Better Diameter Bounds for Efficient Shortcuts and a Structural Criterion for Constructiveness

TL;DR

The paper advances the theory of parallel graph algorithms for directed reachability by introducing certified shortcuts, a structural notion that aligns constructiveness with algorithmic feasibility. It proves a near-linear-time lower bound on depth, $n^{1/4-o(1)}$, for shortcut-based methods, and shows that certifiability is both necessary and sufficient to capture the practicality of existing shortcut/construction techniques. The authors also bridge the gap between lower bounds for shortcuts and hopsets under the certified framework, delivering tighter diameter bounds that match the best previous results in weighted cases. Moreover, the certification-complexity perspective reveals that all known efficient methods are effectively constructive, while many existential proofs lack efficient certifiability, guiding future design toward certifiable constructions with provable depth and size guarantees. Overall, this work offers a unifying structural tool and a rigorous barrier that clarifies what is achievable with current techniques and how to design next-generation shortcut constructions.

Abstract

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.

Figures (7)

• Figure 1: An illustration of the graph construction for \ref{['lem:noncert-uy-sc']} (red = sampled).
• Figure 2: An illustration of the graph construction for \ref{['rmk:uy', 'lem:noncert-brr-sc']} (red = sampled).
• Figure 3: An illustration of the graph construction for \ref{['lem:noncert-kp']} (red = sampled).
• Figure 4: An illustration of the graph construction for \ref{['lem:noncert-cc']} (red: a possible chain).
• Figure 5: Two treaps on sequences ABCDE and FGHIJ respectively, with the priorities represented by y-coordinates, vertices of smaller priority being higher up.

Theorems & Definitions (45)

• Theorem 1.1
• Definition 1.2
• Theorem 1.4
• Definition 1.5
• Theorem 1.6: Informal
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Lemma 2.1: barany1998convex
• Lemma 2.2