Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

TL;DR

This paper establishes tight bounds for distributed verification of minimum-weight spanning trees, showing that verification can be more efficient than construction, with matching upper and lower bounds on time and message complexities.

Contribution

It provides the first tight bounds for distributed MST verification without preprocessing, revealing that verification may be easier than construction.

Findings

MST verification can be achieved with O(m) messages and O(√n+D) time.

Any MST verification algorithm requires at least Ω(m) messages and Ω(√n+D) time.

Verification complexity is close to the bounds of MST construction, indicating similar difficulty levels.

Abstract

This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously O(m) messages and O($\sqrt$ n+D) time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G's diameter. On the other hand, we show that any MST verification algorithm must send {\Omega}(m) messages and incur {\Omega}($\sqrt$ n + D) time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of {\Omega}(m) messages and {\Omega}($\sqrt$ n+D time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously O(m) messages and O($\sqrt$ n + D) time. Specifically, the best known time-optimal algorithm (using O($\sqrt$ n + D) time) requires O(m + n 3/2) messages, and the best known message-optimal algorithm (using O(m) messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.