Randomized Lower Bound for Distributed Spanning-Tree Verification

Abstract

The distributed verification is the problem of deciding whether the subgraph induced by an input edge set L has a desired property (e.g., spanning trees, connectivity, cycle containment, and so on) or not. In this paper, we consider the lower bounds for the distributed verification of spanning trees and Hamiltonian paths. While the original work of the distributed verification by Das Sarma et al. [1] has shown their $\tilde{\Omega}(\sqrt{n})$-round lower bounds, that result is applied only for deterministic algorithms. Recently, their randomized lower bounds are proved by Elkin et al. [3], but the proof strategy is quite complicated. The primary contribution of this paper is that the same randomizied lower bounds are obtained by a simple and elementary reduction from the well-known two-party communication complexity of the set-disjointness function. We also show a tight lower bound for the verification problem of low-diameter spanning trees. By a simple modification of our proof, we can show that the randomized $\Omega(\min\{\sqrt{n} / \log n, h\})$-round lower bound holds for the verification of spanning trees with diameter h. This result implies that the naive approach (i.e., the breadth-first search along the edges in L) is the best possible for the verification of low-diameter spanning trees.

This work is supported in part by KAKENHI No.25106507 and No.25289114.