All parallel algorithms for directed connectivity and shortest paths crucially rely on efficient shortcut constructions that add a linear number of transitive closure edges to a given DAG to reduce its diameter. A long sequence of works has studied both (efficient) shortcut constructions and impossibility results on the best diameter and therefore the best parallelism that can be achieved with this approach. This paper introduces a new conceptual and technical tool, called certified shortcuts, for this line of research in the form of a simple and natural structural criterion that holds for any shortcut constructed by an efficient (combinatorial) algorithm. It allows us to drastically simplify and strengthen existing impossibility results by proving that any near-linear-time shortcut-based algorithm cannot reduce a graph's diameter below $n^{1/4-o(1)}$. This greatly improves over the $n^{2/9-o(1)}$ lower bound of [HXX25] and seems to be the best bound one can hope for with current techniques. Our structural criterion also precisely captures the constructiveness of all known shortcut constructions: we show that existing constructions satisfy the criterion if and only if they have known efficient algorithms. We believe our new criterion and perspective of looking for certified shortcuts can provide crucial guidance for designing efficient shortcut constructions in the future.

翻译：所有针对有向连通性和最短路径的并行算法，其核心都依赖于高效的捷径构造，即在给定的有向无环图中添加线性数量的传递闭包边以缩减其直径。一系列研究工作已深入探讨了（高效）捷径构造以及关于最佳直径（进而通过此方法可实现的最佳并行度）的不可能性结果。本文为该研究方向引入了一种新的概念与技术工具——认证捷径，其形式为一个简单而自然的**结构性准则**，该准则适用于任何由高效（组合）算法构造的捷径。利用这一工具，我们能够极大简化并强化现有的不可能性结果：证明任何基于捷径的近似线性时间算法都无法将图的直径降至 $n^{1/4-o(1)}$ 以下。这显著改进了 [HXX25] 中 $n^{2/9-o(1)}$ 的下界，且似乎是当前技术下可期望的最佳界。我们的结构性准则也精确刻画了所有已知捷径构造的可构造性：我们证明，现有构造满足该准则当且仅当其存在已知的高效算法。我们相信，这一新准则以及寻找认证捷径的视角，能为未来设计高效的捷径构造提供关键指导。