AI 中文总结
本研究提出随机化d-捷径构造方法，绕过并行可达性的传递闭包壁垒，实现了深度为Õ(1)、总工作量低于现有条件下界的并行可达性算法，证明捷径虽密度接近传递闭包但计算更快。
AI 中文摘要
有向图$G=(V,E)$的$d$-捷径是取自传递闭包$TC(G)$的一个边子集，加入该边子集后可将图的直径降至至多$d$。在$d=1$的特殊情形下，计算1-捷径与计算传递闭包是等价的。对于更大的$d$值，[Hesse, SODA 2003]的下界表明，对于小常数$δ>0$，$n^δ$-捷径可能仍包含$TC(G)$的大部分边，这意味着即使在该参数范围内，捷径构造的难度可能仍与传递闭包相当。因此，由于深度为$\tilde{O}(d)$的并行可达性算法依赖于$d$-捷径的计算，此前通过该方法实现$\tilde{O}(1)$深度都需要计算完整的传递闭包。假设$ω=2$，[Abboud, Bringmann, Fischer, and Künnemann, SODA 2024]提出的PS-AE-Triangle假设给出了条件下界：当$T≤n^{3/2}$（其中$T=|TC(G)|$）时，计算传递闭包存在$T^{4/3-o(1)}$的时间壁垒。\n在本研究中，我们绕过了深度为$\tilde{O}(1)$的并行可达性问题的传递闭包壁垒。我们提出了随机化$d$-捷径构造方法，该方法在$d=3$时即可突破上述壁垒，更一般地，对于所有满足$4≤d≤O(\text{log}n)$的偶数$d$均有效。我们的方法得到了一种随机化的、深度为$\tilde{O}(1)$的并行可达性算法，其总工作量为$\tilde{O}(T^{ω/2})$；当$ω=2$时，总工作量为$\tilde{O}(T)$，在整个参数范围内均低于上述$T^{4/3-o(1)}$的条件壁垒。在当前的$ω$取值下，该算法的工作量为$\tilde{O}(T^{1.186})$，优于Abboud等人提出的传递闭包顺序时间上界$T^{1.3459+o(1)}$。因此，尽管$\tilde{O}(1)$-捷径的密度可能几乎与完整传递闭包相当，但其计算速度却可以快得多。
英文摘要
A $d$-shortcut of a directed graph $G=(V,E)$ is a subset of edges drawn from the transitive closure $TC(G)$ whose addition reduces the graph diameter to at most $d$. In the special case $d=1$, computing a $1$-shortcut is \emph{equivalent} to computing the transitive closure. For larger values of $d$, a lower bound of [Hesse, SODA 2003] shows that $n^δ$-shortcuts, for small constants $δ>0$, may still contain a large fraction of the edges of $TC(G)$, suggesting that shortcut construction may remain as hard as transitive closure even in this regime. Consequently, since $\widetilde{O}(d)$-depth parallel reachability algorithms rely on computing $d$-shortcuts, achieving $\widetilde{O}(1)$ depth by this approach has so far required computing the full transitive closure. Assuming $ω=2$, the PS-AE-Triangle hypothesis of [Abboud, Bringmann, Fischer, and Künnemann, SODA 2024] yields a conditional $T^{4/3-o(1)}$ time barrier for computing transitive closure when $T\leq n^{3/2}$, where $T=|TC(G)|$. In this work, we bypass the transitive-closure barrier for $\widetilde{O}(1)$-depth parallel reachability. We introduce randomized $d$-shortcut constructions that already circumvent this barrier for $d=3$ and, more generally, for every even $d\geq4$ up to $O(\log n)$. Our approach yields a randomized $\widetilde{O}(1)$-depth parallel reachability algorithm with total work $\widetilde{O}(T^{ω/2})$, which becomes $\widetilde{O}(T)$ when $ω=2$, falling below this conditional $T^{4/3-o(1)}$ barrier throughout that regime. Under the current bound of $ω$, this gives $\widetilde{O}(T^{1.186})$ work, improving on the current $T^{1.3459+o(1)}$ sequential-time bound for transitive closure due to Abboud et al. Thus, although $\widetilde{O}(1)$-shortcuts might be almost as dense as the full transitive closure, they can nevertheless be computed substantially faster.
CommentsAccepted to FOCS 2026
