# Almost Shortest Paths with Near-Additive Error in Weighted Graphs

## TL;DR

This paper introduces new algorithms for almost shortest paths in weighted graphs that achieve near-additive error bounds with improved efficiency in both centralized and parallel models, depending on local maximum edge weights.

## Contribution

It presents a novel centralized and parallel algorithm for ASP with near-additive error depending on local max edge weight, extending prior work with improved bounds and new constructions.

## Key findings

- Centralized algorithm runs in $O((m+ ns)n^
ho)$ time.
- PRAM algorithm has polylogarithmic depth and similar work complexity.
- Provides additive approximations for all pairs shortest paths with error depending on local maximum edge weight.

## Abstract

Let $G=(V,E,w)$ be a weighted undirected graph with $n$ vertices and $m$ edges, and fix a set of $s$ sources $S\subseteq V$. We study the problem of computing {\em almost shortest paths} (ASP) for all pairs in $S \times V$ in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of $1+\epsilon$, for an arbitrarily small constant $\epsilon > 0$ . In this regime existing centralized algorithms require $\Omega(\min\{|E|s,n^\omega\})$ time, where $\omega < 2.372$ is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work $\Omega(\min\{|E|s,n^\omega\})$. Our centralized algorithm has running time $O((m+ ns)n^\rho)$, and its PRAM counterpart has polylogarithmic depth and work $O((m + ns)n^\rho)$, for an arbitrarily small constant $\rho > 0$. For a pair $(s,v) \in S\times V$, it provides a path of length $\hat{d}(s,v)$ that satisfies $\hat{d}(s,v) \le (1+\epsilon)d_G(s,v) + \beta \cdot W(s,v)$, where $W(s,v)$ is the weight of the heaviest edge on some shortest $s-v$ path. Hence our additive term depends linearly on a {\em local} maximum edge weight, as opposed to the global maximum edge weight in previous works. Finally, our $\beta = (1/\rho)^{O(1/\rho)}$. We also extend a centralized algorithm of Dor et al. \cite{DHZ00}. For a parameter $\kappa = 1,2,\ldots$, this algorithm provides for {\em unweighted} graphs a purely additive approximation of $2(\kappa -1)$ for {\em all pairs shortest paths} (APASP) in time $\tilde{O}(n^{2+1/\kappa})$. Within the same running time, our algorithm for {\em weighted} graphs provides a purely additive error of $2(\kappa - 1) W(u,v)$, for every vertex pair $(u,v) \in {V \choose 2}$, with $W(u,v)$ defined as above. On the way to these results we devise a suit of novel constructions of spanners, emulators and hopsets.