  The limited energy capacity of individual robotic agents in a swarm often
limits the possible cooperative tasks they can perform. In this work, we
investigate the problem of covering an unknown connected grid environment (e.g.
a maze or connected corridors) with a robotic swarm so as to minimize the
maximal number of steps that each member of the swarm makes and their activity
time before their work is finished, thereby minimizing the energy requirements.
The robots are autonomous, anonymous and identical, with local sensors and
finite memory, and possess no communication capabilities. They are assumed to
disperse over time from a fixed location, and to move synchronously. The robots
are tasked with occupying every cell of the environment, while avoiding
collisions.
  In the literature such topics are known as \textit{uniform dispersal
problems}. The goal of minimizing the number of steps traveled by the robots
has previously been studied in this context. Our contribution is a local
robotic strategy for simply connected grid environments that, by exploiting
their topology, achieves optimal makespan (the amount of time it takes to cover
the environment) and minimizes the maximal number of steps taken by the
individual robots before their deactivation. The robots succeed in discovering
optimal paths to their eventual destinations, and finish the covering process
in $2V-1$ time steps, where $V$ is the number of cells in the environment.
