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

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 the distance 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 [F. L. Gall and F. Magniez, Sublinear-time quantum computation of the diameter in CONGEST networks, in Proc. 2018 ACM Symp. Principles of Distributed Computing (PODC), (2018), pp. 337–346; F. Magniez and A. Nayak, arXiv:2002.11795 ]. We obtain the lower bounds via a reduction from the two-way quantum communication complexity of the set intersection [A. A. Razborov, Izv. Math. 159 (2003)].