Fort Formation by an Automaton

Building structures by low capability robots is a very recent research development. A robot (or a mobile agent) is designed as a deterministic finite automaton. The objective is to make a structure from a given distribution of materials (\textit{bricks}) in an infinite grid $Z\times Z$. The grid cells may contain a brick (\textit{full cells}) or it may be empty (\textit{empty cells}). The \textit{field}, a sub-graph induced by the full cells, is initially connected. At a given point in time, a robot can carry at most one brick. The robot can move in four directions (north, east, south, and west) and starts from a \textit{full cell}. The \textit{Manhattan distance} between the farthest full cells is the \textit{span} of the field. We consider the construction of a \textit{fort}, a structure with the minimum span and maximum covered area. On a square grid, a fort is a hollow rectangle with bricks on the perimeter. We show that the construction of such a fort can be done in $O(z^2)$ time -- with a matching lower bound $\Omega(z^2)$ -- where $z$ is the number of bricks present in the environment.