The Space Complexity of Undirected Graph Exploration

Yann Disser, Max Klimm

Abstract: We review the space complexity of deterministically exploring undirected graphs. We assume that vertices are indistinguishable and that edges have a locally unique color that guides the traversal of a space-constrained agent. The graph is considered to be explored once the agent has visited all vertices. We visit results for this setting showing that \(\varTheta \,(\log n)\) bits of memory are necessary and sufficient for an agent to explore all n-vertex graphs. We then illustrate that, if agents only have sublogarithmic memory, the number of (distinguishable) agents needed for collaborative exploration is \(\varTheta \,(\log \log n)\).