# Reliable Hubs for Partially-Dynamic All-Pairs Shortest Paths in Directed Graphs

## TL;DR

This paper introduces new deterministic and randomized algorithms for efficiently maintaining approximate all-pairs shortest paths in directed graphs under dynamic updates, significantly improving update times and reliability.

## Contribution

It presents the first deterministic incremental algorithm with subquadratic update time for all-pairs shortest paths in directed graphs and enhances randomized algorithms to Las Vegas guarantees.

## Key findings

- Deterministic incremental algorithm with $ ilde{O}(mn^{4/3} ext{polylog}(W)/ extepsilon)$ total update time.
- Conversion of existing randomized algorithms from Monte Carlo to Las Vegas without increased running time.
- New algorithms for dynamically maintaining hubs that hit shortest paths with certain hop-lengths.

## Abstract

We give new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs. Most importantly, we give a new deterministic incremental algorithm for the problem that handles updates in $\widetilde{O}(mn^{4/3}\log{W}/\epsilon)$ total time (where the edge weights are from $[1,W]$) and explicitly maintains a $(1+\epsilon)$-approximate distance matrix. For a fixed $\epsilon>0$, this is the first deterministic partially dynamic algorithm for all-pairs shortest paths in directed graphs, whose update time is $o(n^2)$ regardless of the number of edges. Furthermore, we also show how to improve the state-of-the-art partially dynamic randomized algorithms for all-pairs shortest paths [Baswana et al. STOC'02, Bernstein STOC'13] from Monte Carlo randomized to Las Vegas randomized without increasing the running time bounds (with respect to the $\widetilde{O}(\cdot)$ notation). Our results are obtained by giving new algorithms for the problem of dynamically maintaining hubs, that is a set of $\widetilde{O}(n/d)$ vertices which hit a shortest path between each pair of vertices, provided it has hop-length $\Omega(d)$. We give new subquadratic deterministic and Las Vegas algorithms for maintenance of hubs under either edge insertions or deletions.