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arXiv · 2506.19207

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

Yang P. Liu
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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].

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Yang P. Liu. 2025-06-24. Incremental Shortest Paths in Almost Linear Time via a Modified Interior Point Method. https://arxiv.org/abs/2506.19207

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