The Nearest Unvisited Vertex Walk on Random Graphs

TL;DR

This paper studies a deterministic walk on graphs that visits the nearest unvisited vertex, linking cover time to metric entropy, and uses first passage percolation to estimate cover times in random graphs.

Contribution

It connects cover time analysis with metric entropy and applies first passage percolation to estimate cover times in random graph models.

Findings

Cover time relates to ball-covering measures.

First passage percolation estimates can determine cover time order.

Sharper bounds on cover time remain an open challenge.

Abstract

We revisit an old minor topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and ball-covering (metric entropy) measures. For some familiar models of random graphs, this connection allows the order of magnitude of the cover time to be deduced from first passage percolation estimates. Establishing sharper results seems a challenging problem.