A General Lower Bound for Collaborative Tree Exploration

Abstract

We consider collaborative graph exploration with a set of k agents. All agents start at a common vertex of an initially unknown graph with n vertices and need to collectively visit all other vertices. We assume agents are deterministic, moves are simultaneous, and we allow agents to communicate globally. For this setting, we give the first non-trivial lower bounds that bridge the gap between small ($k \le \sqrt{n}$) and large ($k \ge n$) teams of agents. Remarkably, our bounds tightly connect to existing results in both domains.

First, we significantly extend a lower bound of $\varOmega (\log k / \log \log k)$ by Dynia et al. on the competitive ratio of a collaborative tree exploration strategy to the range $k \le n \log ^c n$ for any $c \in \mathbb {N}$. Second, we provide a tight lower bound on the number of agents needed for any competitive exploration algorithm. In particular, we show that any collaborative tree exploration algorithm with $k=Dn^{1+o(1)}$ agents has a competitive ratio of $\omega (1)$, while Dereniowski et al. gave an algorithm with $k=Dn^{1+\varepsilon }$ agents and competitive ratio $\mathcal {O}(1)$, for any $\varepsilon > 0$ and with D denoting the diameter of the graph. Lastly, we show that, for any exploration algorithm using $k=n$ agents, there exist trees of arbitrarily large height D that require $\varOmega (D^2)$ rounds, and we provide a simple algorithm that matches this bound for all trees.

Y. Disser—Supported by the ‘Excellence Initiative’ of the German Federal and State Governments and the Graduate School CE at TU Darmstadt.

F. Mousset—Supported by grant no. 6910960 of the Fonds National de la Recherche, Luxembourg.

A. Noever—Supported by grant no. 200021 143338 of the Swiss National Science Foundation.