# Faster Algorithms for All Pairs Non-decreasing Paths Problem

## TL;DR

This paper introduces faster algorithms for the All Pairs Non-decreasing Paths problem, achieving near-optimal time complexity on weighted digraphs and optimal time on undirected graphs, advancing the efficiency of path computations.

## Contribution

It presents an improved algorithm with a time complexity of O(n^{2.686}) for weighted digraphs and an O(n^2) algorithm for undirected graphs, matching or surpassing previous bounds.

## Key findings

- Achieves O(n^{2.686}) time for weighted digraphs.
- Provides an O(n^2) time algorithm for undirected graphs.
- Matches the best known bounds for related matrix product problems.

## Abstract

In this paper, we present an improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs, which has running time $\tilde{O}(n^{\frac{3 + \omega}{2}}) = \tilde{O}(n^{2.686})$. Here $n$ is the number of vertices, and $\omega < 2.373$ is the exponent of time complexity of fast matrix multiplication [Williams 2012, Le Gall 2014]. This matches the current best upper bound for $(\max, \min)$-matrix product [Duan, Pettie 2009] which is reducible to APNP. Thus, further improvement for APNP will imply a faster algorithm for $(\max, \min)$-matrix product. The previous best upper bound for APNP on weighted digraphs was $\tilde{O}(n^{\frac{1}{2}(3 + \frac{3 - \omega}{\omega + 1} + \omega)}) = \tilde{O}(n^{2.78})$ [Duan, Gu, Zhang 2018]. We also show an $\tilde{O}(n^2)$ time algorithm for APNP in undirected graphs which also reaches optimal within logarithmic factors.