Quantum algorithms and lower bounds for eccentricity, radius, and diameter in undirected graphs

TL;DR

This paper introduces quantum algorithms for efficiently computing and approximating the diameter and radius of undirected graphs, along with establishing fundamental lower bounds on their quantum query complexity.

Contribution

It presents novel quantum algorithms for diameter and radius computation, including an approximation method, and proves lower bounds via reductions from minima finding.

Findings

Quantum algorithms run in rac{n ilde{O}(m)}{} time for diameter and radius.

An approximation algorithm achieves a 2/3 ratio in rac{ ilde{O}(\

Lower bounds of rac{\

Abstract

The problems of computing eccentricity, radius, and diameter are fundamental to graph theory. These parameters are intrinsically defined based on the distance metric of the graph. In this work, we propose quantum algorithms for the diameter and radius of undirected, weighted graphs in the adjacency list model. The algorithms output diameter and radius with the corresponding paths in $\widetilde{O}(n\sqrt{m})$ time. Additionally, for the diameter, we present a quantum algorithm that approximates the diameter within a $2/3$ ratio in $\widetilde{O}(\sqrt{m}n^{3/4})$ time. We also establish quantum query lower bounds of $\Omega(\sqrt{nm})$ for all the aforementioned problems through a reduction from the minima finding problem.