Distributed Searching of Partial Grids

We consider the following distributed pursuit-evasion problem. A team of mobile agents called searchers starts at an arbitrary node of an unknown $n$-node network. Their goal is to execute a search strategy that guarantees capturing a fast and invisible intruder regardless of its movements using as few agents as possible. We restrict our attention to networks that are embedded into partial grids: nodes are placed on the plane at integer coordinates and only nodes at distance one can be adjacent. We give a distributed algorithm for the searchers that allow them to compute a connected and monotone strategy that guarantees searching any unknown partial grid with the use of $O(\sqrt{n})$ searchers. As for a lower bound, not only there exist partial grids that require $Ω(\sqrt{n})$ searchers, but we prove that for each distributed searching algorithm there is a partial grid that forces the algorithm to use $Ω(\sqrt{n})$ searchers but $O(\log n)$ searchers are sufficient in the offline scenario. This gives a lower bound of $Ω(\sqrt{n}/\log n)$ in terms of achievable competitive ratio of any distributed algorithm.