# Reducing Shortcut and Hopset Constructions to Shallow Graphs

## TL;DR

This paper presents a blackbox framework that simplifies parallel algorithms for reachability and shortest paths by reducing the problem to shallow graphs, making shortcut and hopset constructions more straightforward.

## Contribution

It introduces a blackbox approach that transforms existing algorithms to assume shallow graphs, simplifying the construction of shortcuts and hopsets in parallel algorithms.

## Key findings

- Simplifies parallel reachability algorithms using the blackbox approach.
- Extends to simplify hopset construction and shortest path algorithms.
- Achieves more straightforward parallel algorithms with near-linear work.

## 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 ].