Reducing Shortcut and Hopset Constructions to Shallow Graphs

2025-08-27

AI Summary

This paper addresses the high computational complexity of parallel algorithms for single-source reachability and shortest paths on directed graphs. We propose a generic black-box framework that applies a shallow graph transformation to convert any input graph into an equivalent, structurally simpler, and shallower graph—enabling existing shortcut- and hopset-based parallel algorithms to run directly and efficiently on the transformed instance. Crucially, our framework decouples shortcut and hopset construction while weakening structural assumptions previously required on the original graph. Leveraging near-linear-work parallel graph primitives, our approach significantly simplifies the design and analysis of multiple classical algorithms, enhancing both theoretical interpretability and practical implementability. The framework is model-agnostic, supporting diverse parallel computing models including PRAM and MPC.

Technology Category

Search and Optimization: Heuristic SearchMachine Learning: Graph-based Machine LearningData Mining & Knowledge Management: Scalability, Parallel & Distributed Systems

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsResponsible Web: Human-perceived consequences of algorithmic deployment on the webSystems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We introduce a blackbox framework that simplifies all known parallel algorithms with near-linear work for single-source reachability and shortest paths in directed graphs. Specifically, existing reachability algorithms rely on constructing shortcuts; our blackbox allows these algorithms that construct shortcuts with hopbound $h$ to assume the input graph $G$ is ``shallow'', meaning if vertex $s$ can reach vertex $t$, it can do so in approximately $h$ hops. This assumption significantly simplifies shortcut construction [Fin18, JLS19], resulting in simpler parallel reachability algorithms. Furthermore, our blackbox extends naturally to simplify parallel algorithms for constructing hopsets and, consequently, for computing shortest paths [CFR20 , CF23 , RHM+23 ].

Problem

• Simplifying parallel algorithms for single-source reachability
• Reducing shortcut and hopset constructions to shallow graphs
• Enabling simpler computation of shortest paths in directed graphs

Innovation

• Blackbox framework simplifies parallel reachability algorithms
• Reduces shortcut construction to shallow graph assumptions
• Extends naturally to simplify parallel hopset algorithms