction showing $\beta = O(n^{1/2})$, but in a breakthrough result Kogan and Parter [SODA 2022] improve this to $\beta = \tilde{O}(n^{1/3})$ for shortcut sets and $\tilde{O}(n^{2/5})$ for hopsets. Our result is to close the gap between shortcut sets and hopsets. That is, we show that for any graph $G$ and any fixed $\epsilon$ there is a $(\tilde{O}(n^{1/3}),\epsilon)$ hopset with $O(n)$ edges. More generally, we achieve a smooth tradeoff between hopset size and $\beta$ which exactly matches the tradeoff of Kogan and Parter for shortcut sets (up to polylog factors). Using a very recent black-box reduction of Kogan and Parter, our new hopset implies improved bounds for approximate distance preservers.

翻译：对于 $n$ 个顶点的有向图 $G = (V,E)$，一个 $\beta$-\textit{捷径集} $H$ 是添加边集 $H \subseteq V \times V$，使得 $G \cup H$ 与 $G$ 具有相同的传递闭包，且对于任意 $u,v \in V$，在 $G \cup H$ 中存在一条不超过 $\beta$ 条边的 $uv$ 路径。捷径集到距离的自然推广是 $(\beta,\epsilon)$-\textit{跳集} $H \subseteq V \times V$，要求 $H$ 与 $G \cup H$ 具有相同的最短路径距离，且对于任意 $u,v \in V$，在 $G \cup H$ 中存在一条边数不超过 $\beta$ 的 $(1+\epsilon)$ 近似最短路径。关于捷径集/跳集大小与 $\beta$ 值之间的权衡已有大量文献。我们关注这一权衡中最自然的问题：对于任意图 $G$，存在一个包含 $O(n)$ 条边的 $\beta$ 捷径集（或 $(\beta,\epsilon)$ 跳集）时，$\beta$ 的最小值是多少？这不仅本身是一个自然的结构性问题，而且捷径集/跳集构成了许多分布式、并行和动态可达性/最短路径算法的核心。直到最近，已知最佳上界仍是经典构造得到的 $\beta = O(n^{1/2})$，但在突破性成果中，Kogan 和 Parter [SODA 2022] 将捷径集改进为 $\beta = \tilde{O}(n^{1/3})$，跳集改进为 $\tilde{O}(n^{2/5})$。我们的结果是弥合捷径集与跳集之间的差距。即，我们证明对于任意图 $G$ 和任意固定 $\epsilon$，存在一个包含 $O(n)$ 条边的 $(\tilde{O}(n^{1/3}),\epsilon)$ 跳集。更一般地，我们实现了跳集大小与 $\beta$ 之间的平滑权衡，该权衡与 Kogan 和 Parter 关于捷径集的权衡完全匹配（至多相差多对数因子）。利用 Kogan 和 Parter 最近的黑盒简化，我们的新跳集改进了近似距离保持器的界限。