Time, Travel, and Energy in the Uniform Dispersion Problem

We investigate the algorithmic problem of uniformly dispersing a swarm of robots in an unknown, gridlike environment. In this setting, our goal is to comprehensively study the relationships between performance metrics and robot capabilities. We introduce a formal model comparing dispersion algorithms based on makespan, traveled distance, energy consumption, sensing, communication, and memory. Using this framework, we classify several uniform dispersion algorithms according to their capability requirements and performance. We prove that while makespan and travel can be minimized in all environments, energy cannot, as long as the swarm's sensing range is bounded. In contrast, we show that energy can be minimized even by simple, ``ant-like" robots in synchronous settings and asymptotically minimized in asynchronous settings, provided the environment is topologically simply connected. Our findings offer insights into fundamental limitations that arise when designing swarm robotics systems for exploring unknown environments, highlighting the impact of environment's topology on the feasibility of energy-efficient dispersion.