Complexity of Eccentricities and All-Pairs Shortest Paths in the Quantum CONGEST Model

TL;DR

This paper establishes that quantum communication does not significantly speed up the computation of eccentricities and all-pairs shortest paths in distributed networks, matching classical lower bounds.

Contribution

It provides almost linear lower bounds for eccentricities and APSP in the quantum CONGEST model, showing no quantum speedup for these problems.

Findings

Quantum lower bounds match classical bounds for eccentricities and APSP.

Quantum communication can reduce complexity for diameter and radius in low-diameter networks.

No quantum advantage exists for computing eccentricities and APSP in distributed settings.

Abstract

Computing the distance parameters of a network, including the diameter, radius, eccentricities and the all-pairs shortest paths (APSP) is a central problem in distributed computing. This paper investigates he dtistance parameters in the quantum CONGEST models and establishes almost linear lower bounds on eccentricities and APSP, which match the classical upper bounds. Our results imply that there is not quantum speedup for these two problems. In contrast with the diameter and radius, exchanging quantum messages is able to save the communication when the networks have low diameters [Le Gall and Magniez, PODC 2018]. We obtain the lower bounds via a reduction from the two-way quantum communication complexity of the set intersection [Razborov, Izvestiya Mathematics 2003].