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.

Patrolling Grids with a Bit of Memory

We study the following problem in elementary robotics: can a mobile agent with $b$ bits of memory, which is able to sense only locations at Manhattan distance $V$ or less from itself, patrol a $d$-dimensional grid graph? We show that it is impossible to patrol some grid graphs with $0$ bits of memory, regardless of $V$, and give an exact characterization of those grid graphs that can be patrolled with $0$ bits of memory and visibility range $V$. On the other hand, we show that, surprisingly, an algorithm exists using $1$ bit of memory and $V=1$ that patrols any $d$-dimensional grid graph.

Probabilistic Pursuits on Graphs

We consider discrete dynamical systems of "ant-like" agents engaged in a sequence of pursuits on a graph environment. The agents emerge one by one at equal time intervals from a source vertex $s$ and pursue each other by greedily attempting to close the distance to their immediate predecessor, the agent that emerged just before them from $s$, until they arrive at the destination point $t$. Such pursuits have been investigated before in the continuous setting and in discrete time when the underlying environment is a regular grid. In both these settings the agents' walks provably converge to a shortest path from $s$ to $t$. Furthermore, assuming a certain natural probability distribution over the move choices of the agents on the grid (in case there are multiple shortest paths between an agent and its predecessor), the walks converge to the uniform distribution over all shortest paths from $s$ to $t$. We study the evolution of agent walks over a general finite graph environment $G$. Our model is a natural generalization of the pursuit rule proposed for the case of the grid. The main results are as follows. We show that "convergence" to the shortest paths in the sense of previous work extends to all pseudo-modular graphs (i.e. graphs in which every three pairwise intersecting disks have a nonempty intersection), and also to environments obtained by taking graph products, generalizing previous results in two different ways. We show that convergence to the shortest paths is also obtained by chordal graphs, and discuss some further positive and negative results for planar graphs. In the most general case, convergence to the shortest paths is not guaranteed, and the agents may get stuck on sets of recurrent, non-optimal walks from $s$ to $t$. However, we show that the limiting distributions of the agents' walks will always be uniform distributions over some set of walks of equal length.