Incremental Shortest Paths in Almost Linear Time via a Modified Interior Point Method

TL;DR

This paper presents a deterministic algorithm for maintaining approximate shortest paths in a dynamically changing directed graph with edge insertions, achieving near-linear total running time using a novel interior point method.

Contribution

It introduces a nonstandard interior point method combined with a deterministic min-ratio cycle data structure for efficient dynamic shortest path maintenance.

Findings

Runs in total time $m^{1+o(1)}\, ext{log}\,W$ for $m$ edge insertions.

Maintains $(1+ ext{epsilon})$-approximate shortest paths efficiently.

Works for graphs with edge lengths in $[1, W]$ and any $ ext{epsilon} > ext{exp}(-( ext{log} ext{ }m)^{0.99})$.

Abstract

We give an algorithm that takes a directed graph $G$ undergoing $m$ edge insertions with lengths in $[1, W]$, and maintains $(1+\epsilon)$-approximate shortest path distances from a fixed source $s$ to all other vertices. The algorithm is deterministic and runs in total time $m^{1+o(1)}\log W$, for any $\epsilon > \exp(-(\log m)^{0.99})$. This is achieved by designing a nonstandard interior point method to crudely detect when the distances from $s$ other vertices $v$ have decreased by a $(1+\epsilon)$ factor, and implementing it using the deterministic min-ratio cycle data structure of [Chen-Kyng-Liu-Meierhans-Probst, STOC 2024].