Lower Bounds for Adaptive Relaxation-Based Algorithms for Single-Source Shortest Paths

TL;DR

This paper establishes an lower bound for adaptive algorithms solving the single-source shortest path problem, extending previous results to include algorithms that perform linear inequality queries, thus broadening the understanding of algorithmic limitations.

Contribution

The paper generalizes existing lower bounds to adaptive algorithms with linear inequality queries, encompassing Dijkstra-like operations, revealing fundamental computational constraints.

Findings

Lower bound applies to a broad class of adaptive algorithms.

The results unify and extend previous lower bounds for oblivious algorithms.

Implications for the design of shortest path algorithms under adaptive query models.

Abstract

We consider the classical single-source shortest path problem in directed weighted graphs. D.~Eppstein proved recently an $\Omega(n^3)$ lower bound for oblivious algorithms that use relaxation operations to update the tentative distances from the source vertex. We generalize this result by extending this $\Omega(n^3)$ lower bound to \emph{adaptive} algorithms that, in addition to relaxations, can perform queries involving some simple types of linear inequalities between edge weights and tentative distances. Our model captures as a special case the operations on tentative distances used by Dijkstra's algorithm.