Fort Formation by an Automaton

Building structures by low capability robots is a very recent research development [3]. 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 (bricks) in an infinite grid Z × Z. The grid cells may contain a brick (full cells) or it may be empty (empty cells). The 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 full cell. The Manhattan distance between the farthest full cells is the span of the field. We consider the construction of a 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 Ω(z
²
) where z is the number of bricks present in the environment.